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a/parse/train/3g5KdPD8LIL/3g5KdPD8LIL.md b/parse/train/3g5KdPD8LIL/3g5KdPD8LIL.md new file mode 100644 index 0000000000000000000000000000000000000000..bac4a613a430eb17dc3f70df79ec6ac98c051558 --- /dev/null +++ b/parse/train/3g5KdPD8LIL/3g5KdPD8LIL.md @@ -0,0 +1,504 @@ +# CUD-NET: Color Universal Design Neural Filter for the Color Weakness + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 Information on images should be visually understood to anyone, including the color +2 weakness. However, it is not recognizable if color that seems distorted to the color +3 weakness meets an adjacent object. We suggest CUD-NET1 based on convolutional +4 deep neural network to generate color universal design (CUD) images that satisfy +5 both color preservation and distinguishment of color for input images. CUD-NET +6 regresses the node point of the piecewise linear function based on information of +7 input images and comprises a specific filter per image. We present the following +8 methods to generate CUD images for the color weakness. First, we refine the CUD +9 dataset on specific criteria by color experts. Second, the input image information +10 is expanded through the pre-processing specialized on the color weakness vision. +11 Third, we suggest a multi-modal feature fusion architecture that combines features +12 to process expanded images. Finally, we suggest a deformable loss function by the +13 composition of the predicted image through the model to avoid the one-to-many +14 problems of the dataset. + +# 15 1 Introduction + +# 16 1.1 Motivation + +17 The green and red color blindness are made up of $8 \%$ of males and $0 . 5 \%$ of females in Northern +18 European descent[Won11], which is almost up to rate of one person in 20 people. Green and red +19 blindness is the most common pattern, followed by blue, yellow, and total color blindness. In this +20 paper, we generate Color Universal Design (CUD) images, which are color weakness friendly design +21 forms, through deep learning around the aspect of the red color weakness (protanopia) and green +22 color weakness (deuteranopia) vision. Protanopia is insensitive to red color and deuteranopia is +23 insensitive to green color, although it varies depending on individual color weakness extent. + +There are studies that help color discrimination to the color weakness, including wearable devices and surgeries[VZCR20]. However, since these research require time and cost, we simply generate CUD images with an image enhancement method based on deep learning to make the corresponding color visible for the color weakness. For an example of the left-above image $I$ in Figure 1, the people who are not color weakness can distinguish the letter $\cdot 5 '$ in the image. But as a deuteranopia vision in left-below image $I ^ { d }$ , the surrounding color and the letter $\cdot 5 '$ are very analogous, making it ambiguous to distinguish the bound of adjacent object. The right-bottom target image ${ \bf \bar { \boldsymbol { T } } } ^ { d }$ , refined image by color expert designers, shows that the letter $\cdot 5 '$ appeared well at the deuteranopia vision. Here, we define the non-CUD objects as the letter $\cdot 5 '$ and surroundings invisible to deuteranopia vision in the image $I$ , and define the CUD objects as the letter $\bullet _ { 5 } ,$ and surroundings visible to deuteranopia vision in the image $T$ . In other words, CUD object means that adjacent objects are distinguishable on both the + +![](images/8e40a88186fe991a1f5b33f0d69f552be1b201d8e8b9ad49435e47363052c85c.jpg) +Figure 1: Comparisons of the non-CUD image, our CUD-NET’s predicted image, and CUD image. The above row is represented in normal vision, and below row is represented in deuteranopia vision. + +35 normal vision and the color weakness vision. The non-CUD object means that adjacent objects are +36 distinguishable on normal vision but not the color weakness vision. Consequently, we generate $\hat { I }$ that +37 satisfies CUD with a specific filter to the image $I$ . +38 We want to apply as weak filter as possible to CUD objects to preserve color, which requires a +39 certain level of object comprehension mechanism to do so. There are various studies from classic +40 PCA[WEG87] to machine learning-based object segmentation methods[TSC20, $Z \mathrm { G L } ^ { * } 2 0 ]$ to define +41 specific objects or areas in image. The research on semantic segmentation, which even provides labels +42 between objects, seems that deep learning still does not have a complete comprehension of all objects +43 in the real-world. The visual question answering to arbitrary questions about object’s interactions, +44 the most general issue on comprehension of object, does not have high transmission power to be +45 practical uses $[ \mathrm { A H B ^ { * } } 1 8$ , $\mathrm { K Z G ^ { * } } 1 7$ , $\mathrm { L Y L } ^ { * } 2 0 ]$ . Therefore, we expand feature of the input image around +46 the information of color weakness vision and define the robust neural filter. In summary, we suggest +47 a CUD-NET that generates an image suitable for CUD, while complying with the color preservation +48 for the source image. +49 In this paper, we suggest the Color Universal Design Network (CUD-NET) to satisfy both color +50 preservation and contrast of non-CUD objects (CUD suitability). We introduce 4 core contributions +51 of CUD-NET. + +• Dataset refinement criteria for CUD image We refine training data into two groups, the one with a simple color tone image based on H and $\mathrm { v }$ in the HSV color space, the other with two or more non-CUD objects that must be distinguished in publications. • Image pre-processing for CUD-NET We carry out pre-processing to expand the information of the input image. Input image $I$ is reconstructed with three expanded feature information with noise removed. Multi-modal feature fusion architecture We define a feature layer, the fusion layer, and a regression layer to handle pre-processed images. The three features from the feature extracting layer are combined into the one fusion feature, and finally a filter is constructed by regressing the node point of the piecewise linear function, or indicator of filter. • Variational loss function We suggest a deformable loss function by the composition of the predicted image through the model. Our data have a problem of one-to-many, where the specific color in input image $I$ is mapped into multiple colors in target image $T$ . + +# 65 1.2 Related Works + +66 Image-to-Image translation based on GAN GAN is used in various image translation areas, +67 including image generation, style transfer, and colorization[KWK21, IZZE17]. In a preliminary +68 experiment, Cycle-GAN[PEZZ20] has reached the best performance in maximizing the contrast of +69 non-CUD objects. However, our goal is to keep the color preservation of the input image as well, +70 so in the case of black color, which has lost all its color of the input image, it is considered the +71 worst case for color preservation. Enlighten-GAN[JGL $^ { * } 2 1$ ] complements those instability, enabling +72 them to generate more stable results on color preservation. But since most of the GAN-based image +73 translation fixes the size of the predicted image, reshaping a high-resolution image causes information +74 loss of source image. Also, it is difficult to reconstruct the complete geometry for the source image +75 as it generates images through the dilated convolution layer. +76 Image enhancement based on neural filter estimation Unlike GAN, there are researches that +77 scale the pixel values of images based on neural filter estimation $[ \mathrm { W } \mathrm { Z F ^ { * } } 1 9 $ , DLT18, BCPS19]. Zero +78 $\mathrm { D C E } [ \mathrm { G L G } ^ { * } 2 0 ]$ is a low-light image enhancement research that provides a brighter visual display +79 of input image. It estimates pixel-wise and high-order filter for dynamic range adjustment of input +80 images with lightweight deep network, DCE-Net. DeepLPF $\mathbf { M M M } ^ { * } 2 0 ]$ tried to solve the problem by +81 using a graduated filter, elliptical filter, and polynomial filter. The authors not only tried to visually +82 enhance the contrast of images but also to comprise stable filters that are easy to understand for +83 the spectators while keeping the color preservation. In our problem, however, the contrast factor is +84 almost same results as the input image in both visions, while complying the high color preservation, +85 resulting over-stable filter. It is assumed that the inability in comprehension of object’s interaction +86 leads to over-stable filter. + +![](images/b08f9cfa25e49147de6bd5c99c32e6d5058eeef27cd23f658d79d4db5b7758a1.jpg) +Figure 2: The ideal color conversion of predicted image between contrast and color preservation. Non-CUD object $a$ should increase the gap compared to the input image and preserve its original color, while the CUD object $b$ maintain both contrast and color. + +# 87 2 Methodology + +88 We define the ideal predicted image as an increase in the contrast between non-CUD objects and the +89 color preservation for the input image. The non-CUD object $a$ should be mapped into $\acute { a }$ and CUD +90 object $b$ should preserve its color and contrast like an ideal example of Figure 2. However, as our +91 neural filter affects the whole pixels throughout the image, we have the constraint of applying the +92 same filter to objects $a$ and $b$ . It is very hard to make the contrast and color of object $b$ exactly the +93 same as before the filter adjustment while maximizing the contrast of object $a$ . Therefore, we propose +94 a deep learning-based regression to comprise the specific filter per image that maximizes the contrast +95 of object $a$ while minimizing the adjustment of features on object $b$ . +96 First, we propose a solution to maximize the contrast of the $\mathrm { L }$ channel values in CIELab color +97 space[RG19]. We empirically confirmed that protanopia and deuteranopia, which account for the +98 most proportion of color weakness, can distinguish the difference by $\mathrm { L }$ channel values in common +99 when the non-CUD objects are adjacent to each other. To illustrate Figure 1 again, the $\mathrm { L }$ channel +100 value of letter $^ { \bullet } 5 ^ { \bullet }$ in image $I$ is 61 and the surrounding color is 61. The distinguishment between +101 the two objects is easy to normal vision, however the image $I ^ { d }$ , the deuteranopia vision, is very +102 ambiguous. On the contrary, the CUD target image $T$ and $T ^ { d }$ have a difference of $\mathrm { L }$ channel value 75 +103 for the letter $\bullet _ { 5 } ,$ and 45 for the surroundings, making it easy to distinguish between the normal and +104 the deuteranopia vision. Due to the characteristics of these data, we refine a data pair by defining a +105 criterion that separates two invisible non-CUD objects by $\mathrm { L }$ channel values. +106 Secondly, we propose a variational loss function and multi-modal feature fusion network for color +107 preservation. It can be said that the increase in the contrast of $\mathrm { L }$ channel values between non-CUD +108 objects is quantitatively superior, but not in the case of increasing the differences in color preservation +109 of input images. When non-CUD objects exist, as a simple example, the most likely way to maximize +110 contrast is to polarize the color of the object black and white. But it is the result of complete ignorance +111 for color preservation, so just enabling to distinguish between non-CUD objects is not always a good +112 answer. A strong filter must be applied to distinguish non-CUD object, but its impact should not be +113 too extensive to leading the loss of information in CUD objects. In this paper, we solve this problem +114 by taking an appropriate trade-off of color preservation and contrast of L channel value. + +# 115 2.1 Dataset refinement criteria for CUD image + +116 The training data is refined by two groups. The one is vectorized image with two colors divided +117 by value $\mathrm { v }$ and hue degree H in HSV color space[HMKO19], and the other is image with two +118 or more objects that must be distinguished while preserving the color of non-CUD objects. The +119 training data is grouped about 1,600 color combinations into the same V and then simulates them +120 with the deuteranopia vision, converting to the adjacent color family to comply color preservation. +121 All conversions are scaled within only S and V in HSV color space to increase at least 15 difference +122 in the L channel value of selected non-CUD objects. The colors are combined with the 10 essential H +123 and tones, and the similar color simulated with the deuteranopia vision was converted. Consequently, +124 the key part of refining training data is preservation of color, allowing the models to comply with the +125 same approach on learning. + +# 2.2 Image pre-processing + +127 Our model regresses node points of piecewise linear function, which will be described in the model +128 architecture section, and the final filter is a multiplication operation for the input image. Therefore, +129 the multiplication operations of less than the number 1 tend to fade the color saturation. The image +130 without color inversion converges the white color value to 1, so if the multiplying value is in the [0, +131 1] range, the white color is shifted to the black. By inverting the color of input image, it ignores the +132 multiplication operations for white value with 0. +133 We generate the map image $I ^ { m }$ based on original RGB input images calculating the difference value +134 between the image with an aspect of normal vision and the image with an aspect of deuteranopia vision. +135 Recent studies have been conducted to augment the information or expanded the models’ perspective +136 through transformer models[JSZK16, RFB15]. In our experiment, however, the transformer model +137 tends to generate the predicted image ignoring the source color, which result in the polarized color to +138 black and white like Cycle-GAN’s. + +$$ +{ \cal I } ^ { m } = \vert i n v e r t ( I ^ { n } ) - i n v e r t ( I ^ { d } ) \vert +$$ + +$$ +{ \cal I } ~ = ~ \delta \left( c a t ^ { c h a n n e l } \left( I ^ { n } , I ^ { d } , I ^ { m } \right) \right) +$$ + +139 After applying color inversion from the original RGB input image $I ^ { n }$ , we generate the image $I ^ { d }$ with +140 an aspect of deuteranopia vision in equation 1. From these two generated images, we can get the +141 absolute difference value to compose the map image $I ^ { m }$ . In equation 2, the final input $I$ concatenated +142 with $9 \times H \times W$ dimensions passes through the model. The $\bar { \delta } ( . )$ clips output to a range of [0, 1]. + +# 2.3 Model Architecture + +144 CUD-NET regresses the node points of piecewise linear filter function from the input. The value +145 of each node points computes the multiplication operation and generates the predicted image. The +146 input $I$ is compressed into 3 feature blocks matching each input through convolution layer, pooling +147 layer, and global pooling layer. The input with 9 channels is separated into $3 \times 3$ channels before +148 passing the model. The first 3 channels are literally used as the main inputs, where the multiplication +149 operation takes place, while the remaining 6 channels are used as features. +150 First of all, we use multi-modal fusion architecture for three separate inputs to extract expanded +151 features. The three inputs converted to the HSV color space pass through a weights-sharing convolu +152 tion layer to extract a feature block corresponding to the inputs. Each convolution layer consists of +153 kernel size $^ { = 3 }$ , stride $^ { : = 1 }$ , and padding=1, reducing dimension through average pooling. We empirically +154 noticed that the most values of output feature have distribution within the range of [-1, 1] with valid +155 values for constructing the node points, so we use hyperbolic tan for activation function. Since we +156 use inputs with unstructured image size, the last global pooling block holds the size of the feature +157 instead of the average pooling block[LCY14]. +158 The three feature blocks are combined through the multi-modal compact bilinear pooling gate +159 $( \mathbf { M C B } ) [ \mathbf { F P Y ^ { * } } 1 6 ]$ , following the fusion process shown in Figure 3. The MCB gate allows both +160 features to interact in a multiplicative way with low memory consumption and computation +161 times. The fusion features are complemented to enhanced feature through the split attention +162 mechanism $[ Z \mathrm { W } Z ^ { \ast } 2 0 ]$ . At the beginning of the experiment, we have applied the convolutional +163 block attention mechanism[WPLK18] of each MCB gate, but we found that it does not make sense +164 of understanding the feature itself, so we apply only one attention block to the last fusion feature. +165 The enhanced feature pass through the fully-connected regression layer. We picked the 64 points +166 to be regressed to compose the piecewise linear function, which is empirically confirmed to the +167 optimized number of points in this research. The first half of the values construct the node points of +168 the S channel and the other half comprise the $\mathrm { v }$ channel in HSV color space. Finally, node points +169 become the scaling factors to generate predicted image in equation 3[MMS19]. + +![](images/6fe42bfba41420f38aecf4a17ab1e4d9b4816d2107844d698cd5beb3f6b072ab.jpg) +Figure 3: Overview structure of CUD image generation + +$$ +S \left( I _ { i } ^ { s , v } \right) = k _ { 0 } + \sum _ { m = 0 } ^ { M - 1 } \left( k _ { m + 1 } - k _ { m } \right) \delta \left( M I _ { i } ^ { s , v } - m \right) +$$ + +170 The total number of node point $M$ , each pixel values of S, V channel in input image $I _ { i } ^ { s , v }$ are +171 multiplicated with the slope of actual regressed value $k _ { m }$ , the $m - t h$ generated node point. The +172 specific node points $M$ is scaled through a multiplication operation to pixel value of the input image +173 according to each node point. + +# 174 2.4 Loss function + +175 Our dataset has one-to-many problems between input and target data. In dataset pair +176 $( I _ { 1 } , T _ { 1 } )$ , $( I _ { 2 } , T _ { 2 } )$ , . . . , $( I _ { n } , T _ { n } )$ , for example, the red color in $I _ { 1 }$ can be targeted to purple color in +177 $T _ { 1 }$ , and the red color in $I _ { 2 }$ can be targeted to orange color in $T _ { 2 }$ . With these one-to-many dataset +178 structures, we design the loss function $\mathcal { L }$ that expresses the potential and the diversity of predicted +179 image in equation 4. + +$$ +\mathcal { L } = \sum _ { i = 1 } ^ { N } L a b _ { l o s s } \left( V \left( \Phi \left( \hat { I } _ { i } \right) \right) \right) + H _ { l o s s } \left( \Phi \left( \hat { I } _ { i } \right) \right) +$$ + +180 Stencil Masking As explained in the dataset refining criteria, we do not proceed with color +181 conversion for all areas in the target images, but only for areas with color combinations that are +182 invisible to the deuteranopia (non-CUD object). For this reason, the input image has color regions of +183 converting color and unconverting color, which also can be referred to as non-CUD object and CUD +184 object. To imply the color bound to model, the stencil masking method is introduced. + +$$ +\Phi \left( { \hat { I } } _ { i } \right) = { \hat { I } } _ { i j } \parallel ( I _ { i j } \cdot T _ { i j } ) +$$ + +185 We consist a stencil maps through the logical and operations ’·’ of each pixel value $I _ { i j }$ , $T _ { i j }$ . Stencil +186 map can specify the non-CUD area and be computed with predicted image $\hat { I } _ { i j }$ of logical or operation +187 ’ $| |$ ’in equation 5. Consequently, CUD object of the image adjusted with a stencil mask does not carry +188 out the neural filter computation, such as the same way we refine the target image. This refined image +189 is calculated on the loss function. +190 CIELab Loss We use the CIELab channel loss function to maximize the contrast of color on +191 deuteranopia vision. To stabilize the contrast and brightness of the predicted image, we calculate the +192 MS· SSIM(multi-scale structural similarity[WSB03]) of $\mathrm { L }$ channel. + +$$ +L a b _ { l o s s } = \left\| L a b \left( \hat { I } _ { i } ^ { r g b } \right) - L a b \left( T _ { i } ^ { r g b } \right) \right\| _ { 1 } + M S \cdot S S I M \left( L a b \left( \hat { I } _ { i } ^ { L } \right) , L a b \left( T _ { i } ^ { L } \right) \right) +$$ + +193 The $L a b \left( . \right)$ expression in equation 6 returns the CIELab channel corresponding to the RGB channel, +194 and all calculations are made only on the L channel. +195 Histogram Loss We use the histogram loss function to comply with the color preservation of the +196 image. The RGB channel is used to preserve its color, contrary to using only the $\mathrm { L }$ channel in other +197 loss functions. Handling the RGB channel as a loss function rather than using Lab’s ab channels has +198 shown better results on color preservation. + +$$ +H _ { l o s s } = - { \omega _ { h i s t } } \int N \left( \hat { I } _ { i } ^ { r g b } ; \sigma \right) - { \cal N } \left( T _ { i } ^ { r g b } ; \sigma \right) +$$ + +199 When simply designing a loss function with the L1 distance of the RGB channel pixel values, it was +200 very sensitive to certain values and the gradients are diverged, resulting in an untrainable experiment. +201 Therefore, we used a gaussian expansion method $[ S \mathbf { A } C ^ { * } 1 7 ]$ denoted by $N ( . )$ to infer a differentiable +202 histogram loss function in equation 7. We compute the difference of the RGB channel of the +203 differentiable histogram function, which can be altered to mean squared error or cosine similarity. +204 The scaler $\omega _ { h i s t }$ is determined in inverse proportion to the size of the input image. By maintaining +205 the RGB similarity between the predicted image and the target image, we can comply with the color +206 preservation. +207 Variational Prediction There are various ways to maximize difference of the L channel in the +208 image. And the target image is converted at least two colors compared to the input image. However, +209 the predicted image of the model is generated by the neural filter, so it is unpredictable which area +210 of color is modified. Therefore, if the color in predicted image is over-shifted or in the color value +211 of opposite shifts to the target, the loss will rather increase. In addition to one-to-many problem +212 that the data pair itself does not matches one-to-one in a particular color, it is necessary to generate +213 alternative predicted image with the same aspect of the data pair. We calculate the loss function with +214 a variational prediction based on the predicted image for potential color shifts. +215 The first potential is the case of excessive shifts. Assume that $I _ { i } ^ { L } = \{ 7 4 , 4 1 , 7 9 \}$ , $\hat { I } _ { i } ^ { L } = \{ 9 7 , 1 0 , 7 0 \}$ , +216 $T _ { i } ^ { L } \ = \ \{ 5 0 , 4 1 , 8 0 \}$ in L channel value. The first and third components of each image is non-CUD +217 objects, and second component is CUD object. Therefore, we refined data paired with a difference of +218 15 on L channel. Here, we clip the excessive $\mathrm { L }$ channel value in $\hat { I } _ { i } ^ { L }$ by equation 8. Up to this point, +219 no calculation is made as no value is exceeded in this example. The second potential is the case of +220 opposite shifts. It can be said that a complete neural filter has been proceeded for value 97, 10, 70 +221 where $\mathrm { L }$ channel difference is 27. However, if we actually calculate the mean square error between +222 $\hat { I } _ { i } ^ { L }$ and $T _ { i } ^ { L }$ , it will be an large value over 1k. Here we can generate alternative predicted image from +223 equation 9 and 10. + +$$ +c l i p \left( \hat { I } _ { i j } \right) \ = \ \left\{ { m a x } { \left( \hat { I } _ { i j } , { T } _ { i j } \right) } , \qquad I _ { i j } \ > T _ { i j } \right. +$$ + +$$ +R _ { 1 } = \ 2 I _ { i j } - \ \hat { I } _ { i j } , R _ { 2 } = \ \hat { I } _ { i j } +$$ + +$$ +V \left( { \hat { I } } _ { i j } \right) = a r g m i n \left( \left\| c l i p \left( R _ { 1 , 2 } \right) - T _ { i j } \right\| _ { 2 } \right) +$$ + +224 As mentioned above, we define thresholds by the maximum and minimum value of each corresponding +225 pixel position of $\hat { I } _ { i } ^ { L }$ and $T _ { i } ^ { L }$ . By computing a difference of residual map and the input image, we +226 induce the alternative two images $R _ { 1 }$ , $R _ { 2 }$ . As a result, $\hat { I } _ { i } ^ { L }$ with a smaller L2 distance is selected to +227 alternative predicted image in equation 10, and it is finally computed with loss function compared to +228 the $T _ { i } ^ { L }$ . The above equation establishes $V \left( \Phi \left( \hat { I } _ { i } \right) \right) = \{ \stackrel { \cdot } { 5 4 } , 4 1 , \stackrel { \cdot } { 8 0 } \}$ and the mean square error to the +229 target image is approximately 5, which is agreeable loss value respect to $\hat { I } _ { i } ^ { L }$ itself. +230 Identity Loss[ZPIE17, TPW16] We use $\mathcal { L } _ { i d e n t i t y } \left( T _ { i } \right)$ to apprehend the CUD object to the model. +231 In the case of target image that already satisfy the CUD, the filter should be relatively weakly applied +232 than input image. The input of identity loss is target image $T _ { i j }$ instead of input image $I _ { i j }$ , and the +233 reference of the loss function is also target image $T _ { i j }$ to maintain the value itself. In computing +234 identity loss, we do not require variational prediction as we cannot judge the potential region by +235 equation 10. + +# 236 3 Experiments + +237 The experiment was performed with Tesla V100 SXM2 and Intel Xeon Gold 5120 and the computation +238 speed was about 40 images per minutes. We refined a dataset with Adobe Photoshop to maximize +239 contrast in the L channel by adjusting saturation and brightness for areas that require color conversion +240 based on deuteranopia vision simulation. Color experts has refined about 1,500 vectorized image for +241 the training data and 300 publication images for the validation data. All the comparative experimental +242 models used the same train, test, validation data in this paper. We used the inference data in +243 publications, which is almost composed of vectorized images, as colors often appear distorted in a +244 gradation-rich image. The Figure 4 is arranged in descending order of the number of combinations in +245 colors from the top image. +246 Both structure similarity (SSIM)[ZBSS04] and peak signal to noise ratio (PSNR) in Table 1 can +247 indicate whether the image is suitable for CUD or not. As a notable aspect, the result has shown that +248 comparative models with lower metrics are sensitive to high-gradation input images, which generated +249 color-heterogeneous image. SSIM and PSNR itself can determine the increase in contrast compared +250 to the target image but do not determine whether the color preservation complied. Therefore, we +251 evaluated SSIM and PSNR with three references, inputs images $I$ , predicted images $\hat { I }$ , and target +252 images $T$ . The higher the estimation of the $\hat { I }$ and $I$ , the more color preservation factor worked. +253 The higher the estimation of the $\hat { I }$ and $T$ , the more increase in contrast can be considered. We also +254 define SSIM mean absolute error, PSNR mean absolute error to measure the extent of the conversion +255 between the $F : I T$ and $F : \dot { I } \hat { I }$ in equation 11 and 12, respectively. The $N$ is total number +256 of inference data. + +$$ +S S I M \cdot M A E = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } | S S I M ( \hat { I } _ { i } , \mathcal { T } _ { i } ) - S S I M ( I _ { i } , T _ { i } ) | +$$ + +$$ +P S N R \cdot M A E = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } | P S N R ( \hat { I } _ { i } , \mathcal { T } _ { i } ) - P S N R ( I _ { i } , T _ { i } ) | +$$ + +![](images/ee74ef15b1c4bf5acba45467aa3b157cb56fa3118986b351ff3cd98fe81ae34d.jpg) +Figure 4: Comparisons of predicted images in deuteranopia vision. The color experts selected the validation data that do not satisfy the CUD in publications. + +Table 1: Evaluation table of comparison experiment. The CUD-NET with a low bottle-neck feature achieves better results in the experiment of the deuteranopia and the protanopia subjects(Figure 5), although the evaluation metrics are lower than that of CUD-NET. + +
ArchitecturesSIM(i,1)SSIM(i,T)PSNR(i,1)PSNR(i,T)SSIM-MAEPSNR·MAE
Cycle-GANZero-DCE0.6300.63413.2813.830.31918.4430
0.9240.88821.9518.670.06613.7300
DeepLPFEnlighten-GAN0.8500.83126.3120.340.12202.0566
0.8200.80821.8519.580.14703.9983
Enlighten-GAN(scaled)0.9660.92124.8621.360.03923.4937
CUD-NET(low bottle-neck feature)CUD-NET0.8970.86627.7721.010.09012.0826
0.9620.92429.5421.190.03121.4760
+ +257 Cycle-GAN and Zero-DCE showed worse result than others. Cycle-GAN model had difficulty +258 reconstructing a geometry of a particular object, and overall color had low saturation and brightness, +259 resulting in color conversion into an almost grey scale image. Zero-DCE is faded in color, and the +260 contrast was not much different from the input image. The overall image lost its color preservation, +261 which we focused to solve in this paper. +262 DeepLPF meets both the color preservation and contrast that we deal with for. However, DeepLPF +263 tends to color be over-stably filtered for the images with fewer color combinations. Although the +264 color preservation has complied better than other experiments, there were many failed results from +265 the perspective of contrast, which the over-stable filter leads to by DeepLPF. +266 Remarkably, predicted images of Enlighten-GAN showed reasonable results. However, simple color +267 combinations or the images with already satisfying the CUD often showed results degenerated with +268 low CUD suitability. Enlighten-GAN was able to generate the results we targeted, but its deviation of +269 filter is so high that it sometimes failed to satisfy the contrast even on simple images or decreased +270 the contrast. As the problem of GAN-based method including Enlighten-GAN, moreover, model +271 fixes the width and height of the predicted image. If width and height of $T$ and $I$ down-scaled + +to size of Enlighten-GAN 272 $\hat { I }$ (approximately 25K pixels in this experiment), the $S S I M \left( \hat { I } , I \right)$ and $P S N R \left( \hat { I } , T \right)$ showed higher estimation in some metrics than CUD-NET. In the opposite case of $\hat { I }$ up-scaled to size of $T$ and $I$ , the significantly low estimation was recorded due to the information loss of up-scaling problem. + +CUD-NET showed stable and robust predicted images in both color preservation and increase in the contrast compared to other experiments. In comparing the values in the same region of $I$ and $\hat { I }$ , the model scaled two $\mathrm { L }$ channel values with opposite side in the most of case, the one goes up and the other goes down. When we reduced the number of bottle-neck feature of model, it tends to record relatively high deviation of filter scales according to the number of combinations of colors. In summary, the CUD-NET showed the highest estimations for 4 evaluation metrics. Moreover, as our model adopted a neural filter unlike generation models, there is no loss of information regarding the scaling of predicted images. + +![](images/9a2ce8b4db1667e120d3e391edc4e605f10711786bdbd70de95e46dfcb47ad39.jpg) +Figure 5: The box bar is ordered to the left side, input image $I$ , Enlighten-GAN, DeepLPF, CUDNET. The y position of box bar represents a mean and length of the box bar represents a deviation of each experiment. The lower the graph is, the higher the rank is. + +284 The figure 5 shows the evaluation of the deuteranopia and the protanopia. The evaluation metrics +285 consist of object distinguishability and color harmony in order of input image I, predicted image +286 of Enlighten GAN, DeepLPF, and CUD-NET. User study has tested upon the total of 6 subjects, 4 +287 deuteranomaly and 2 protanomaly. The subjects were asked to list the ranks of object distinguishability +288 and color harmony of 4-paired-image for each model-blinded item. As the experimental results, +289 the deuteranopia subject ranked the 1-st in the object distinguishability of CUD-NET at an average +290 rank of 1.821, followed by Enlighten-GAN at an average rank of 2.512. Similarly, the protanopia +291 subject also ranked the 1-st in CUD, followed by Enlighten-GAN, DeepLPF, and input images. The +292 evaluation of color harmony showed that the subjects tend to assume that the image with a good +293 object distinguishability has good color harmony preferentially. For a total of six subjects, the five +294 subjects chose the CUD-NET, with the exception of one who ranked Enlighten-GAN by a subtle gap + +# 295 4 Conclusion + +296 In this paper, we proposed deep network to generate CUD images from non-CUD input images. The +297 pre-processing and multi-modal fusion layer could comprehend the information for color weakness, +298 and the variational loss function makes the model further adapt to CUD dataset. 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[Yes] See on github: https://github.com/Anonymous68864576/CUD-NET-anonymous + +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] on github repository +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See on section 2.3 +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See on Experiment section + +3. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] Cooperated with Co-author +(b) Did you mention the license of the assets? [Yes] on github repository +(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] on github repository +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] on github repository +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] \ No newline at end of file diff --git a/parse/train/3g5KdPD8LIL/3g5KdPD8LIL_content_list.json b/parse/train/3g5KdPD8LIL/3g5KdPD8LIL_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..8f75a807f86960a72156f6f72c032da0c74a0c27 --- /dev/null +++ b/parse/train/3g5KdPD8LIL/3g5KdPD8LIL_content_list.json @@ -0,0 +1,1021 @@ +[ + { + "type": "text", + "text": "CUD-NET: Color Universal Design Neural Filter for the Color Weakness ", + "text_level": 1, + "bbox": [ + 181, + 122, + 816, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 222, + 578, + 276 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 313, + 535, + 329 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Information on images should be visually understood to anyone, including the color \n2 weakness. However, it is not recognizable if color that seems distorted to the color \n3 weakness meets an adjacent object. We suggest CUD-NET1 based on convolutional \n4 deep neural network to generate color universal design (CUD) images that satisfy \n5 both color preservation and distinguishment of color for input images. CUD-NET \n6 regresses the node point of the piecewise linear function based on information of \n7 input images and comprises a specific filter per image. We present the following \n8 methods to generate CUD images for the color weakness. First, we refine the CUD \n9 dataset on specific criteria by color experts. Second, the input image information \n10 is expanded through the pre-processing specialized on the color weakness vision. \n11 Third, we suggest a multi-modal feature fusion architecture that combines features \n12 to process expanded images. Finally, we suggest a deformable loss function by the \n13 composition of the predicted image through the model to avoid the one-to-many \n14 problems of the dataset. ", + "bbox": [ + 148, + 344, + 766, + 539 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "15 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 564, + 312, + 580 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "16 1.1 Motivation ", + "text_level": 1, + "bbox": [ + 153, + 594, + 289, + 609 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "17 The green and red color blindness are made up of $8 \\%$ of males and $0 . 5 \\%$ of females in Northern \n18 European descent[Won11], which is almost up to rate of one person in 20 people. Green and red \n19 blindness is the most common pattern, followed by blue, yellow, and total color blindness. In this \n20 paper, we generate Color Universal Design (CUD) images, which are color weakness friendly design \n21 forms, through deep learning around the aspect of the red color weakness (protanopia) and green \n22 color weakness (deuteranopia) vision. Protanopia is insensitive to red color and deuteranopia is \n23 insensitive to green color, although it varies depending on individual color weakness extent. ", + "bbox": [ + 147, + 621, + 825, + 718 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There are studies that help color discrimination to the color weakness, including wearable devices and surgeries[VZCR20]. However, since these research require time and cost, we simply generate CUD images with an image enhancement method based on deep learning to make the corresponding color visible for the color weakness. For an example of the left-above image $I$ in Figure 1, the people who are not color weakness can distinguish the letter $\\cdot 5 '$ in the image. But as a deuteranopia vision in left-below image $I ^ { d }$ , the surrounding color and the letter $\\cdot 5 '$ are very analogous, making it ambiguous to distinguish the bound of adjacent object. The right-bottom target image ${ \\bf \\bar { \\boldsymbol { T } } } ^ { d }$ , refined image by color expert designers, shows that the letter $\\cdot 5 '$ appeared well at the deuteranopia vision. Here, we define the non-CUD objects as the letter $\\cdot 5 '$ and surroundings invisible to deuteranopia vision in the image $I$ , and define the CUD objects as the letter $\\bullet _ { 5 } ,$ and surroundings visible to deuteranopia vision in the image $T$ . In other words, CUD object means that adjacent objects are distinguishable on both the ", + "bbox": [ + 168, + 723, + 825, + 876 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/8e40a88186fe991a1f5b33f0d69f552be1b201d8e8b9ad49435e47363052c85c.jpg", + "image_caption": [ + "Figure 1: Comparisons of the non-CUD image, our CUD-NET’s predicted image, and CUD image. The above row is represented in normal vision, and below row is represented in deuteranopia vision. " + ], + "image_footnote": [], + "bbox": [ + 341, + 90, + 660, + 292 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "35 normal vision and the color weakness vision. The non-CUD object means that adjacent objects are \n36 distinguishable on normal vision but not the color weakness vision. Consequently, we generate $\\hat { I }$ that \n37 satisfies CUD with a specific filter to the image $I$ . \n38 We want to apply as weak filter as possible to CUD objects to preserve color, which requires a \n39 certain level of object comprehension mechanism to do so. There are various studies from classic \n40 PCA[WEG87] to machine learning-based object segmentation methods[TSC20, $Z \\mathrm { G L } ^ { * } 2 0 ]$ to define \n41 specific objects or areas in image. The research on semantic segmentation, which even provides labels \n42 between objects, seems that deep learning still does not have a complete comprehension of all objects \n43 in the real-world. The visual question answering to arbitrary questions about object’s interactions, \n44 the most general issue on comprehension of object, does not have high transmission power to be \n45 practical uses $[ \\mathrm { A H B ^ { * } } 1 8$ , $\\mathrm { K Z G ^ { * } } 1 7$ , $\\mathrm { L Y L } ^ { * } 2 0 ]$ . Therefore, we expand feature of the input image around \n46 the information of color weakness vision and define the robust neural filter. In summary, we suggest \n47 a CUD-NET that generates an image suitable for CUD, while complying with the color preservation \n48 for the source image. \n49 In this paper, we suggest the Color Universal Design Network (CUD-NET) to satisfy both color \n50 preservation and contrast of non-CUD objects (CUD suitability). We introduce 4 core contributions \n51 of CUD-NET. ", + "bbox": [ + 148, + 364, + 825, + 410 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 416, + 825, + 568 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 574, + 826, + 616 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Dataset refinement criteria for CUD image We refine training data into two groups, the one with a simple color tone image based on H and $\\mathrm { v }$ in the HSV color space, the other with two or more non-CUD objects that must be distinguished in publications. • Image pre-processing for CUD-NET We carry out pre-processing to expand the information of the input image. Input image $I$ is reconstructed with three expanded feature information with noise removed. Multi-modal feature fusion architecture We define a feature layer, the fusion layer, and a regression layer to handle pre-processed images. The three features from the feature extracting layer are combined into the one fusion feature, and finally a filter is constructed by regressing the node point of the piecewise linear function, or indicator of filter. • Variational loss function We suggest a deformable loss function by the composition of the predicted image through the model. Our data have a problem of one-to-many, where the specific color in input image $I$ is mapped into multiple colors in target image $T$ . ", + "bbox": [ + 217, + 630, + 825, + 828 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "65 1.2 Related Works ", + "text_level": 1, + "bbox": [ + 148, + 843, + 316, + 858 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "66 Image-to-Image translation based on GAN GAN is used in various image translation areas, \n67 including image generation, style transfer, and colorization[KWK21, IZZE17]. In a preliminary \n68 experiment, Cycle-GAN[PEZZ20] has reached the best performance in maximizing the contrast of \n69 non-CUD objects. However, our goal is to keep the color preservation of the input image as well, \n70 so in the case of black color, which has lost all its color of the input image, it is considered the \n71 worst case for color preservation. Enlighten-GAN[JGL $^ { * } 2 1$ ] complements those instability, enabling \n72 them to generate more stable results on color preservation. But since most of the GAN-based image \n73 translation fixes the size of the predicted image, reshaping a high-resolution image causes information \n74 loss of source image. Also, it is difficult to reconstruct the complete geometry for the source image \n75 as it generates images through the dilated convolution layer. \n76 Image enhancement based on neural filter estimation Unlike GAN, there are researches that \n77 scale the pixel values of images based on neural filter estimation $[ \\mathrm { W } \\mathrm { Z F ^ { * } } 1 9 $ , DLT18, BCPS19]. Zero \n78 $\\mathrm { D C E } [ \\mathrm { G L G } ^ { * } 2 0 ]$ is a low-light image enhancement research that provides a brighter visual display \n79 of input image. It estimates pixel-wise and high-order filter for dynamic range adjustment of input \n80 images with lightweight deep network, DCE-Net. DeepLPF $\\mathbf { M M M } ^ { * } 2 0 ]$ tried to solve the problem by \n81 using a graduated filter, elliptical filter, and polynomial filter. The authors not only tried to visually \n82 enhance the contrast of images but also to comprise stable filters that are easy to understand for \n83 the spectators while keeping the color preservation. In our problem, however, the contrast factor is \n84 almost same results as the input image in both visions, while complying the high color preservation, \n85 resulting over-stable filter. It is assumed that the inability in comprehension of object’s interaction \n86 leads to over-stable filter. ", + "bbox": [ + 148, + 869, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/b08f9cfa25e49147de6bd5c99c32e6d5058eeef27cd23f658d79d4db5b7758a1.jpg", + "image_caption": [ + "Figure 2: The ideal color conversion of predicted image between contrast and color preservation. Non-CUD object $a$ should increase the gap compared to the input image and preserve its original color, while the CUD object $b$ maintain both contrast and color. " + ], + "image_footnote": [], + "bbox": [ + 308, + 90, + 669, + 224 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 306, + 825, + 404 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 419, + 825, + 570 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "87 2 Methodology ", + "text_level": 1, + "bbox": [ + 150, + 590, + 313, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "88 We define the ideal predicted image as an increase in the contrast between non-CUD objects and the \n89 color preservation for the input image. The non-CUD object $a$ should be mapped into $\\acute { a }$ and CUD \n90 object $b$ should preserve its color and contrast like an ideal example of Figure 2. However, as our \n91 neural filter affects the whole pixels throughout the image, we have the constraint of applying the \n92 same filter to objects $a$ and $b$ . It is very hard to make the contrast and color of object $b$ exactly the \n93 same as before the filter adjustment while maximizing the contrast of object $a$ . Therefore, we propose \n94 a deep learning-based regression to comprise the specific filter per image that maximizes the contrast \n95 of object $a$ while minimizing the adjustment of features on object $b$ . \n96 First, we propose a solution to maximize the contrast of the $\\mathrm { L }$ channel values in CIELab color \n97 space[RG19]. We empirically confirmed that protanopia and deuteranopia, which account for the \n98 most proportion of color weakness, can distinguish the difference by $\\mathrm { L }$ channel values in common \n99 when the non-CUD objects are adjacent to each other. To illustrate Figure 1 again, the $\\mathrm { L }$ channel \n100 value of letter $^ { \\bullet } 5 ^ { \\bullet }$ in image $I$ is 61 and the surrounding color is 61. The distinguishment between \n101 the two objects is easy to normal vision, however the image $I ^ { d }$ , the deuteranopia vision, is very \n102 ambiguous. On the contrary, the CUD target image $T$ and $T ^ { d }$ have a difference of $\\mathrm { L }$ channel value 75 \n103 for the letter $\\bullet _ { 5 } ,$ and 45 for the surroundings, making it easy to distinguish between the normal and \n104 the deuteranopia vision. Due to the characteristics of these data, we refine a data pair by defining a \n105 criterion that separates two invisible non-CUD objects by $\\mathrm { L }$ channel values. \n106 Secondly, we propose a variational loss function and multi-modal feature fusion network for color \n107 preservation. It can be said that the increase in the contrast of $\\mathrm { L }$ channel values between non-CUD \n108 objects is quantitatively superior, but not in the case of increasing the differences in color preservation \n109 of input images. When non-CUD objects exist, as a simple example, the most likely way to maximize \n110 contrast is to polarize the color of the object black and white. But it is the result of complete ignorance \n111 for color preservation, so just enabling to distinguish between non-CUD objects is not always a good \n112 answer. A strong filter must be applied to distinguish non-CUD object, but its impact should not be \n113 too extensive to leading the loss of information in CUD objects. In this paper, we solve this problem \n114 by taking an appropriate trade-off of color preservation and contrast of L channel value. ", + "bbox": [ + 147, + 621, + 825, + 732 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 738, + 825, + 877 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 883, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 92, + 825, + 189 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "115 2.1 Dataset refinement criteria for CUD image ", + "text_level": 1, + "bbox": [ + 142, + 204, + 509, + 219 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "116 The training data is refined by two groups. The one is vectorized image with two colors divided \n117 by value $\\mathrm { v }$ and hue degree H in HSV color space[HMKO19], and the other is image with two \n118 or more objects that must be distinguished while preserving the color of non-CUD objects. The \n119 training data is grouped about 1,600 color combinations into the same V and then simulates them \n120 with the deuteranopia vision, converting to the adjacent color family to comply color preservation. \n121 All conversions are scaled within only S and V in HSV color space to increase at least 15 difference \n122 in the L channel value of selected non-CUD objects. The colors are combined with the 10 essential H \n123 and tones, and the similar color simulated with the deuteranopia vision was converted. Consequently, \n124 the key part of refining training data is preservation of color, allowing the models to comply with the \n125 same approach on learning. ", + "bbox": [ + 140, + 229, + 826, + 368 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 Image pre-processing ", + "text_level": 1, + "bbox": [ + 169, + 385, + 362, + 400 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "127 Our model regresses node points of piecewise linear function, which will be described in the model \n128 architecture section, and the final filter is a multiplication operation for the input image. Therefore, \n129 the multiplication operations of less than the number 1 tend to fade the color saturation. The image \n130 without color inversion converges the white color value to 1, so if the multiplying value is in the [0, \n131 1] range, the white color is shifted to the black. By inverting the color of input image, it ignores the \n132 multiplication operations for white value with 0. \n133 We generate the map image $I ^ { m }$ based on original RGB input images calculating the difference value \n134 between the image with an aspect of normal vision and the image with an aspect of deuteranopia vision. \n135 Recent studies have been conducted to augment the information or expanded the models’ perspective \n136 through transformer models[JSZK16, RFB15]. In our experiment, however, the transformer model \n137 tends to generate the predicted image ignoring the source color, which result in the polarized color to \n138 black and white like Cycle-GAN’s. ", + "bbox": [ + 140, + 410, + 825, + 493 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 498, + 825, + 583 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2145ab9b3bc8178553796fba897c230aa364a9743d0b54f31a575a34b3004279.jpg", + "text": "$$\n{ \\cal I } ^ { m } = \\vert i n v e r t ( I ^ { n } ) - i n v e r t ( I ^ { d } ) \\vert\n$$", + "text_format": "latex", + "bbox": [ + 351, + 599, + 599, + 619 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/112dfe3d49a6bf56b784a4733b751cf0a34de8798cb4de036bc65b74280c3606.jpg", + "text": "$$\n{ \\cal I } ~ = ~ \\delta \\left( c a t ^ { c h a n n e l } \\left( I ^ { n } , I ^ { d } , I ^ { m } \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 352, + 638, + 593, + 659 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "139 After applying color inversion from the original RGB input image $I ^ { n }$ , we generate the image $I ^ { d }$ with \n140 an aspect of deuteranopia vision in equation 1. From these two generated images, we can get the \n141 absolute difference value to compose the map image $I ^ { m }$ . In equation 2, the final input $I$ concatenated \n142 with $9 \\times H \\times W$ dimensions passes through the model. The $\\bar { \\delta } ( . )$ clips output to a range of [0, 1]. ", + "bbox": [ + 142, + 667, + 825, + 724 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 Model Architecture ", + "text_level": 1, + "bbox": [ + 160, + 741, + 349, + 756 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "144 CUD-NET regresses the node points of piecewise linear filter function from the input. The value \n145 of each node points computes the multiplication operation and generates the predicted image. The \n146 input $I$ is compressed into 3 feature blocks matching each input through convolution layer, pooling \n147 layer, and global pooling layer. The input with 9 channels is separated into $3 \\times 3$ channels before \n148 passing the model. The first 3 channels are literally used as the main inputs, where the multiplication \n149 operation takes place, while the remaining 6 channels are used as features. \n150 First of all, we use multi-modal fusion architecture for three separate inputs to extract expanded \n151 features. The three inputs converted to the HSV color space pass through a weights-sharing convolu \n152 tion layer to extract a feature block corresponding to the inputs. Each convolution layer consists of \n153 kernel size $^ { = 3 }$ , stride $^ { : = 1 }$ , and padding=1, reducing dimension through average pooling. We empirically \n154 noticed that the most values of output feature have distribution within the range of [-1, 1] with valid \n155 values for constructing the node points, so we use hyperbolic tan for activation function. Since we \n156 use inputs with unstructured image size, the last global pooling block holds the size of the feature \n157 instead of the average pooling block[LCY14]. \n158 The three feature blocks are combined through the multi-modal compact bilinear pooling gate \n159 $( \\mathbf { M C B } ) [ \\mathbf { F P Y ^ { * } } 1 6 ]$ , following the fusion process shown in Figure 3. The MCB gate allows both \n160 features to interact in a multiplicative way with low memory consumption and computation \n161 times. The fusion features are complemented to enhanced feature through the split attention \n162 mechanism $[ Z \\mathrm { W } Z ^ { \\ast } 2 0 ]$ . At the beginning of the experiment, we have applied the convolutional \n163 block attention mechanism[WPLK18] of each MCB gate, but we found that it does not make sense \n164 of understanding the feature itself, so we apply only one attention block to the last fusion feature. \n165 The enhanced feature pass through the fully-connected regression layer. We picked the 64 points \n166 to be regressed to compose the piecewise linear function, which is empirically confirmed to the \n167 optimized number of points in this research. The first half of the values construct the node points of \n168 the S channel and the other half comprise the $\\mathrm { v }$ channel in HSV color space. Finally, node points \n169 become the scaling factors to generate predicted image in equation 3[MMS19]. ", + "bbox": [ + 140, + 765, + 825, + 849 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 856, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/6fe42bfba41420f38aecf4a17ab1e4d9b4816d2107844d698cd5beb3f6b072ab.jpg", + "image_caption": [ + "Figure 3: Overview structure of CUD image generation " + ], + "image_footnote": [], + "bbox": [ + 189, + 88, + 799, + 281 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 337, + 825, + 393 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 398, + 825, + 497 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 502, + 825, + 573 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0cf2ca09adb19d88f99a939c9cae489ced9b2a5be9f1843f20a31b298dbe24b6.jpg", + "text": "$$\nS \\left( I _ { i } ^ { s , v } \\right) = k _ { 0 } + \\sum _ { m = 0 } ^ { M - 1 } \\left( k _ { m + 1 } - k _ { m } \\right) \\delta \\left( M I _ { i } ^ { s , v } - m \\right)\n$$", + "text_format": "latex", + "bbox": [ + 299, + 588, + 661, + 632 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "170 The total number of node point $M$ , each pixel values of S, V channel in input image $I _ { i } ^ { s , v }$ are \n171 multiplicated with the slope of actual regressed value $k _ { m }$ , the $m - t h$ generated node point. The \n172 specific node points $M$ is scaled through a multiplication operation to pixel value of the input image \n173 according to each node point. ", + "bbox": [ + 142, + 642, + 825, + 699 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "174 2.4 Loss function ", + "text_level": 1, + "bbox": [ + 142, + 715, + 307, + 731 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "175 Our dataset has one-to-many problems between input and target data. In dataset pair \n176 $( I _ { 1 } , T _ { 1 } )$ , $( I _ { 2 } , T _ { 2 } )$ , . . . , $( I _ { n } , T _ { n } )$ , for example, the red color in $I _ { 1 }$ can be targeted to purple color in \n177 $T _ { 1 }$ , and the red color in $I _ { 2 }$ can be targeted to orange color in $T _ { 2 }$ . With these one-to-many dataset \n178 structures, we design the loss function $\\mathcal { L }$ that expresses the potential and the diversity of predicted \n179 image in equation 4. ", + "bbox": [ + 140, + 739, + 825, + 810 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0dd2b39a24bcc3ade8b7c1b5e7351c7f059bac8249985049f6a955060c23ea56.jpg", + "text": "$$\n\\mathcal { L } = \\sum _ { i = 1 } ^ { N } L a b _ { l o s s } \\left( V \\left( \\Phi \\left( \\hat { I } _ { i } \\right) \\right) \\right) + H _ { l o s s } \\left( \\Phi \\left( \\hat { I } _ { i } \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 300, + 825, + 653, + 869 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "180 Stencil Masking As explained in the dataset refining criteria, we do not proceed with color \n181 conversion for all areas in the target images, but only for areas with color combinations that are \n182 invisible to the deuteranopia (non-CUD object). For this reason, the input image has color regions of \n183 converting color and unconverting color, which also can be referred to as non-CUD object and CUD \n184 object. To imply the color bound to model, the stencil masking method is introduced. ", + "bbox": [ + 138, + 882, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 90, + 825, + 133 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/4b789ddb2c78b08153055c11a351b43a49a73b9ef706ae3b51d1a596f2a2b26a.jpg", + "text": "$$\n\\Phi \\left( { \\hat { I } } _ { i } \\right) = { \\hat { I } } _ { i j } \\parallel ( I _ { i j } \\cdot T _ { i j } )\n$$", + "text_format": "latex", + "bbox": [ + 379, + 148, + 573, + 176 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "185 We consist a stencil maps through the logical and operations ’·’ of each pixel value $I _ { i j }$ , $T _ { i j }$ . Stencil \n186 map can specify the non-CUD area and be computed with predicted image $\\hat { I } _ { i j }$ of logical or operation \n187 ’ $| |$ ’in equation 5. Consequently, CUD object of the image adjusted with a stencil mask does not carry \n188 out the neural filter computation, such as the same way we refine the target image. This refined image \n189 is calculated on the loss function. \n190 CIELab Loss We use the CIELab channel loss function to maximize the contrast of color on \n191 deuteranopia vision. To stabilize the contrast and brightness of the predicted image, we calculate the \n192 MS· SSIM(multi-scale structural similarity[WSB03]) of $\\mathrm { L }$ channel. ", + "bbox": [ + 142, + 184, + 825, + 258 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 271, + 825, + 315 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/42432145446d15826e0912ad9a582adecc8b903cb263d306487137489724f86e.jpg", + "text": "$$\nL a b _ { l o s s } = \\left\\| L a b \\left( \\hat { I } _ { i } ^ { r g b } \\right) - L a b \\left( T _ { i } ^ { r g b } \\right) \\right\\| _ { 1 } + M S \\cdot S S I M \\left( L a b \\left( \\hat { I } _ { i } ^ { L } \\right) , L a b \\left( T _ { i } ^ { L } \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 191, + 338, + 753, + 367 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "193 The $L a b \\left( . \\right)$ expression in equation 6 returns the CIELab channel corresponding to the RGB channel, \n194 and all calculations are made only on the L channel. \n195 Histogram Loss We use the histogram loss function to comply with the color preservation of the \n196 image. The RGB channel is used to preserve its color, contrary to using only the $\\mathrm { L }$ channel in other \n197 loss functions. Handling the RGB channel as a loss function rather than using Lab’s ab channels has \n198 shown better results on color preservation. ", + "bbox": [ + 140, + 380, + 826, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 422, + 825, + 479 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/03e0b4da2dd1716cbf0f48130a0e3e9d6b152fda36171fe2a70ef2ee68884792.jpg", + "text": "$$\nH _ { l o s s } = - { \\omega _ { h i s t } } \\int N \\left( \\hat { I } _ { i } ^ { r g b } ; \\sigma \\right) - { \\cal N } \\left( T _ { i } ^ { r g b } ; \\sigma \\right)\n$$", + "text_format": "latex", + "bbox": [ + 303, + 494, + 640, + 527 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "199 When simply designing a loss function with the L1 distance of the RGB channel pixel values, it was \n200 very sensitive to certain values and the gradients are diverged, resulting in an untrainable experiment. \n201 Therefore, we used a gaussian expansion method $[ S \\mathbf { A } C ^ { * } 1 7 ]$ denoted by $N ( . )$ to infer a differentiable \n202 histogram loss function in equation 7. We compute the difference of the RGB channel of the \n203 differentiable histogram function, which can be altered to mean squared error or cosine similarity. \n204 The scaler $\\omega _ { h i s t }$ is determined in inverse proportion to the size of the input image. By maintaining \n205 the RGB similarity between the predicted image and the target image, we can comply with the color \n206 preservation. \n207 Variational Prediction There are various ways to maximize difference of the L channel in the \n208 image. And the target image is converted at least two colors compared to the input image. However, \n209 the predicted image of the model is generated by the neural filter, so it is unpredictable which area \n210 of color is modified. Therefore, if the color in predicted image is over-shifted or in the color value \n211 of opposite shifts to the target, the loss will rather increase. In addition to one-to-many problem \n212 that the data pair itself does not matches one-to-one in a particular color, it is necessary to generate \n213 alternative predicted image with the same aspect of the data pair. We calculate the loss function with \n214 a variational prediction based on the predicted image for potential color shifts. \n215 The first potential is the case of excessive shifts. Assume that $I _ { i } ^ { L } = \\{ 7 4 , 4 1 , 7 9 \\}$ , $\\hat { I } _ { i } ^ { L } = \\{ 9 7 , 1 0 , 7 0 \\}$ , \n216 $T _ { i } ^ { L } \\ = \\ \\{ 5 0 , 4 1 , 8 0 \\}$ in L channel value. The first and third components of each image is non-CUD \n217 objects, and second component is CUD object. Therefore, we refined data paired with a difference of \n218 15 on L channel. Here, we clip the excessive $\\mathrm { L }$ channel value in $\\hat { I } _ { i } ^ { L }$ by equation 8. Up to this point, \n219 no calculation is made as no value is exceeded in this example. The second potential is the case of \n220 opposite shifts. It can be said that a complete neural filter has been proceeded for value 97, 10, 70 \n221 where $\\mathrm { L }$ channel difference is 27. However, if we actually calculate the mean square error between \n222 $\\hat { I } _ { i } ^ { L }$ and $T _ { i } ^ { L }$ , it will be an large value over 1k. Here we can generate alternative predicted image from \n223 equation 9 and 10. ", + "bbox": [ + 138, + 535, + 826, + 647 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 661, + 825, + 773 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 780, + 826, + 912 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/689434e91b4dde1ece15a75047f8b53515cfedbab3ace44521bfc1117625af75.jpg", + "text": "$$\nc l i p \\left( \\hat { I } _ { i j } \\right) \\ = \\ \\left\\{ { m a x } { \\left( \\hat { I } _ { i j } , { T } _ { i j } \\right) } , \\qquad I _ { i j } \\ > T _ { i j } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 300, + 104, + 648, + 152 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/9b2597f442abd0c904edd84c873c006247936ef70233e08802bcb5c81bd995f5.jpg", + "text": "$$\nR _ { 1 } = \\ 2 I _ { i j } - \\ \\hat { I } _ { i j } , R _ { 2 } = \\ \\hat { I } _ { i j }\n$$", + "text_format": "latex", + "bbox": [ + 380, + 167, + 571, + 188 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/c3a32688718fc2498034dc1d8c73a0871eedd77b075fe8697acaf10411eb48b5.jpg", + "text": "$$\nV \\left( { \\hat { I } } _ { i j } \\right) = a r g m i n \\left( \\left\\| c l i p \\left( R _ { 1 , 2 } \\right) - T _ { i j } \\right\\| _ { 2 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 328, + 209, + 624, + 237 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "224 As mentioned above, we define thresholds by the maximum and minimum value of each corresponding \n225 pixel position of $\\hat { I } _ { i } ^ { L }$ and $T _ { i } ^ { L }$ . By computing a difference of residual map and the input image, we \n226 induce the alternative two images $R _ { 1 }$ , $R _ { 2 }$ . As a result, $\\hat { I } _ { i } ^ { L }$ with a smaller L2 distance is selected to \n227 alternative predicted image in equation 10, and it is finally computed with loss function compared to \n228 the $T _ { i } ^ { L }$ . The above equation establishes $V \\left( \\Phi \\left( \\hat { I } _ { i } \\right) \\right) = \\{ \\stackrel { \\cdot } { 5 4 } , 4 1 , \\stackrel { \\cdot } { 8 0 } \\}$ and the mean square error to the \n229 target image is approximately 5, which is agreeable loss value respect to $\\hat { I } _ { i } ^ { L }$ itself. \n230 Identity Loss[ZPIE17, TPW16] We use $\\mathcal { L } _ { i d e n t i t y } \\left( T _ { i } \\right)$ to apprehend the CUD object to the model. \n231 In the case of target image that already satisfy the CUD, the filter should be relatively weakly applied \n232 than input image. The input of identity loss is target image $T _ { i j }$ instead of input image $I _ { i j }$ , and the \n233 reference of the loss function is also target image $T _ { i j }$ to maintain the value itself. In computing \n234 identity loss, we do not require variational prediction as we cannot judge the potential region by \n235 equation 10. ", + "bbox": [ + 140, + 247, + 825, + 349 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 364, + 825, + 449 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "236 3 Experiments ", + "text_level": 1, + "bbox": [ + 143, + 468, + 312, + 486 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "237 The experiment was performed with Tesla V100 SXM2 and Intel Xeon Gold 5120 and the computation \n238 speed was about 40 images per minutes. We refined a dataset with Adobe Photoshop to maximize \n239 contrast in the L channel by adjusting saturation and brightness for areas that require color conversion \n240 based on deuteranopia vision simulation. Color experts has refined about 1,500 vectorized image for \n241 the training data and 300 publication images for the validation data. All the comparative experimental \n242 models used the same train, test, validation data in this paper. We used the inference data in \n243 publications, which is almost composed of vectorized images, as colors often appear distorted in a \n244 gradation-rich image. The Figure 4 is arranged in descending order of the number of combinations in \n245 colors from the top image. \n246 Both structure similarity (SSIM)[ZBSS04] and peak signal to noise ratio (PSNR) in Table 1 can \n247 indicate whether the image is suitable for CUD or not. As a notable aspect, the result has shown that \n248 comparative models with lower metrics are sensitive to high-gradation input images, which generated \n249 color-heterogeneous image. SSIM and PSNR itself can determine the increase in contrast compared \n250 to the target image but do not determine whether the color preservation complied. Therefore, we \n251 evaluated SSIM and PSNR with three references, inputs images $I$ , predicted images $\\hat { I }$ , and target \n252 images $T$ . The higher the estimation of the $\\hat { I }$ and $I$ , the more color preservation factor worked. \n253 The higher the estimation of the $\\hat { I }$ and $T$ , the more increase in contrast can be considered. We also \n254 define SSIM mean absolute error, PSNR mean absolute error to measure the extent of the conversion \n255 between the $F : I T$ and $F : \\dot { I } \\hat { I }$ in equation 11 and 12, respectively. The $N$ is total number \n256 of inference data. ", + "bbox": [ + 143, + 501, + 825, + 627 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 631, + 826, + 791 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/b5c3ae0112d4775e1d17c9295e5537664ae043d2a3d125584282c129b55680db.jpg", + "text": "$$\nS S I M \\cdot M A E = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } | S S I M ( \\hat { I } _ { i } , \\mathcal { T } _ { i } ) - S S I M ( I _ { i } , T _ { i } ) |\n$$", + "text_format": "latex", + "bbox": [ + 259, + 808, + 689, + 852 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/ab9235c819339abb3bd3d756dfaf8257af88464cf4bb5c06397fd49952b333df.jpg", + "text": "$$\nP S N R \\cdot M A E = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } | P S N R ( \\hat { I } _ { i } , \\mathcal { T } _ { i } ) - P S N R ( I _ { i } , T _ { i } ) |\n$$", + "text_format": "latex", + "bbox": [ + 254, + 866, + 694, + 910 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/ee74ef15b1c4bf5acba45467aa3b157cb56fa3118986b351ff3cd98fe81ae34d.jpg", + "image_caption": [ + "Figure 4: Comparisons of predicted images in deuteranopia vision. The color experts selected the validation data that do not satisfy the CUD in publications. " + ], + "image_footnote": [], + "bbox": [ + 191, + 92, + 805, + 395 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/54f033e20437b2470146b6ad167747c161c7dea49907ec6e1075a5ad9b5bc028.jpg", + "table_caption": [ + "Table 1: Evaluation table of comparison experiment. The CUD-NET with a low bottle-neck feature achieves better results in the experiment of the deuteranopia and the protanopia subjects(Figure 5), although the evaluation metrics are lower than that of CUD-NET. " + ], + "table_footnote": [], + "table_body": "
ArchitecturesSIM(i,1)SSIM(i,T)PSNR(i,1)PSNR(i,T)SSIM-MAEPSNR·MAE
Cycle-GANZero-DCE0.6300.63413.2813.830.31918.4430
0.9240.88821.9518.670.06613.7300
DeepLPFEnlighten-GAN0.8500.83126.3120.340.12202.0566
0.8200.80821.8519.580.14703.9983
Enlighten-GAN(scaled)0.9660.92124.8621.360.03923.4937
CUD-NET(low bottle-neck feature)CUD-NET0.8970.86627.7721.010.09012.0826
0.9620.92429.5421.190.03121.4760
", + "bbox": [ + 191, + 458, + 803, + 606 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "257 Cycle-GAN and Zero-DCE showed worse result than others. Cycle-GAN model had difficulty \n258 reconstructing a geometry of a particular object, and overall color had low saturation and brightness, \n259 resulting in color conversion into an almost grey scale image. Zero-DCE is faded in color, and the \n260 contrast was not much different from the input image. The overall image lost its color preservation, \n261 which we focused to solve in this paper. \n262 DeepLPF meets both the color preservation and contrast that we deal with for. However, DeepLPF \n263 tends to color be over-stably filtered for the images with fewer color combinations. Although the \n264 color preservation has complied better than other experiments, there were many failed results from \n265 the perspective of contrast, which the over-stable filter leads to by DeepLPF. \n266 Remarkably, predicted images of Enlighten-GAN showed reasonable results. However, simple color \n267 combinations or the images with already satisfying the CUD often showed results degenerated with \n268 low CUD suitability. Enlighten-GAN was able to generate the results we targeted, but its deviation of \n269 filter is so high that it sometimes failed to satisfy the contrast even on simple images or decreased \n270 the contrast. As the problem of GAN-based method including Enlighten-GAN, moreover, model \n271 fixes the width and height of the predicted image. If width and height of $T$ and $I$ down-scaled ", + "bbox": [ + 140, + 689, + 825, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 766, + 825, + 821 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 827, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "to size of Enlighten-GAN 272 $\\hat { I }$ (approximately 25K pixels in this experiment), the $S S I M \\left( \\hat { I } , I \\right)$ and $P S N R \\left( \\hat { I } , T \\right)$ showed higher estimation in some metrics than CUD-NET. In the opposite case of $\\hat { I }$ up-scaled to size of $T$ and $I$ , the significantly low estimation was recorded due to the information loss of up-scaling problem. ", + "bbox": [ + 142, + 88, + 825, + 164 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "CUD-NET showed stable and robust predicted images in both color preservation and increase in the contrast compared to other experiments. In comparing the values in the same region of $I$ and $\\hat { I }$ , the model scaled two $\\mathrm { L }$ channel values with opposite side in the most of case, the one goes up and the other goes down. When we reduced the number of bottle-neck feature of model, it tends to record relatively high deviation of filter scales according to the number of combinations of colors. In summary, the CUD-NET showed the highest estimations for 4 evaluation metrics. Moreover, as our model adopted a neural filter unlike generation models, there is no loss of information regarding the scaling of predicted images. ", + "bbox": [ + 140, + 170, + 826, + 284 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/9a2ce8b4db1667e120d3e391edc4e605f10711786bdbd70de95e46dfcb47ad39.jpg", + "image_caption": [ + "Figure 5: The box bar is ordered to the left side, input image $I$ , Enlighten-GAN, DeepLPF, CUDNET. The y position of box bar represents a mean and length of the box bar represents a deviation of each experiment. The lower the graph is, the higher the rank is. " + ], + "image_footnote": [], + "bbox": [ + 302, + 310, + 702, + 449 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "284 The figure 5 shows the evaluation of the deuteranopia and the protanopia. The evaluation metrics \n285 consist of object distinguishability and color harmony in order of input image I, predicted image \n286 of Enlighten GAN, DeepLPF, and CUD-NET. User study has tested upon the total of 6 subjects, 4 \n287 deuteranomaly and 2 protanomaly. The subjects were asked to list the ranks of object distinguishability \n288 and color harmony of 4-paired-image for each model-blinded item. As the experimental results, \n289 the deuteranopia subject ranked the 1-st in the object distinguishability of CUD-NET at an average \n290 rank of 1.821, followed by Enlighten-GAN at an average rank of 2.512. Similarly, the protanopia \n291 subject also ranked the 1-st in CUD, followed by Enlighten-GAN, DeepLPF, and input images. The \n292 evaluation of color harmony showed that the subjects tend to assume that the image with a good \n293 object distinguishability has good color harmony preferentially. For a total of six subjects, the five \n294 subjects chose the CUD-NET, with the exception of one who ranked Enlighten-GAN by a subtle gap ", + "bbox": [ + 138, + 531, + 825, + 685 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "295 4 Conclusion ", + "text_level": 1, + "bbox": [ + 142, + 714, + 299, + 731 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "296 In this paper, we proposed deep network to generate CUD images from non-CUD input images. The \n297 pre-processing and multi-modal fusion layer could comprehend the information for color weakness, \n298 and the variational loss function makes the model further adapt to CUD dataset. Compared to other \n299 research, we are able to maintain high-resolution images and both stable color preservation and \n300 contrast with neural filter per images. \n301 Our current research shows a robust filter for a single color, such as vectorized images, but it is \n302 difficult to expect stable results in the case of a real-world image with high gradation in hues. We \n303 consider the same limitation of our work when the certain pixel values react sensitively, making noise \n304 appear more prominent in the predicted image. In the future, we plan to create additional datasets \n305 with gradation on the vectorized image and focus on the fusion layer to improve performance of the \n306 model. \n307 References \n308 [AHB∗18] ANDERSON P., HE X., BUEHLER C., TENEY D., JOHNSON M., GOULD S., ZHANG L.: Bottom \n309 up and top-down attention for image captioning and visual question answering. 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In Proceedings of the IEEE International Conference on \n399 Computer Vision (ICCV) (Oct 2017). \n400 [ZWZ∗20] ZHANG H., WU C., ZHANG Z., ZHU Y., LIN H., ZHANG Z., SUN Y., HE T., MUELLER \n401 J., MANMATHA R., LI M., SMOLA A.: Resnest: Split-attention networks, 2020. arXiv: \n402 2004.08955. ", + "bbox": [ + 140, + 752, + 825, + 821 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 828, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 88, + 828, + 916 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 92, + 826, + 676 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "403 Checklist ", + "text_level": 1, + "bbox": [ + 143, + 691, + 254, + 708 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 719, + 339, + 733 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] See on conclusion section \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work is for positive societal impacts \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 738, + 825, + 844 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "2. If you ran experiments... ", + "bbox": [ + 214, + 849, + 393, + 864 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See on github: https://github.com/Anonymous68864576/CUD-NET-anonymous ", + "bbox": [ + 240, + 869, + 825, + 911 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] on github repository \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See on section 2.3 \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See on Experiment section ", + "bbox": [ + 238, + 92, + 825, + 180 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "3. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 210, + 184, + 823, + 199 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] Cooperated with Co-author \n(b) Did you mention the license of the assets? [Yes] on github repository \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] on github repository \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] on github repository \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 203, + 825, + 339 + ], + "page_idx": 11 + } +] \ No newline at end of file diff --git a/parse/train/3g5KdPD8LIL/3g5KdPD8LIL_middle.json b/parse/train/3g5KdPD8LIL/3g5KdPD8LIL_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..a9b730251cea1acda246e01a25d0bd5008fa4e1c --- /dev/null +++ b/parse/train/3g5KdPD8LIL/3g5KdPD8LIL_middle.json @@ -0,0 +1,40004 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 111, + 97, + 500, + 136 + ], + "lines": [ + { + "bbox": [ + 113, + 97, + 500, + 118 + ], + "spans": [ + { + "bbox": [ + 113, + 97, + 500, + 118 + ], + "score": 1.0, + "content": "CUD-NET: Color Universal Design Neural Filter for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 230, + 117, + 381, + 138 + ], + "spans": [ + { + "bbox": [ + 230, + 117, + 381, + 138 + ], + "score": 1.0, + "content": "the Color Weakness", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 259, + 176, + 354, + 219 + ], + "lines": [ + { + "bbox": [ + 258, + 176, + 355, + 187 + ], + "spans": [ + { + "bbox": [ + 258, + 176, + 355, + 187 + ], + "score": 1.0, + "content": "Anonymous Author(s)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 283, + 186, + 328, + 199 + ], + "spans": [ + { + "bbox": [ + 283, + 186, + 328, + 199 + ], + "score": 1.0, + "content": "Affiliation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 286, + 197, + 324, + 209 + ], + "spans": [ + { + "bbox": [ + 286, + 197, + 324, + 209 + ], + "score": 1.0, + "content": "Address", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 290, + 210, + 320, + 219 + ], + "spans": [ + { + "bbox": [ + 290, + 210, + 320, + 219 + ], + "score": 1.0, + "content": "email", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 283, + 248, + 328, + 261 + ], + "lines": [ + { + "bbox": [ + 281, + 247, + 330, + 263 + ], + "spans": [ + { + "bbox": [ + 281, + 247, + 330, + 263 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 91, + 273, + 469, + 427 + ], + "lines": [ + { + "bbox": [ + 93, + 273, + 469, + 286 + ], + "spans": [ + { + "bbox": [ + 93, + 277, + 99, + 284 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 141, + 273, + 469, + 286 + ], + "score": 1.0, + "content": "Information on images should be visually understood to anyone, including the color", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 93, + 285, + 470, + 296 + ], + "spans": [ + { + "bbox": [ + 93, + 287, + 99, + 296 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 141, + 285, + 470, + 296 + ], + "score": 1.0, + "content": "weakness. 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The right-bottom target image", + "type": "text" + }, + { + "bbox": [ + 398, + 639, + 411, + 649 + ], + "score": 0.87, + "content": "{ \\bf \\bar { \\boldsymbol { T } } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 637, + 506, + 652 + ], + "score": 1.0, + "content": ", refined image by color", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 650, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 259, + 662 + ], + "score": 1.0, + "content": "expert designers, shows that the letter", + "type": "text" + }, + { + "bbox": [ + 260, + 650, + 272, + 660 + ], + "score": 0.51, + "content": "\\cdot 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 650, + 506, + 662 + ], + "score": 1.0, + "content": "appeared well at the deuteranopia vision. Here, we define", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 659, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 242, + 675 + ], + "score": 1.0, + "content": "the non-CUD objects as the letter", + "type": "text" + }, + { + "bbox": [ + 243, + 661, + 254, + 671 + ], + "score": 0.65, + "content": "\\cdot 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 659, + 506, + 675 + ], + "score": 1.0, + "content": "and surroundings invisible to deuteranopia vision in the image", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 113, + 682 + ], + "score": 0.69, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 672, + 277, + 684 + ], + "score": 1.0, + "content": ", and define the CUD objects as the letter", + "type": "text" + }, + { + "bbox": [ + 277, + 672, + 289, + 682 + ], + "score": 0.39, + "content": "\\bullet _ { 5 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "and surroundings visible to deuteranopia vision in the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 133, + 695 + ], + "score": 1.0, + "content": "image", + "type": "text" + }, + { + "bbox": [ + 134, + 683, + 142, + 693 + ], + "score": 0.62, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 683, + 505, + 695 + ], + "score": 1.0, + "content": ". 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Do not distribute.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 111, + 97, + 500, + 136 + ], + "lines": [ + { + "bbox": [ + 113, + 97, + 500, + 118 + ], + "spans": [ + { + "bbox": [ + 113, + 97, + 500, + 118 + ], + "score": 1.0, + "content": "CUD-NET: Color Universal Design Neural Filter for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 230, + 117, + 381, + 138 + ], + "spans": [ + { + "bbox": [ + 230, + 117, + 381, + 138 + ], + "score": 1.0, + "content": "the Color Weakness", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "list", + "bbox": [ + 259, + 176, + 354, + 219 + ], + "lines": [ + { + "bbox": [ + 258, + 176, + 355, + 187 + ], + "spans": [ + { + "bbox": [ + 258, + 176, + 355, + 187 + ], + "score": 1.0, + "content": "Anonymous Author(s)", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 283, + 186, + 328, + 199 + ], + "spans": [ + { + "bbox": [ + 283, + 186, + 328, + 199 + ], + "score": 1.0, + "content": "Affiliation", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 286, + 197, + 324, + 209 + ], + "spans": [ + { + "bbox": [ + 286, + 197, + 324, + 209 + ], + "score": 1.0, + "content": "Address", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 290, + 210, + 320, + 219 + ], + "spans": [ + { + "bbox": [ + 290, + 210, + 320, + 219 + ], + "score": 1.0, + "content": "email", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + } + ], + "index": 3.5, + "bbox_fs": [ + 258, + 176, + 355, + 219 + ] + }, + { + "type": "title", + "bbox": [ + 283, + 248, + 328, + 261 + ], + "lines": [ + { + "bbox": [ + 281, + 247, + 330, + 263 + ], + "spans": [ + { + "bbox": [ + 281, + 247, + 330, + 263 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "index", + "bbox": [ + 91, + 273, + 469, + 427 + ], + "lines": [ + { + "bbox": [ + 93, + 273, + 469, + 286 + ], + "spans": [ + { + "bbox": [ + 93, + 277, + 99, + 284 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 141, + 273, + 469, + 286 + ], + "score": 1.0, + "content": "Information on images should be visually understood to anyone, including the color", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 285, + 470, + 296 + ], + "spans": [ + { + "bbox": [ + 93, + 287, + 99, + 296 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 141, + 285, + 470, + 296 + ], + "score": 1.0, + "content": "weakness. However, it is not recognizable if color that seems distorted to the color", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 295, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 93, + 298, + 99, + 307 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 141, + 295, + 470, + 308 + ], + "score": 1.0, + "content": "weakness meets an adjacent object. We suggest CUD-NET1 based on convolutional", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 306, + 470, + 320 + ], + "spans": [ + { + "bbox": [ + 92, + 309, + 100, + 318 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 141, + 306, + 470, + 320 + ], + "score": 1.0, + "content": "deep neural network to generate color universal design (CUD) images that satisfy", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 317, + 470, + 330 + ], + "spans": [ + { + "bbox": [ + 93, + 320, + 100, + 329 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 141, + 317, + 470, + 330 + ], + "score": 1.0, + "content": "both color preservation and distinguishment of color for input images. CUD-NET", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 329, + 470, + 340 + ], + "spans": [ + { + "bbox": [ + 93, + 330, + 99, + 340 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 141, + 329, + 470, + 340 + ], + "score": 1.0, + "content": "regresses the node point of the piecewise linear function based on information of", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 340, + 469, + 352 + ], + "spans": [ + { + "bbox": [ + 92, + 342, + 100, + 351 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 141, + 340, + 469, + 352 + ], + "score": 1.0, + "content": "input images and comprises a specific filter per image. We present the following", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 351, + 469, + 362 + ], + "spans": [ + { + "bbox": [ + 92, + 353, + 99, + 361 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 141, + 351, + 469, + 362 + ], + "score": 1.0, + "content": "methods to generate CUD images for the color weakness. First, we refine the CUD", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 360, + 470, + 374 + ], + "spans": [ + { + "bbox": [ + 92, + 363, + 100, + 372 + ], + "score": 1.0, + "content": "9", + "type": "text" + }, + { + "bbox": [ + 141, + 360, + 470, + 374 + ], + "score": 1.0, + "content": "dataset on specific criteria by color experts. Second, the input image information", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 371, + 470, + 385 + ], + "spans": [ + { + "bbox": [ + 90, + 375, + 100, + 383 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 141, + 371, + 470, + 385 + ], + "score": 1.0, + "content": "is expanded through the pre-processing specialized on the color weakness vision.", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 382, + 470, + 396 + ], + "spans": [ + { + "bbox": [ + 89, + 385, + 100, + 395 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 141, + 382, + 470, + 396 + ], + "score": 1.0, + "content": "Third, we suggest a multi-modal feature fusion architecture that combines features", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 394, + 469, + 406 + ], + "spans": [ + { + "bbox": [ + 89, + 395, + 100, + 406 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 141, + 394, + 469, + 406 + ], + "score": 1.0, + "content": "to process expanded images. Finally, we suggest a deformable loss function by the", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 405, + 469, + 417 + ], + "spans": [ + { + "bbox": [ + 90, + 407, + 99, + 416 + ], + "score": 1.0, + "content": "13", + "type": "text" + }, + { + "bbox": [ + 142, + 405, + 469, + 417 + ], + "score": 1.0, + "content": "composition of the predicted image through the model to avoid the one-to-many", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 416, + 240, + 428 + ], + "spans": [ + { + "bbox": [ + 90, + 418, + 99, + 427 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 141, + 416, + 240, + 428 + ], + "score": 1.0, + "content": "problems of the dataset.", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + } + ], + "index": 13.5, + "bbox_fs": [ + 89, + 273, + 470, + 428 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 447, + 191, + 460 + ], + "lines": [ + { + "bbox": [ + 87, + 446, + 192, + 462 + ], + "spans": [ + { + "bbox": [ + 87, + 446, + 192, + 462 + ], + "score": 1.0, + "content": "15 1 Introduction", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 94, + 471, + 177, + 483 + ], + "lines": [ + { + "bbox": [ + 90, + 470, + 178, + 484 + ], + "spans": [ + { + "bbox": [ + 90, + 470, + 178, + 484 + ], + "score": 1.0, + "content": "16 1.1 Motivation", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "index", + "bbox": [ + 90, + 492, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 91, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 91, + 495, + 99, + 503 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 105, + 492, + 313, + 504 + ], + "score": 1.0, + "content": "The green and red color blindness are made up of", + "type": "text" + }, + { + "bbox": [ + 313, + 492, + 328, + 502 + ], + "score": 0.86, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 492, + 385, + 504 + ], + "score": 1.0, + "content": "of males and", + "type": "text" + }, + { + "bbox": [ + 385, + 492, + 408, + 502 + ], + "score": 0.86, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "of females in Northern", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 90, + 505, + 100, + 514 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 106, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "European descent[Won11], which is almost up to rate of one person in 20 people. Green and red", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 90, + 516, + 100, + 525 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "blindness is the most common pattern, followed by blue, yellow, and total color blindness. In this", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 89, + 527, + 100, + 536 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "paper, we generate Color Universal Design (CUD) images, which are color weakness friendly design", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 89, + 538, + 100, + 547 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "forms, through deep learning around the aspect of the red color weakness (protanopia) and green", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 89, + 549, + 100, + 558 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "color weakness (deuteranopia) vision. Protanopia is insensitive to red color and deuteranopia is", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 558, + 474, + 569 + ], + "spans": [ + { + "bbox": [ + 89, + 560, + 100, + 569 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 558, + 474, + 569 + ], + "score": 1.0, + "content": "insensitive to green color, although it varies depending on individual color weakness extent.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + } + ], + "index": 26, + "bbox_fs": [ + 89, + 492, + 506, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 573, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 585 + ], + "score": 1.0, + "content": "There are studies that help color discrimination to the color weakness, including wearable devices", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "and surgeries[VZCR20]. However, since these research require time and cost, we simply generate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "CUD images with an image enhancement method based on deep learning to make the corresponding", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 406, + 619 + ], + "score": 1.0, + "content": "color visible for the color weakness. For an example of the left-above image", + "type": "text" + }, + { + "bbox": [ + 407, + 607, + 414, + 616 + ], + "score": 0.69, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "in Figure 1, the people", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 316, + 630 + ], + "score": 1.0, + "content": "who are not color weakness can distinguish the letter", + "type": "text" + }, + { + "bbox": [ + 316, + 618, + 327, + 627 + ], + "score": 0.67, + "content": "\\cdot 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 616, + 506, + 630 + ], + "score": 1.0, + "content": "in the image. But as a deuteranopia vision in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 174, + 642 + ], + "score": 1.0, + "content": "left-below image", + "type": "text" + }, + { + "bbox": [ + 175, + 627, + 186, + 638 + ], + "score": 0.85, + "content": "I ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 627, + 329, + 642 + ], + "score": 1.0, + "content": ", the surrounding color and the letter", + "type": "text" + }, + { + "bbox": [ + 329, + 629, + 341, + 639 + ], + "score": 0.66, + "content": "\\cdot 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 627, + 506, + 642 + ], + "score": 1.0, + "content": "are very analogous, making it ambiguous", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 398, + 652 + ], + "score": 1.0, + "content": "to distinguish the bound of adjacent object. The right-bottom target image", + "type": "text" + }, + { + "bbox": [ + 398, + 639, + 411, + 649 + ], + "score": 0.87, + "content": "{ \\bf \\bar { \\boldsymbol { T } } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 637, + 506, + 652 + ], + "score": 1.0, + "content": ", refined image by color", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 650, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 259, + 662 + ], + "score": 1.0, + "content": "expert designers, shows that the letter", + "type": "text" + }, + { + "bbox": [ + 260, + 650, + 272, + 660 + ], + "score": 0.51, + "content": "\\cdot 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 650, + 506, + 662 + ], + "score": 1.0, + "content": "appeared well at the deuteranopia vision. Here, we define", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 659, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 242, + 675 + ], + "score": 1.0, + "content": "the non-CUD objects as the letter", + "type": "text" + }, + { + "bbox": [ + 243, + 661, + 254, + 671 + ], + "score": 0.65, + "content": "\\cdot 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 659, + 506, + 675 + ], + "score": 1.0, + "content": "and surroundings invisible to deuteranopia vision in the image", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 113, + 682 + ], + "score": 0.69, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 672, + 277, + 684 + ], + "score": 1.0, + "content": ", and define the CUD objects as the letter", + "type": "text" + }, + { + "bbox": [ + 277, + 672, + 289, + 682 + ], + "score": 0.39, + "content": "\\bullet _ { 5 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "and surroundings visible to deuteranopia vision in the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 133, + 695 + ], + "score": 1.0, + "content": "image", + "type": "text" + }, + { + "bbox": [ + 134, + 683, + 142, + 693 + ], + "score": 0.62, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 683, + 505, + 695 + ], + "score": 1.0, + "content": ". 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The non-CUD object means that adjacent objects are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 89, + 301, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 89, + 305, + 100, + 314 + ], + "score": 1.0, + "content": "36", + "type": "text" + }, + { + "bbox": [ + 105, + 302, + 480, + 316 + ], + "score": 1.0, + "content": "distinguishable on normal vision but not the color weakness vision. Consequently, we generate", + "type": "text" + }, + { + "bbox": [ + 480, + 301, + 487, + 312 + ], + "score": 0.82, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 90, + 314, + 307, + 326 + ], + "spans": [ + { + "bbox": [ + 90, + 316, + 101, + 325 + ], + "score": 1.0, + "content": "37", + "type": "text" + }, + { + "bbox": [ + 105, + 314, + 297, + 326 + ], + "score": 1.0, + "content": "satisfies CUD with a specific filter to the image", + "type": "text" + }, + { + "bbox": [ + 297, + 314, + 304, + 324 + ], + "score": 0.47, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 314, + 307, + 326 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 89, + 330, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 89, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 89, + 332, + 99, + 342 + ], + "score": 1.0, + "content": "38", + "type": "text" + }, + { + "bbox": [ + 106, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "We want to apply as weak filter as possible to CUD objects to preserve color, which requires a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 88, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 88, + 343, + 99, + 353 + ], + "score": 1.0, + "content": "39", + "type": "text" + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "certain level of object comprehension mechanism to do so. There are various studies from classic", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 89, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 89, + 352, + 100, + 365 + ], + "score": 1.0, + "content": "40", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 428, + 365 + ], + "score": 1.0, + "content": "PCA[WEG87] to machine learning-based object segmentation methods[TSC20,", + "type": "text" + }, + { + "bbox": [ + 428, + 352, + 466, + 363 + ], + "score": 0.51, + "content": "Z \\mathrm { G L } ^ { * } 2 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "to define", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 88, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 88, + 365, + 99, + 374 + ], + "score": 1.0, + "content": "41", + "type": "text" + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "specific objects or areas in image. The research on semantic segmentation, which even provides labels", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 88, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 88, + 375, + 99, + 385 + ], + "score": 1.0, + "content": "42", + "type": "text" + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "between objects, seems that deep learning still does not have a complete comprehension of all objects", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 88, + 384, + 507, + 398 + ], + "spans": [ + { + "bbox": [ + 88, + 387, + 99, + 396 + ], + "score": 1.0, + "content": "43", + "type": "text" + }, + { + "bbox": [ + 104, + 384, + 507, + 398 + ], + "score": 1.0, + "content": "in the real-world. The visual question answering to arbitrary questions about object’s interactions,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 88, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 88, + 397, + 100, + 407 + ], + "score": 1.0, + "content": "44", + "type": "text" + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "the most general issue on comprehension of object, does not have high transmission power to be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 88, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 88, + 408, + 100, + 418 + ], + "score": 1.0, + "content": "45", + "type": "text" + }, + { + "bbox": [ + 105, + 406, + 161, + 419 + ], + "score": 1.0, + "content": "practical uses", + "type": "text" + }, + { + "bbox": [ + 162, + 406, + 199, + 417 + ], + "score": 0.73, + "content": "[ \\mathrm { A H B ^ { * } } 1 8", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 406, + 203, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 203, + 406, + 239, + 417 + ], + "score": 0.71, + "content": "\\mathrm { K Z G ^ { * } } 1 7", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 406, + 242, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 243, + 406, + 279, + 417 + ], + "score": 0.52, + "content": "\\mathrm { L Y L } ^ { * } 2 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 406, + 506, + 419 + ], + "score": 1.0, + "content": ". Therefore, we expand feature of the input image around", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 88, + 416, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 88, + 419, + 99, + 429 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 105, + 416, + 506, + 431 + ], + "score": 1.0, + "content": "the information of color weakness vision and define the robust neural filter. In summary, we suggest", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 88, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 88, + 430, + 99, + 440 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 104, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "a CUD-NET that generates an image suitable for CUD, while complying with the color preservation", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 89, + 438, + 193, + 452 + ], + "spans": [ + { + "bbox": [ + 89, + 441, + 100, + 450 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 193, + 452 + ], + "score": 1.0, + "content": "for the source image.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 90, + 455, + 506, + 488 + ], + "lines": [ + { + "bbox": [ + 88, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 88, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "49 In this paper, we suggest the Color Universal Design Network (CUD-NET) to satisfy both color", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 89, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 89, + 469, + 100, + 477 + ], + "score": 1.0, + "content": "50", + "type": "text" + }, + { + "bbox": [ + 104, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "preservation and contrast of non-CUD objects (CUD suitability). We introduce 4 core contributions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 88, + 477, + 165, + 489 + ], + "spans": [ + { + "bbox": [ + 88, + 477, + 165, + 489 + ], + "score": 1.0, + "content": "51 of CUD-NET.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 133, + 499, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 133, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 133, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "• Dataset refinement criteria for CUD image We refine training data into two groups, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 141, + 510, + 344, + 522 + ], + "score": 1.0, + "content": "one with a simple color tone image based on H and", + "type": "text" + }, + { + "bbox": [ + 344, + 510, + 353, + 520 + ], + "score": 0.26, + "content": "\\mathrm { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "in the HSV color space, the other with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 521, + 437, + 533 + ], + "spans": [ + { + "bbox": [ + 141, + 521, + 437, + 533 + ], + "score": 1.0, + "content": "two or more non-CUD objects that must be distinguished in publications.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 133, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 133, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "• Image pre-processing for CUD-NET We carry out pre-processing to expand the infor-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 141, + 546, + 311, + 560 + ], + "score": 1.0, + "content": "mation of the input image. 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There are various studies from classic", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 89, + 352, + 100, + 365 + ], + "score": 1.0, + "content": "40", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 428, + 365 + ], + "score": 1.0, + "content": "PCA[WEG87] to machine learning-based object segmentation methods[TSC20,", + "type": "text" + }, + { + "bbox": [ + 428, + 352, + 466, + 363 + ], + "score": 0.51, + "content": "Z \\mathrm { G L } ^ { * } 2 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "to define", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 88, + 365, + 99, + 374 + ], + "score": 1.0, + "content": "41", + "type": "text" + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "specific objects or areas in image. The research on semantic segmentation, which even provides labels", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 88, + 375, + 99, + 385 + ], + "score": 1.0, + "content": "42", + "type": "text" + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "between objects, seems that deep learning still does not have a complete comprehension of all objects", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 384, + 507, + 398 + ], + "spans": [ + { + "bbox": [ + 88, + 387, + 99, + 396 + ], + "score": 1.0, + "content": "43", + "type": "text" + }, + { + "bbox": [ + 104, + 384, + 507, + 398 + ], + "score": 1.0, + "content": "in the real-world. The visual question answering to arbitrary questions about object’s interactions,", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 88, + 397, + 100, + 407 + ], + "score": 1.0, + "content": "44", + "type": "text" + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "the most general issue on comprehension of object, does not have high transmission power to be", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 88, + 408, + 100, + 418 + ], + "score": 1.0, + "content": "45", + "type": "text" + }, + { + "bbox": [ + 105, + 406, + 161, + 419 + ], + "score": 1.0, + "content": "practical uses", + "type": "text" + }, + { + "bbox": [ + 162, + 406, + 199, + 417 + ], + "score": 0.73, + "content": "[ \\mathrm { A H B ^ { * } } 1 8", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 406, + 203, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 203, + 406, + 239, + 417 + ], + "score": 0.71, + "content": "\\mathrm { K Z G ^ { * } } 1 7", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 406, + 242, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 243, + 406, + 279, + 417 + ], + "score": 0.52, + "content": "\\mathrm { L Y L } ^ { * } 2 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 406, + 506, + 419 + ], + "score": 1.0, + "content": ". Therefore, we expand feature of the input image around", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 416, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 88, + 419, + 99, + 429 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 105, + 416, + 506, + 431 + ], + "score": 1.0, + "content": "the information of color weakness vision and define the robust neural filter. In summary, we suggest", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 88, + 430, + 99, + 440 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 104, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "a CUD-NET that generates an image suitable for CUD, while complying with the color preservation", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 438, + 193, + 452 + ], + "spans": [ + { + "bbox": [ + 89, + 441, + 100, + 450 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 193, + 452 + ], + "score": 1.0, + "content": "for the source image.", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 88, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "49 In this paper, we suggest the Color Universal Design Network (CUD-NET) to satisfy both color", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 89, + 469, + 100, + 477 + ], + "score": 1.0, + "content": "50", + "type": "text" + }, + { + "bbox": [ + 104, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "preservation and contrast of non-CUD objects (CUD suitability). 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It estimates pixel-wise and high-order filter for dynamic range adjustment of input", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 88, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 88, + 378, + 100, + 388 + ], + "score": 1.0, + "content": "80", + "type": "text" + }, + { + "bbox": [ + 105, + 375, + 343, + 388 + ], + "score": 1.0, + "content": "images with lightweight deep network, DCE-Net. DeepLPF", + "type": "text" + }, + { + "bbox": [ + 343, + 376, + 389, + 387 + ], + "score": 0.54, + "content": "\\mathbf { M M M } ^ { * } 2 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "tried to solve the problem by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 88, + 386, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 88, + 389, + 99, + 398 + ], + "score": 1.0, + "content": "81", + "type": "text" + }, + { + "bbox": [ + 105, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "using a graduated filter, elliptical filter, and polynomial filter. The authors not only tried to visually", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 88, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 88, + 399, + 100, + 409 + ], + "score": 1.0, + "content": "82", + "type": "text" + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "enhance the contrast of images but also to comprise stable filters that are easy to understand for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 88, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 88, + 410, + 100, + 421 + ], + "score": 1.0, + "content": "83", + "type": "text" + }, + { + "bbox": [ + 106, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "the spectators while keeping the color preservation. In our problem, however, the contrast factor is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 88, + 418, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 88, + 421, + 100, + 431 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 105, + 418, + 506, + 433 + ], + "score": 1.0, + "content": "almost same results as the input image in both visions, while complying the high color preservation,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 88, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 88, + 432, + 100, + 443 + ], + "score": 1.0, + "content": "85", + "type": "text" + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "resulting over-stable filter. It is assumed that the inability in comprehension of object’s interaction", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 88, + 441, + 209, + 453 + ], + "spans": [ + { + "bbox": [ + 88, + 443, + 100, + 452 + ], + "score": 1.0, + "content": "86", + "type": "text" + }, + { + "bbox": [ + 105, + 441, + 209, + 453 + ], + "score": 1.0, + "content": "leads to over-stable filter.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 92, + 468, + 192, + 482 + ], + "lines": [ + { + "bbox": [ + 87, + 464, + 195, + 486 + ], + "spans": [ + { + "bbox": [ + 87, + 464, + 195, + 486 + ], + "score": 1.0, + "content": "87 2 Methodology", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 90, + 492, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 89, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 89, + 495, + 100, + 504 + ], + "score": 1.0, + "content": "88", + "type": "text" + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "We define the ideal predicted image as an increase in the contrast between non-CUD objects and the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 89, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 89, + 505, + 100, + 515 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 106, + 504, + 353, + 515 + ], + "score": 1.0, + "content": "color preservation for the input image. The non-CUD object", + "type": "text" + }, + { + "bbox": [ + 354, + 505, + 360, + 513 + ], + "score": 0.65, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 504, + 456, + 515 + ], + "score": 1.0, + "content": "should be mapped into", + "type": "text" + }, + { + "bbox": [ + 456, + 504, + 463, + 513 + ], + "score": 0.77, + "content": "\\acute { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "and CUD", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 89, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 89, + 516, + 100, + 526 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 106, + 515, + 134, + 527 + ], + "score": 1.0, + "content": "object", + "type": "text" + }, + { + "bbox": [ + 134, + 515, + 140, + 524 + ], + "score": 0.59, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "should preserve its color and contrast like an ideal example of Figure 2. However, as our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 89, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 89, + 527, + 100, + 537 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "neural filter affects the whole pixels throughout the image, we have the constraint of applying the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 89, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 89, + 538, + 100, + 548 + ], + "score": 1.0, + "content": "92", + "type": "text" + }, + { + "bbox": [ + 106, + 536, + 193, + 549 + ], + "score": 1.0, + "content": "same filter to objects", + "type": "text" + }, + { + "bbox": [ + 194, + 538, + 200, + 546 + ], + "score": 0.68, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 536, + 218, + 549 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 537, + 224, + 546 + ], + "score": 0.54, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 536, + 451, + 549 + ], + "score": 1.0, + "content": ". It is very hard to make the contrast and color of object", + "type": "text" + }, + { + "bbox": [ + 451, + 537, + 457, + 546 + ], + "score": 0.68, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "exactly the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 89, + 546, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 89, + 549, + 100, + 559 + ], + "score": 1.0, + "content": "93", + "type": "text" + }, + { + "bbox": [ + 104, + 546, + 403, + 561 + ], + "score": 1.0, + "content": "same as before the filter adjustment while maximizing the contrast of object", + "type": "text" + }, + { + "bbox": [ + 404, + 549, + 410, + 557 + ], + "score": 0.62, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 546, + 506, + 561 + ], + "score": 1.0, + "content": ". Therefore, we propose", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 89, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 89, + 560, + 100, + 570 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "a deep learning-based regression to comprise the specific filter per image that maximizes the contrast", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 89, + 569, + 379, + 581 + ], + "spans": [ + { + "bbox": [ + 89, + 571, + 100, + 580 + ], + "score": 1.0, + "content": "95", + "type": "text" + }, + { + "bbox": [ + 105, + 569, + 144, + 581 + ], + "score": 1.0, + "content": "of object", + "type": "text" + }, + { + "bbox": [ + 144, + 571, + 151, + 579 + ], + "score": 0.61, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 569, + 370, + 581 + ], + "score": 1.0, + "content": "while minimizing the adjustment of features on object", + "type": "text" + }, + { + "bbox": [ + 370, + 569, + 375, + 579 + ], + "score": 0.55, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 569, + 379, + 581 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 87, + 585, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 89, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 89, + 587, + 100, + 596 + ], + "score": 1.0, + "content": "96", + "type": "text" + }, + { + "bbox": [ + 104, + 585, + 360, + 597 + ], + "score": 1.0, + "content": "First, we propose a solution to maximize the contrast of the", + "type": "text" + }, + { + "bbox": [ + 360, + 586, + 369, + 595 + ], + "score": 0.28, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "channel values in CIELab color", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 89, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 89, + 598, + 99, + 608 + ], + "score": 1.0, + "content": "97", + "type": "text" + }, + { + "bbox": [ + 104, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "space[RG19]. We empirically confirmed that protanopia and deuteranopia, which account for the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 89, + 609, + 100, + 619 + ], + "score": 1.0, + "content": "98", + "type": "text" + }, + { + "bbox": [ + 105, + 607, + 385, + 620 + ], + "score": 1.0, + "content": "most proportion of color weakness, can distinguish the difference by", + "type": "text" + }, + { + "bbox": [ + 386, + 608, + 394, + 617 + ], + "score": 0.41, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "channel values in common", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 89, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 89, + 620, + 100, + 629 + ], + "score": 1.0, + "content": "99", + "type": "text" + }, + { + "bbox": [ + 106, + 618, + 461, + 630 + ], + "score": 1.0, + "content": "when the non-CUD objects are adjacent to each other. To illustrate Figure 1 again, the", + "type": "text" + }, + { + "bbox": [ + 461, + 618, + 470, + 628 + ], + "score": 0.27, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "channel", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 629, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 86, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "100", + "type": "text" + }, + { + "bbox": [ + 105, + 629, + 167, + 642 + ], + "score": 1.0, + "content": "value of letter", + "type": "text" + }, + { + "bbox": [ + 167, + 629, + 178, + 639 + ], + "score": 0.47, + "content": "^ { \\bullet } 5 ^ { \\bullet }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 629, + 218, + 642 + ], + "score": 1.0, + "content": "in image", + "type": "text" + }, + { + "bbox": [ + 218, + 630, + 225, + 639 + ], + "score": 0.65, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 629, + 506, + 642 + ], + "score": 1.0, + "content": "is 61 and the surrounding color is 61. The distinguishment between", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 86, + 637, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 86, + 642, + 99, + 651 + ], + "score": 1.0, + "content": "101", + "type": "text" + }, + { + "bbox": [ + 104, + 637, + 357, + 654 + ], + "score": 1.0, + "content": "the two objects is easy to normal vision, however the image", + "type": "text" + }, + { + "bbox": [ + 358, + 639, + 369, + 650 + ], + "score": 0.86, + "content": "I ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 637, + 507, + 654 + ], + "score": 1.0, + "content": ", the deuteranopia vision, is very", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 86, + 653, + 100, + 663 + ], + "score": 1.0, + "content": "102", + "type": "text" + }, + { + "bbox": [ + 105, + 650, + 308, + 663 + ], + "score": 1.0, + "content": "ambiguous. 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It can be said that the increase in the contrast of", + "type": "text" + }, + { + "bbox": [ + 356, + 712, + 363, + 721 + ], + "score": 0.52, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "channel values between non-CUD", + "type": "text" + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "108", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "objects is quantitatively superior, but not in the case of increasing the differences in color preservation", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 100, + 96 + ], + "score": 1.0, + "content": "109", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "of input images. When non-CUD objects exist, as a simple example, the most likely way to maximize", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 86, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "110", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "contrast is to polarize the color of the object black and white. But it is the result of complete ignorance", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "111", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "for color preservation, so just enabling to distinguish between non-CUD objects is not always a good", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 129 + ], + "score": 1.0, + "content": "112", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "answer. A strong filter must be applied to distinguish non-CUD object, but its impact should not be", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 86, + 129, + 100, + 139 + ], + "score": 1.0, + "content": "113", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "too extensive to leading the loss of information in CUD objects. In this paper, we solve this problem", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 137, + 459, + 151 + ], + "spans": [ + { + "bbox": [ + 86, + 140, + 100, + 150 + ], + "score": 1.0, + "content": "114", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 137, + 459, + 151 + ], + "score": 1.0, + "content": "by taking an appropriate trade-off of color preservation and contrast of L channel value.", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + } + ], + "index": 33.5, + "bbox_fs": [ + 89, + 493, + 506, + 581 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 585, + 505, + 695 + ], + "lines": [], + "index": 42.5, + "bbox_fs": [ + 86, + 585, + 507, + 695 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 90, + 700, + 505, + 722 + ], + "lines": [], + "index": 48.5, + "bbox_fs": [ + 88, + 699, + 505, + 723 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 73, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 86, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 104, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "objects is quantitatively superior, but not in the case of increasing the differences in color preservation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 85, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 100, + 96 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "of input images. When non-CUD objects exist, as a simple example, the most likely way to maximize", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 86, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "contrast is to polarize the color of the object black and white. But it is the result of complete ignorance", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 86, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "111", + "type": "text" + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "for color preservation, so just enabling to distinguish between non-CUD objects is not always a good", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 85, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 129 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 106, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "answer. 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In this paper, we solve this problem", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 137, + 459, + 151 + ], + "spans": [ + { + "bbox": [ + 86, + 140, + 100, + 150 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 105, + 137, + 459, + 151 + ], + "score": 1.0, + "content": "by taking an appropriate trade-off of color preservation and contrast of L channel value.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 87, + 162, + 312, + 174 + ], + "lines": [ + { + "bbox": [ + 84, + 160, + 314, + 177 + ], + "spans": [ + { + "bbox": [ + 84, + 160, + 314, + 177 + ], + "score": 1.0, + "content": "115 2.1 Dataset refinement criteria for CUD image", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 86, + 182, + 506, + 292 + ], + "lines": [ + { + "bbox": [ + 86, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 86, + 184, + 100, + 194 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "The training data is refined by two groups. The one is vectorized image with two colors divided", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 193, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 86, + 195, + 99, + 205 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 105, + 193, + 145, + 206 + ], + "score": 1.0, + "content": "by value", + "type": "text" + }, + { + "bbox": [ + 145, + 194, + 154, + 204 + ], + "score": 0.25, + "content": "\\mathrm { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 193, + 506, + 206 + ], + "score": 1.0, + "content": "and hue degree H in HSV color space[HMKO19], and the other is image with two", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 86, + 206, + 100, + 216 + ], + "score": 1.0, + "content": "118", + "type": "text" + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "or more objects that must be distinguished while preserving the color of non-CUD objects. The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 86, + 217, + 100, + 227 + ], + "score": 1.0, + "content": "119", + "type": "text" + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "training data is grouped about 1,600 color combinations into the same V and then simulates them", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 225, + 507, + 239 + ], + "spans": [ + { + "bbox": [ + 86, + 228, + 100, + 238 + ], + "score": 1.0, + "content": "120", + "type": "text" + }, + { + "bbox": [ + 105, + 225, + 507, + 239 + ], + "score": 1.0, + "content": "with the deuteranopia vision, converting to the adjacent color family to comply color preservation.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 86, + 239, + 99, + 249 + ], + "score": 1.0, + "content": "121", + "type": "text" + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "All conversions are scaled within only S and V in HSV color space to increase at least 15 difference", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 86, + 249, + 100, + 259 + ], + "score": 1.0, + "content": "122", + "type": "text" + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "in the L channel value of selected non-CUD objects. The colors are combined with the 10 essential H", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 86, + 261, + 99, + 270 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "and tones, and the similar color simulated with the deuteranopia vision was converted. 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Therefore,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 345, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 86, + 348, + 100, + 358 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 104, + 345, + 506, + 361 + ], + "score": 1.0, + "content": "the multiplication operations of less than the number 1 tend to fade the color saturation. The image", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 85, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 85, + 359, + 100, + 369 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "without color inversion converges the white color value to 1, so if the multiplying value is in the [0,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 85, + 370, + 100, + 380 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "1] range, the white color is shifted to the black. 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The colors are combined with the 10 essential H", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 86, + 261, + 99, + 270 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "and tones, and the similar color simulated with the deuteranopia vision was converted. 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The image", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 85, + 359, + 100, + 369 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "without color inversion converges the white color value to 1, so if the multiplying value is in the [0,", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 85, + 370, + 100, + 380 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "1] range, the white color is shifted to the black. By inverting the color of input image, it ignores the", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 379, + 302, + 392 + ], + "spans": [ + { + "bbox": [ + 86, + 381, + 100, + 392 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 105, + 379, + 302, + 392 + ], + "score": 1.0, + "content": "multiplication operations for white value with 0.", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 394, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 86, + 397, + 100, + 407 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 104, + 394, + 217, + 409 + ], + "score": 1.0, + "content": "We generate the map image", + "type": "text" + }, + { + "bbox": [ + 218, + 396, + 231, + 406 + ], + "score": 0.83, + "content": "I ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 394, + 506, + 409 + ], + "score": 1.0, + "content": "based on original RGB input images calculating the difference value", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 85, + 408, + 100, + 419 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 106, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "between the image with an aspect of normal vision and the image with an aspect of deuteranopia vision.", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 86, + 419, + 100, + 429 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "Recent studies have been conducted to augment the information or expanded the models’ perspective", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 85, + 430, + 100, + 441 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "through transformer models[JSZK16, RFB15]. 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For this reason, the input image has color regions of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 86, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 99, + 95 + ], + "score": 1.0, + "content": "183", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "converting color and unconverting color, which also can be referred to as non-CUD object and CUD", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 87, + 95, + 449, + 107 + ], + "spans": [ + { + "bbox": [ + 87, + 97, + 99, + 105 + ], + "score": 1.0, + "content": "184", + "type": "text" + }, + { + "bbox": [ + 106, + 95, + 449, + 107 + ], + "score": 1.0, + "content": "object. To imply the color bound to model, the stencil masking method is introduced.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 118, + 351, + 140 + ], + "lines": [ + { + "bbox": [ + 232, + 118, + 351, + 140 + ], + "spans": [ + { + "bbox": [ + 232, + 118, + 351, + 140 + ], + "score": 0.93, + "content": "\\Phi \\left( { \\hat { I } } _ { i } \\right) = { \\hat { I } } _ { i j } \\parallel ( I _ { i j } \\cdot T _ { i j } )", + "type": "interline_equation", + "image_path": "4b789ddb2c78b08153055c11a351b43a49a73b9ef706ae3b51d1a596f2a2b26a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 232, + 118, + 351, + 140 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 146, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 86, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 86, + 149, + 99, + 158 + ], + "score": 1.0, + "content": "185", + "type": "text" + }, + { + "bbox": [ + 106, + 147, + 439, + 159 + ], + "score": 1.0, + "content": "We consist a stencil maps through the logical and operations ’·’ of each pixel value", + "type": "text" + }, + { + "bbox": [ + 440, + 147, + 452, + 159 + ], + "score": 0.56, + "content": "I _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 147, + 455, + 159 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 456, + 147, + 470, + 159 + ], + "score": 0.52, + "content": "T _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 147, + 505, + 159 + ], + "score": 1.0, + "content": ". Stencil", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 86, + 162, + 100, + 172 + ], + "score": 1.0, + "content": "186", + "type": "text" + }, + { + "bbox": [ + 105, + 160, + 401, + 173 + ], + "score": 1.0, + "content": "map can specify the non-CUD area and be computed with predicted image", + "type": "text" + }, + { + "bbox": [ + 401, + 158, + 414, + 172 + ], + "score": 0.9, + "content": "\\hat { I } _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "of logical or operation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 86, + 172, + 100, + 183 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 105, + 171, + 109, + 185 + ], + "score": 1.0, + "content": "’", + "type": "text" + }, + { + "bbox": [ + 109, + 172, + 116, + 183 + ], + "score": 0.26, + "content": "| |", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "’in equation 5. Consequently, CUD object of the image adjusted with a stencil mask does not carry", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 86, + 180, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 86, + 184, + 100, + 194 + ], + "score": 1.0, + "content": "188", + "type": "text" + }, + { + "bbox": [ + 104, + 180, + 506, + 196 + ], + "score": 1.0, + "content": "out the neural filter computation, such as the same way we refine the target image. This refined image", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 86, + 192, + 242, + 205 + ], + "spans": [ + { + "bbox": [ + 86, + 195, + 100, + 204 + ], + "score": 1.0, + "content": "189", + "type": "text" + }, + { + "bbox": [ + 104, + 192, + 242, + 205 + ], + "score": 1.0, + "content": "is calculated on the loss function.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 87, + 215, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 87, + 214, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 87, + 218, + 99, + 227 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 105, + 214, + 506, + 229 + ], + "score": 1.0, + "content": "CIELab Loss We use the CIELab channel loss function to maximize the contrast of color on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 86, + 230, + 99, + 238 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "deuteranopia vision. To stabilize the contrast and brightness of the predicted image, we calculate the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 238, + 376, + 250 + ], + "spans": [ + { + "bbox": [ + 86, + 240, + 100, + 249 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 105, + 238, + 332, + 250 + ], + "score": 1.0, + "content": "MS· SSIM(multi-scale structural similarity[WSB03]) of", + "type": "text" + }, + { + "bbox": [ + 333, + 239, + 340, + 248 + ], + "score": 0.34, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 238, + 376, + 250 + ], + "score": 1.0, + "content": "channel.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 268, + 461, + 291 + ], + "lines": [ + { + "bbox": [ + 117, + 268, + 461, + 291 + ], + "spans": [ + { + "bbox": [ + 117, + 268, + 461, + 291 + ], + "score": 0.9, + "content": "L a b _ { l o s s } = \\left\\| L a b \\left( \\hat { I } _ { i } ^ { r g b } \\right) - L a b \\left( T _ { i } ^ { r g b } \\right) \\right\\| _ { 1 } + M S \\cdot S S I M \\left( L a b \\left( \\hat { I } _ { i } ^ { L } \\right) , L a b \\left( T _ { i } ^ { L } \\right) \\right)", + "type": "interline_equation", + "image_path": "42432145446d15826e0912ad9a582adecc8b903cb263d306487137489724f86e.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 117, + 268, + 461, + 291 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 301, + 506, + 324 + ], + "lines": [ + { + "bbox": [ + 86, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 86, + 303, + 101, + 313 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 104, + 301, + 124, + 314 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 124, + 302, + 154, + 313 + ], + "score": 0.88, + "content": "L a b \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "expression in equation 6 returns the CIELab channel corresponding to the RGB channel,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 84, + 312, + 317, + 325 + ], + "spans": [ + { + "bbox": [ + 84, + 312, + 317, + 325 + ], + "score": 1.0, + "content": "194 and all calculations are made only on the L channel.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 86, + 335, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 86, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 86, + 338, + 100, + 347 + ], + "score": 1.0, + "content": "195", + "type": "text" + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "Histogram Loss We use the histogram loss function to comply with the color preservation of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 86, + 349, + 100, + 358 + ], + "score": 1.0, + "content": "196", + "type": "text" + }, + { + "bbox": [ + 106, + 347, + 429, + 360 + ], + "score": 1.0, + "content": "image. The RGB channel is used to preserve its color, contrary to using only the", + "type": "text" + }, + { + "bbox": [ + 429, + 347, + 437, + 357 + ], + "score": 0.33, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "channel in other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 86, + 360, + 100, + 369 + ], + "score": 1.0, + "content": "197", + "type": "text" + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "loss functions. Handling the RGB channel as a loss function rather than using Lab’s ab channels has", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 368, + 278, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 100, + 380 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 368, + 278, + 380 + ], + "score": 1.0, + "content": "shown better results on color preservation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 392, + 392, + 418 + ], + "lines": [ + { + "bbox": [ + 186, + 392, + 392, + 418 + ], + "spans": [ + { + "bbox": [ + 186, + 392, + 392, + 418 + ], + "score": 0.92, + "content": "H _ { l o s s } = - { \\omega _ { h i s t } } \\int N \\left( \\hat { I } _ { i } ^ { r g b } ; \\sigma \\right) - { \\cal N } \\left( T _ { i } ^ { r g b } ; \\sigma \\right)", + "type": "interline_equation", + "image_path": "03e0b4da2dd1716cbf0f48130a0e3e9d6b152fda36171fe2a70ef2ee68884792.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 186, + 392, + 392, + 418 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 85, + 424, + 506, + 513 + ], + "lines": [ + { + "bbox": [ + 86, + 425, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 86, + 426, + 100, + 437 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "score": 1.0, + "content": "When simply designing a loss function with the L1 distance of the RGB channel pixel values, it was", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 86, + 438, + 99, + 448 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "very sensitive to certain values and the gradients are diverged, resulting in an untrainable experiment.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 86, + 449, + 99, + 458 + ], + "score": 1.0, + "content": "201", + "type": "text" + }, + { + "bbox": [ + 105, + 446, + 302, + 460 + ], + "score": 1.0, + "content": "Therefore, we used a gaussian expansion method", + "type": "text" + }, + { + "bbox": [ + 303, + 447, + 341, + 458 + ], + "score": 0.78, + "content": "[ S \\mathbf { A } C ^ { * } 1 7 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 446, + 389, + 460 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 389, + 447, + 410, + 459 + ], + "score": 0.9, + "content": "N ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "to infer a differentiable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 85, + 459, + 100, + 470 + ], + "score": 1.0, + "content": "202", + "type": "text" + }, + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "histogram loss function in equation 7. We compute the difference of the RGB channel of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 467, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 86, + 471, + 99, + 480 + ], + "score": 1.0, + "content": "203", + "type": "text" + }, + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "score": 1.0, + "content": "differentiable histogram function, which can be altered to mean squared error or cosine similarity.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 86, + 482, + 100, + 491 + ], + "score": 1.0, + "content": "204", + "type": "text" + }, + { + "bbox": [ + 105, + 478, + 150, + 493 + ], + "score": 1.0, + "content": "The scaler", + "type": "text" + }, + { + "bbox": [ + 150, + 481, + 173, + 491 + ], + "score": 0.88, + "content": "\\omega _ { h i s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 478, + 506, + 493 + ], + "score": 1.0, + "content": "is determined in inverse proportion to the size of the input image. By maintaining", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 85, + 492, + 100, + 502 + ], + "score": 1.0, + "content": "205", + "type": "text" + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "the RGB similarity between the predicted image and the target image, we can comply with the color", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 502, + 159, + 514 + ], + "spans": [ + { + "bbox": [ + 86, + 504, + 100, + 513 + ], + "score": 1.0, + "content": "206", + "type": "text" + }, + { + "bbox": [ + 105, + 502, + 159, + 514 + ], + "score": 1.0, + "content": "preservation.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 85, + 524, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 86, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 86, + 527, + 99, + 536 + ], + "score": 1.0, + "content": "207", + "type": "text" + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "Variational Prediction There are various ways to maximize difference of the L channel in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 534, + 507, + 549 + ], + "spans": [ + { + "bbox": [ + 86, + 537, + 99, + 547 + ], + "score": 1.0, + "content": "208", + "type": "text" + }, + { + "bbox": [ + 105, + 534, + 507, + 549 + ], + "score": 1.0, + "content": "image. And the target image is converted at least two colors compared to the input image. However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 85, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 85, + 548, + 100, + 559 + ], + "score": 1.0, + "content": "209", + "type": "text" + }, + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "the predicted image of the model is generated by the neural filter, so it is unpredictable which area", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 85, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 85, + 559, + 100, + 570 + ], + "score": 1.0, + "content": "210", + "type": "text" + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "of color is modified. Therefore, if the color in predicted image is over-shifted or in the color value", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 85, + 570, + 100, + 581 + ], + "score": 1.0, + "content": "211", + "type": "text" + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "of opposite shifts to the target, the loss will rather increase. In addition to one-to-many problem", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 86, + 581, + 100, + 590 + ], + "score": 1.0, + "content": "212", + "type": "text" + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "that the data pair itself does not matches one-to-one in a particular color, it is necessary to generate", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 86, + 592, + 100, + 601 + ], + "score": 1.0, + "content": "213", + "type": "text" + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "alternative predicted image with the same aspect of the data pair. We calculate the loss function with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 601, + 421, + 613 + ], + "spans": [ + { + "bbox": [ + 86, + 604, + 100, + 612 + ], + "score": 1.0, + "content": "214", + "type": "text" + }, + { + "bbox": [ + 105, + 601, + 421, + 613 + ], + "score": 1.0, + "content": "a variational prediction based on the predicted image for potential color shifts.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 86, + 618, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 86, + 617, + 507, + 632 + ], + "spans": [ + { + "bbox": [ + 86, + 621, + 100, + 630 + ], + "score": 1.0, + "content": "215", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 346, + 632 + ], + "score": 1.0, + "content": "The first potential is the case of excessive shifts. Assume that", + "type": "text" + }, + { + "bbox": [ + 346, + 618, + 415, + 631 + ], + "score": 0.7, + "content": "I _ { i } ^ { L } = \\{ 7 4 , 4 1 , 7 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 617, + 419, + 632 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 419, + 617, + 503, + 631 + ], + "score": 0.82, + "content": "\\hat { I } _ { i } ^ { L } = \\{ 9 7 , 1 0 , 7 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 617, + 507, + 632 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 85, + 626, + 509, + 649 + ], + "spans": [ + { + "bbox": [ + 85, + 632, + 100, + 643 + ], + "score": 1.0, + "content": "216", + "type": "text" + }, + { + "bbox": [ + 106, + 630, + 193, + 643 + ], + "score": 0.9, + "content": "T _ { i } ^ { L } \\ = \\ \\{ 5 0 , 4 1 , 8 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 626, + 509, + 649 + ], + "score": 1.0, + "content": "in L channel value. The first and third components of each image is non-CUD", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 85, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 85, + 643, + 100, + 654 + ], + "score": 1.0, + "content": "217", + "type": "text" + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "objects, and second component is CUD object. Therefore, we refined data paired with a difference of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 85, + 652, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 655, + 100, + 666 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 105, + 652, + 287, + 668 + ], + "score": 1.0, + "content": "15 on L channel. Here, we clip the excessive", + "type": "text" + }, + { + "bbox": [ + 288, + 655, + 295, + 665 + ], + "score": 0.33, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 652, + 364, + 668 + ], + "score": 1.0, + "content": "channel value in", + "type": "text" + }, + { + "bbox": [ + 365, + 653, + 377, + 667 + ], + "score": 0.9, + "content": "\\hat { I } _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 652, + 506, + 668 + ], + "score": 1.0, + "content": "by equation 8. Up to this point,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 665, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 85, + 667, + 100, + 677 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 105, + 665, + 506, + 677 + ], + "score": 1.0, + "content": "no calculation is made as no value is exceeded in this example. The second potential is the case of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 85, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 85, + 677, + 100, + 688 + ], + "score": 1.0, + "content": "220", + "type": "text" + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "opposite shifts. It can be said that a complete neural filter has been proceeded for value 97, 10, 70", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 86, + 689, + 99, + 699 + ], + "score": 1.0, + "content": "221", + "type": "text" + }, + { + "bbox": [ + 105, + 687, + 132, + 700 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 688, + 141, + 698 + ], + "score": 0.29, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "channel difference is 27. However, if we actually calculate the mean square error between", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 85, + 696, + 507, + 714 + ], + "spans": [ + { + "bbox": [ + 85, + 701, + 100, + 712 + ], + "score": 1.0, + "content": "222", + "type": "text" + }, + { + "bbox": [ + 106, + 698, + 119, + 712 + ], + "score": 0.89, + "content": "\\hat { I } _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 696, + 137, + 714 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 699, + 151, + 712 + ], + "score": 0.9, + "content": "T _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 696, + 507, + 714 + ], + "score": 1.0, + "content": ", it will be an large value over 1k. Here we can generate alternative predicted image from", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 711, + 182, + 722 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "223", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 182, + 722 + ], + "score": 1.0, + "content": "equation 9 and 10.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 87, + 72, + 505, + 106 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 86, + 72, + 506, + 107 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 118, + 351, + 140 + ], + "lines": [ + { + "bbox": [ + 232, + 118, + 351, + 140 + ], + "spans": [ + { + "bbox": [ + 232, + 118, + 351, + 140 + ], + "score": 0.93, + "content": "\\Phi \\left( { \\hat { I } } _ { i } \\right) = { \\hat { I } } _ { i j } \\parallel ( I _ { i j } \\cdot T _ { i j } )", + "type": "interline_equation", + "image_path": "4b789ddb2c78b08153055c11a351b43a49a73b9ef706ae3b51d1a596f2a2b26a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 232, + 118, + 351, + 140 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "index", + "bbox": [ + 87, + 146, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 86, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 86, + 149, + 99, + 158 + ], + "score": 1.0, + "content": "185", + "type": "text" + }, + { + "bbox": [ + 106, + 147, + 439, + 159 + ], + "score": 1.0, + "content": "We consist a stencil maps through the logical and operations ’·’ of each pixel value", + "type": "text" + }, + { + "bbox": [ + 440, + 147, + 452, + 159 + ], + "score": 0.56, + "content": "I _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 147, + 455, + 159 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 456, + 147, + 470, + 159 + ], + "score": 0.52, + "content": "T _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 147, + 505, + 159 + ], + "score": 1.0, + "content": ". Stencil", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 86, + 162, + 100, + 172 + ], + "score": 1.0, + "content": "186", + "type": "text" + }, + { + "bbox": [ + 105, + 160, + 401, + 173 + ], + "score": 1.0, + "content": "map can specify the non-CUD area and be computed with predicted image", + "type": "text" + }, + { + "bbox": [ + 401, + 158, + 414, + 172 + ], + "score": 0.9, + "content": "\\hat { I } _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "of logical or operation", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 171, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 86, + 172, + 100, + 183 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 105, + 171, + 109, + 185 + ], + "score": 1.0, + "content": "’", + "type": "text" + }, + { + "bbox": [ + 109, + 172, + 116, + 183 + ], + "score": 0.26, + "content": "| |", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 171, + 505, + 185 + ], + "score": 1.0, + "content": "’in equation 5. Consequently, CUD object of the image adjusted with a stencil mask does not carry", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 180, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 86, + 184, + 100, + 194 + ], + "score": 1.0, + "content": "188", + "type": "text" + }, + { + "bbox": [ + 104, + 180, + 506, + 196 + ], + "score": 1.0, + "content": "out the neural filter computation, such as the same way we refine the target image. This refined image", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 192, + 242, + 205 + ], + "spans": [ + { + "bbox": [ + 86, + 195, + 100, + 204 + ], + "score": 1.0, + "content": "189", + "type": "text" + }, + { + "bbox": [ + 104, + 192, + 242, + 205 + ], + "score": 1.0, + "content": "is calculated on the loss function.", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 214, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 87, + 218, + 99, + 227 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 105, + 214, + 506, + 229 + ], + "score": 1.0, + "content": "CIELab Loss We use the CIELab channel loss function to maximize the contrast of color on", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 86, + 230, + 99, + 238 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "deuteranopia vision. To stabilize the contrast and brightness of the predicted image, we calculate the", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 238, + 376, + 250 + ], + "spans": [ + { + "bbox": [ + 86, + 240, + 100, + 249 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 105, + 238, + 332, + 250 + ], + "score": 1.0, + "content": "MS· SSIM(multi-scale structural similarity[WSB03]) of", + "type": "text" + }, + { + "bbox": [ + 333, + 239, + 340, + 248 + ], + "score": 0.34, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 238, + 376, + 250 + ], + "score": 1.0, + "content": "channel.", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + } + ], + "index": 6, + "bbox_fs": [ + 86, + 147, + 506, + 205 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 215, + 505, + 250 + ], + "lines": [], + "index": 10, + "bbox_fs": [ + 86, + 214, + 506, + 250 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 268, + 461, + 291 + ], + "lines": [ + { + "bbox": [ + 117, + 268, + 461, + 291 + ], + "spans": [ + { + "bbox": [ + 117, + 268, + 461, + 291 + ], + "score": 0.9, + "content": "L a b _ { l o s s } = \\left\\| L a b \\left( \\hat { I } _ { i } ^ { r g b } \\right) - L a b \\left( T _ { i } ^ { r g b } \\right) \\right\\| _ { 1 } + M S \\cdot S S I M \\left( L a b \\left( \\hat { I } _ { i } ^ { L } \\right) , L a b \\left( T _ { i } ^ { L } \\right) \\right)", + "type": "interline_equation", + "image_path": "42432145446d15826e0912ad9a582adecc8b903cb263d306487137489724f86e.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 117, + 268, + 461, + 291 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 301, + 506, + 324 + ], + "lines": [ + { + "bbox": [ + 86, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 86, + 303, + 101, + 313 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 104, + 301, + 124, + 314 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 124, + 302, + 154, + 313 + ], + "score": 0.88, + "content": "L a b \\left( . \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "expression in equation 6 returns the CIELab channel corresponding to the RGB channel,", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 312, + 317, + 325 + ], + "spans": [ + { + "bbox": [ + 84, + 312, + 317, + 325 + ], + "score": 1.0, + "content": "194 and all calculations are made only on the L channel.", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 86, + 338, + 100, + 347 + ], + "score": 1.0, + "content": "195", + "type": "text" + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "Histogram Loss We use the histogram loss function to comply with the color preservation of the", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 86, + 349, + 100, + 358 + ], + "score": 1.0, + "content": "196", + "type": "text" + }, + { + "bbox": [ + 106, + 347, + 429, + 360 + ], + "score": 1.0, + "content": "image. The RGB channel is used to preserve its color, contrary to using only the", + "type": "text" + }, + { + "bbox": [ + 429, + 347, + 437, + 357 + ], + "score": 0.33, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "channel in other", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 86, + 360, + 100, + 369 + ], + "score": 1.0, + "content": "197", + "type": "text" + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "loss functions. Handling the RGB channel as a loss function rather than using Lab’s ab channels has", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 368, + 278, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 100, + 380 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 368, + 278, + 380 + ], + "score": 1.0, + "content": "shown better results on color preservation.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + } + ], + "index": 13.5, + "bbox_fs": [ + 84, + 301, + 506, + 325 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 335, + 505, + 380 + ], + "lines": [], + "index": 16.5, + "bbox_fs": [ + 86, + 335, + 505, + 380 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 392, + 392, + 418 + ], + "lines": [ + { + "bbox": [ + 186, + 392, + 392, + 418 + ], + "spans": [ + { + "bbox": [ + 186, + 392, + 392, + 418 + ], + "score": 0.92, + "content": "H _ { l o s s } = - { \\omega _ { h i s t } } \\int N \\left( \\hat { I } _ { i } ^ { r g b } ; \\sigma \\right) - { \\cal N } \\left( T _ { i } ^ { r g b } ; \\sigma \\right)", + "type": "interline_equation", + "image_path": "03e0b4da2dd1716cbf0f48130a0e3e9d6b152fda36171fe2a70ef2ee68884792.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 186, + 392, + 392, + 418 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "index", + "bbox": [ + 85, + 424, + 506, + 513 + ], + "lines": [ + { + "bbox": [ + 86, + 425, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 86, + 426, + 100, + 437 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "score": 1.0, + "content": "When simply designing a loss function with the L1 distance of the RGB channel pixel values, it was", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 86, + 438, + 99, + 448 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 105, + 436, + 506, + 448 + ], + "score": 1.0, + "content": "very sensitive to certain values and the gradients are diverged, resulting in an untrainable experiment.", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 86, + 449, + 99, + 458 + ], + "score": 1.0, + "content": "201", + "type": "text" + }, + { + "bbox": [ + 105, + 446, + 302, + 460 + ], + "score": 1.0, + "content": "Therefore, we used a gaussian expansion method", + "type": "text" + }, + { + "bbox": [ + 303, + 447, + 341, + 458 + ], + "score": 0.78, + "content": "[ S \\mathbf { A } C ^ { * } 1 7 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 446, + 389, + 460 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 389, + 447, + 410, + 459 + ], + "score": 0.9, + "content": "N ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "to infer a differentiable", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 85, + 459, + 100, + 470 + ], + "score": 1.0, + "content": "202", + "type": "text" + }, + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "histogram loss function in equation 7. We compute the difference of the RGB channel of the", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 467, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 86, + 471, + 99, + 480 + ], + "score": 1.0, + "content": "203", + "type": "text" + }, + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "score": 1.0, + "content": "differentiable histogram function, which can be altered to mean squared error or cosine similarity.", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 86, + 482, + 100, + 491 + ], + "score": 1.0, + "content": "204", + "type": "text" + }, + { + "bbox": [ + 105, + 478, + 150, + 493 + ], + "score": 1.0, + "content": "The scaler", + "type": "text" + }, + { + "bbox": [ + 150, + 481, + 173, + 491 + ], + "score": 0.88, + "content": "\\omega _ { h i s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 478, + 506, + 493 + ], + "score": 1.0, + "content": "is determined in inverse proportion to the size of the input image. By maintaining", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 85, + 492, + 100, + 502 + ], + "score": 1.0, + "content": "205", + "type": "text" + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "the RGB similarity between the predicted image and the target image, we can comply with the color", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 502, + 159, + 514 + ], + "spans": [ + { + "bbox": [ + 86, + 504, + 100, + 513 + ], + "score": 1.0, + "content": "206", + "type": "text" + }, + { + "bbox": [ + 105, + 502, + 159, + 514 + ], + "score": 1.0, + "content": "preservation.", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 86, + 527, + 99, + 536 + ], + "score": 1.0, + "content": "207", + "type": "text" + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "Variational Prediction There are various ways to maximize difference of the L channel in the", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 534, + 507, + 549 + ], + "spans": [ + { + "bbox": [ + 86, + 537, + 99, + 547 + ], + "score": 1.0, + "content": "208", + "type": "text" + }, + { + "bbox": [ + 105, + 534, + 507, + 549 + ], + "score": 1.0, + "content": "image. And the target image is converted at least two colors compared to the input image. However,", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 85, + 548, + 100, + 559 + ], + "score": 1.0, + "content": "209", + "type": "text" + }, + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "the predicted image of the model is generated by the neural filter, so it is unpredictable which area", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 85, + 559, + 100, + 570 + ], + "score": 1.0, + "content": "210", + "type": "text" + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "of color is modified. Therefore, if the color in predicted image is over-shifted or in the color value", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 85, + 570, + 100, + 581 + ], + "score": 1.0, + "content": "211", + "type": "text" + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "of opposite shifts to the target, the loss will rather increase. In addition to one-to-many problem", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 86, + 581, + 100, + 590 + ], + "score": 1.0, + "content": "212", + "type": "text" + }, + { + "bbox": [ + 106, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "that the data pair itself does not matches one-to-one in a particular color, it is necessary to generate", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 86, + 592, + 100, + 601 + ], + "score": 1.0, + "content": "213", + "type": "text" + }, + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "alternative predicted image with the same aspect of the data pair. We calculate the loss function with", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 601, + 421, + 613 + ], + "spans": [ + { + "bbox": [ + 86, + 604, + 100, + 612 + ], + "score": 1.0, + "content": "214", + "type": "text" + }, + { + "bbox": [ + 105, + 601, + 421, + 613 + ], + "score": 1.0, + "content": "a variational prediction based on the predicted image for potential color shifts.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 617, + 507, + 632 + ], + "spans": [ + { + "bbox": [ + 86, + 621, + 100, + 630 + ], + "score": 1.0, + "content": "215", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 346, + 632 + ], + "score": 1.0, + "content": "The first potential is the case of excessive shifts. Assume that", + "type": "text" + }, + { + "bbox": [ + 346, + 618, + 415, + 631 + ], + "score": 0.7, + "content": "I _ { i } ^ { L } = \\{ 7 4 , 4 1 , 7 9 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 617, + 419, + 632 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 419, + 617, + 503, + 631 + ], + "score": 0.82, + "content": "\\hat { I } _ { i } ^ { L } = \\{ 9 7 , 1 0 , 7 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 617, + 507, + 632 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 626, + 509, + 649 + ], + "spans": [ + { + "bbox": [ + 85, + 632, + 100, + 643 + ], + "score": 1.0, + "content": "216", + "type": "text" + }, + { + "bbox": [ + 106, + 630, + 193, + 643 + ], + "score": 0.9, + "content": "T _ { i } ^ { L } \\ = \\ \\{ 5 0 , 4 1 , 8 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 626, + 509, + 649 + ], + "score": 1.0, + "content": "in L channel value. The first and third components of each image is non-CUD", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 85, + 643, + 100, + 654 + ], + "score": 1.0, + "content": "217", + "type": "text" + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "objects, and second component is CUD object. Therefore, we refined data paired with a difference of", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 652, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 655, + 100, + 666 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 105, + 652, + 287, + 668 + ], + "score": 1.0, + "content": "15 on L channel. Here, we clip the excessive", + "type": "text" + }, + { + "bbox": [ + 288, + 655, + 295, + 665 + ], + "score": 0.33, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 652, + 364, + 668 + ], + "score": 1.0, + "content": "channel value in", + "type": "text" + }, + { + "bbox": [ + 365, + 653, + 377, + 667 + ], + "score": 0.9, + "content": "\\hat { I } _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 652, + 506, + 668 + ], + "score": 1.0, + "content": "by equation 8. Up to this point,", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 665, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 85, + 667, + 100, + 677 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 105, + 665, + 506, + 677 + ], + "score": 1.0, + "content": "no calculation is made as no value is exceeded in this example. The second potential is the case of", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 85, + 677, + 100, + 688 + ], + "score": 1.0, + "content": "220", + "type": "text" + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "opposite shifts. It can be said that a complete neural filter has been proceeded for value 97, 10, 70", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 86, + 689, + 99, + 699 + ], + "score": 1.0, + "content": "221", + "type": "text" + }, + { + "bbox": [ + 105, + 687, + 132, + 700 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 688, + 141, + 698 + ], + "score": 0.29, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "channel difference is 27. However, if we actually calculate the mean square error between", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 696, + 507, + 714 + ], + "spans": [ + { + "bbox": [ + 85, + 701, + 100, + 712 + ], + "score": 1.0, + "content": "222", + "type": "text" + }, + { + "bbox": [ + 106, + 698, + 119, + 712 + ], + "score": 0.89, + "content": "\\hat { I } _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 696, + 137, + 714 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 699, + 151, + 712 + ], + "score": 0.9, + "content": "T _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 696, + 507, + 714 + ], + "score": 1.0, + "content": ", it will be an large value over 1k. Here we can generate alternative predicted image from", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 182, + 722 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "223", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 182, + 722 + ], + "score": 1.0, + "content": "equation 9 and 10.", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + } + ], + "index": 23.5, + "bbox_fs": [ + 85, + 425, + 506, + 514 + ] + }, + { + "type": "index", + "bbox": [ + 85, + 524, + 505, + 613 + ], + "lines": [], + "index": 31.5, + "bbox_fs": [ + 85, + 523, + 507, + 613 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 618, + 506, + 723 + ], + "lines": [], + "index": 40, + "bbox_fs": [ + 85, + 617, + 509, + 722 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 184, + 83, + 397, + 121 + ], + "lines": [ + { + "bbox": [ + 184, + 83, + 397, + 121 + ], + "spans": [ + { + "bbox": [ + 184, + 83, + 397, + 121 + ], + "score": 0.9, + "content": "c l i p \\left( \\hat { I } _ { i j } \\right) \\ = \\ \\left\\{ { m a x } { \\left( \\hat { I } _ { i j } , { T } _ { i j } \\right) } , \\qquad I _ { i j } \\ > T _ { i j } \\right.", + "type": "interline_equation", + "image_path": "689434e91b4dde1ece15a75047f8b53515cfedbab3ace44521bfc1117625af75.jpg" + } + ] + } + ], + "index": 0.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 83, + 397, + 102.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 184, + 102.0, + 397, + 121.0 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 133, + 350, + 149 + ], + "lines": [ + { + "bbox": [ + 233, + 133, + 350, + 149 + ], + "spans": [ + { + "bbox": [ + 233, + 133, + 350, + 149 + ], + "score": 0.89, + "content": "R _ { 1 } = \\ 2 I _ { i j } - \\ \\hat { I } _ { i j } , R _ { 2 } = \\ \\hat { I } _ { i j }", + "type": "interline_equation", + "image_path": "9b2597f442abd0c904edd84c873c006247936ef70233e08802bcb5c81bd995f5.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 233, + 133, + 350, + 149 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 166, + 382, + 188 + ], + "lines": [ + { + "bbox": [ + 201, + 166, + 382, + 188 + ], + "spans": [ + { + "bbox": [ + 201, + 166, + 382, + 188 + ], + "score": 0.93, + "content": "V \\left( { \\hat { I } } _ { i j } \\right) = a r g m i n \\left( \\left\\| c l i p \\left( R _ { 1 , 2 } \\right) - T _ { i j } \\right\\| _ { 2 } \\right)", + "type": "interline_equation", + "image_path": "c3a32688718fc2498034dc1d8c73a0871eedd77b075fe8697acaf10411eb48b5.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 201, + 166, + 382, + 188 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 196, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 86, + 195, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 100, + 209 + ], + "score": 1.0, + "content": "224", + "type": "text" + }, + { + "bbox": [ + 103, + 195, + 506, + 210 + ], + "score": 1.0, + "content": "As mentioned above, we define thresholds by the maximum and minimum value of each corresponding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 208, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 86, + 210, + 100, + 221 + ], + "score": 1.0, + "content": "225", + "type": "text" + }, + { + "bbox": [ + 104, + 208, + 176, + 223 + ], + "score": 1.0, + "content": "pixel position of", + "type": "text" + }, + { + "bbox": [ + 176, + 208, + 189, + 222 + ], + "score": 0.9, + "content": "\\hat { I } _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 208, + 207, + 223 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 209, + 221, + 222 + ], + "score": 0.9, + "content": "T _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 208, + 506, + 223 + ], + "score": 1.0, + "content": ". 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The above equation establishes", + "type": "text" + }, + { + "bbox": [ + 265, + 244, + 375, + 264 + ], + "score": 0.94, + "content": "V \\left( \\Phi \\left( \\hat { I } _ { i } \\right) \\right) = \\{ \\stackrel { \\cdot } { 5 4 } , 4 1 , \\stackrel { \\cdot } { 8 0 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 245, + 507, + 262 + ], + "score": 1.0, + "content": "and the mean square error to the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 263, + 435, + 277 + ], + "spans": [ + { + "bbox": [ + 86, + 267, + 100, + 277 + ], + "score": 1.0, + "content": "229", + "type": "text" + }, + { + "bbox": [ + 105, + 264, + 397, + 277 + ], + "score": 1.0, + "content": "target image is approximately 5, which is agreeable loss value respect to", + "type": "text" + }, + { + "bbox": [ + 397, + 263, + 409, + 277 + ], + "score": 0.91, + "content": "\\hat { I } _ { i } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 264, + 435, + 277 + ], + "score": 1.0, + "content": "itself.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 86, + 289, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 86, + 288, + 507, + 302 + ], + "spans": [ + { + "bbox": [ + 86, + 291, + 100, + 301 + ], + "score": 1.0, + "content": "230", + "type": "text" + }, + { + "bbox": [ + 104, + 288, + 280, + 302 + ], + "score": 1.0, + "content": "Identity Loss[ZPIE17, TPW16] We use", + "type": "text" + }, + { + "bbox": [ + 280, + 289, + 335, + 302 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { i d e n t i t y } \\left( T _ { i } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 288, + 507, + 302 + ], + "score": 1.0, + "content": "to apprehend the CUD object to the model.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 86, + 303, + 99, + 311 + ], + "score": 1.0, + "content": "231", + "type": "text" + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "In the case of target image that already satisfy the CUD, the filter should be relatively weakly applied", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 86, + 313, + 100, + 323 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 104, + 310, + 347, + 324 + ], + "score": 1.0, + "content": "than input image. 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We refined a dataset with Adobe Photoshop to maximize", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 86, + 421, + 100, + 431 + ], + "score": 1.0, + "content": "239", + "type": "text" + }, + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "contrast in the L channel by adjusting saturation and brightness for areas that require color conversion", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 86, + 432, + 100, + 442 + ], + "score": 1.0, + "content": "240", + "type": "text" + }, + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "based on deuteranopia vision simulation. Color experts has refined about 1,500 vectorized image for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 86, + 442, + 99, + 453 + ], + "score": 1.0, + "content": "241", + "type": "text" + }, + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "the training data and 300 publication images for the validation data. All the comparative experimental", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 86, + 454, + 100, + 463 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 104, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "models used the same train, test, validation data in this paper. 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The Figure 4 is arranged in descending order of the number of combinations in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 483, + 215, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 487, + 99, + 496 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 105, + 483, + 215, + 498 + ], + "score": 1.0, + "content": "colors from the top image.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 85, + 500, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 86, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 86, + 503, + 99, + 512 + ], + "score": 1.0, + "content": "246", + "type": "text" + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Both structure similarity (SSIM)[ZBSS04] and peak signal to noise ratio (PSNR) in Table 1 can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 86, + 514, + 99, + 523 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "indicate whether the image is suitable for CUD or not. As a notable aspect, the result has shown that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 86, + 524, + 100, + 534 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "comparative models with lower metrics are sensitive to high-gradation input images, which generated", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 86, + 536, + 99, + 545 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "color-heterogeneous image. SSIM and PSNR itself can determine the increase in contrast compared", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 86, + 546, + 100, + 557 + ], + "score": 1.0, + "content": "250", + "type": "text" + }, + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "to the target image but do not determine whether the color preservation complied. 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The higher the estimation of the", + "type": "text" + }, + { + "bbox": [ + 290, + 568, + 297, + 580 + ], + "score": 0.83, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 570, + 316, + 582 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 316, + 570, + 323, + 580 + ], + "score": 0.74, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 570, + 506, + 582 + ], + "score": 1.0, + "content": ", the more color preservation factor worked.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 86, + 585, + 99, + 594 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 106, + 583, + 237, + 594 + ], + "score": 1.0, + "content": "The higher the estimation of the", + "type": "text" + }, + { + "bbox": [ + 237, + 581, + 244, + 592 + ], + "score": 0.82, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 583, + 262, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 582, + 271, + 592 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 583, + 506, + 594 + ], + "score": 1.0, + "content": ", the more increase in contrast can be considered. We also", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 593, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 86, + 596, + 99, + 604 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 593, + 506, + 605 + ], + "score": 1.0, + "content": "define SSIM mean absolute error, PSNR mean absolute error to measure the extent of the conversion", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 86, + 606, + 100, + 616 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 104, + 603, + 157, + 617 + ], + "score": 1.0, + "content": "between the", + "type": "text" + }, + { + "bbox": [ + 157, + 604, + 203, + 614 + ], + "score": 0.92, + "content": "F : I T", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 603, + 221, + 617 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 221, + 603, + 267, + 614 + ], + "score": 0.93, + "content": "F : \\dot { I } \\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 603, + 431, + 617 + ], + "score": 1.0, + "content": "in equation 11 and 12, respectively. 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The Figure 4 is arranged in descending order of the number of combinations in", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 483, + 215, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 487, + 99, + 496 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 105, + 483, + 215, + 498 + ], + "score": 1.0, + "content": "colors from the top image.", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 86, + 503, + 99, + 512 + ], + "score": 1.0, + "content": "246", + "type": "text" + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Both structure similarity (SSIM)[ZBSS04] and peak signal to noise ratio (PSNR) in Table 1 can", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 86, + 514, + 99, + 523 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "indicate whether the image is suitable for CUD or not. As a notable aspect, the result has shown that", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 86, + 524, + 100, + 534 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "comparative models with lower metrics are sensitive to high-gradation input images, which generated", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 86, + 536, + 99, + 545 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "color-heterogeneous image. 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ArchitecturesSIM(i,1)SSIM(i,T)PSNR(i,1)PSNR(i,T)SSIM-MAEPSNR·MAE
Cycle-GANZero-DCE0.6300.63413.2813.830.31918.4430
0.9240.88821.9518.670.06613.7300
DeepLPFEnlighten-GAN0.8500.83126.3120.340.12202.0566
0.8200.80821.8519.580.14703.9983
Enlighten-GAN(scaled)0.9660.92124.8621.360.03923.4937
CUD-NET(low bottle-neck feature)CUD-NET0.8970.86627.7721.010.09012.0826
0.9620.92429.5421.190.03121.4760
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ArchitecturesSIM(i,1)SSIM(i,T)PSNR(i,1)PSNR(i,T)SSIM-MAEPSNR·MAE
Cycle-GANZero-DCE0.6300.63413.2813.830.31918.4430
0.9240.88821.9518.670.06613.7300
DeepLPFEnlighten-GAN0.8500.83126.3120.340.12202.0566
0.8200.80821.8519.580.14703.9983
Enlighten-GAN(scaled)0.9660.92124.8621.360.03923.4937
CUD-NET(low bottle-neck feature)CUD-NET0.8970.86627.7721.010.09012.0826
0.9620.92429.5421.190.03121.4760
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The CUD-NET with a low bottle-neck feature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "score": 1.0, + "content": "achieves better results in the experiment of the deuteranopia and the protanopia subjects(Figure 5),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 508, + 369, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 369, + 519 + ], + "score": 1.0, + "content": "although the evaluation metrics are lower than that of CUD-NET.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "index", + "bbox": [ + 86, + 546, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 86, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 86, + 549, + 99, + 558 + ], + "score": 1.0, + "content": "257", + "type": "text" + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "Cycle-GAN and Zero-DCE showed worse result than others. Cycle-GAN model had difficulty", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 86, + 560, + 99, + 569 + ], + "score": 1.0, + "content": "258", + "type": "text" + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "reconstructing a geometry of a particular object, and overall color had low saturation and brightness,", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 86, + 571, + 100, + 580 + ], + "score": 1.0, + "content": "259", + "type": "text" + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "resulting in color conversion into an almost grey scale image. Zero-DCE is faded in color, and the", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 579, + 507, + 593 + ], + "spans": [ + { + "bbox": [ + 85, + 581, + 100, + 592 + ], + "score": 1.0, + "content": "260", + "type": "text" + }, + { + "bbox": [ + 105, + 579, + 507, + 593 + ], + "score": 1.0, + "content": "contrast was not much different from the input image. The overall image lost its color preservation,", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 590, + 268, + 604 + ], + "spans": [ + { + "bbox": [ + 86, + 593, + 99, + 602 + ], + "score": 1.0, + "content": "261", + "type": "text" + }, + { + "bbox": [ + 105, + 590, + 268, + 604 + ], + "score": 1.0, + "content": "which we focused to solve in this paper.", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 86, + 609, + 100, + 618 + ], + "score": 1.0, + "content": "262", + "type": "text" + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "DeepLPF meets both the color preservation and contrast that we deal with for. However, DeepLPF", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 86, + 620, + 100, + 630 + ], + "score": 1.0, + "content": "263", + "type": "text" + }, + { + "bbox": [ + 106, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "tends to color be over-stably filtered for the images with fewer color combinations. Although the", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 627, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 86, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "264", + "type": "text" + }, + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "score": 1.0, + "content": "color preservation has complied better than other experiments, there were many failed results from", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 640, + 413, + 652 + ], + "spans": [ + { + "bbox": [ + 86, + 641, + 100, + 651 + ], + "score": 1.0, + "content": "265", + "type": "text" + }, + { + "bbox": [ + 105, + 640, + 413, + 652 + ], + "score": 1.0, + "content": "the perspective of contrast, which the over-stable filter leads to by DeepLPF.", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 658, + 100, + 668 + ], + "score": 1.0, + "content": "266", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Remarkably, predicted images of Enlighten-GAN showed reasonable results. However, simple color", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 85, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "267", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "combinations or the images with already satisfying the CUD often showed results degenerated with", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 86, + 680, + 99, + 689 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "low CUD suitability. Enlighten-GAN was able to generate the results we targeted, but its deviation of", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 85, + 690, + 100, + 700 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "filter is so high that it sometimes failed to satisfy the contrast even on simple images or decreased", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 85, + 701, + 100, + 712 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "the contrast. As the problem of GAN-based method including Enlighten-GAN, moreover, model", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 85, + 712, + 100, + 722 + ], + "score": 1.0, + "content": "271", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 414, + 723 + ], + "score": 1.0, + "content": "fixes the width and height of the predicted image. 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When we reduced the number of bottle-neck feature of model, it tends to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "record relatively high deviation of filter scales according to the number of combinations of colors. In", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 203 + ], + "score": 1.0, + "content": "summary, the CUD-NET showed the highest estimations for 4 evaluation metrics. 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The y position of box bar represents a mean and length of the box bar represents a deviation of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 392, + 361, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 361, + 404 + ], + "score": 1.0, + "content": "each experiment. The lower the graph is, the higher the rank is.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 85, + 421, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 86, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 86, + 424, + 99, + 433 + ], + "score": 1.0, + "content": "284", + "type": "text" + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "The figure 5 shows the evaluation of the deuteranopia and the protanopia. The evaluation metrics", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 86, + 435, + 100, + 444 + ], + "score": 1.0, + "content": "285", + "type": "text" + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "consist of object distinguishability and color harmony in order of input image I, predicted image", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 442, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 86, + 446, + 100, + 455 + ], + "score": 1.0, + "content": "286", + "type": "text" + }, + { + "bbox": [ + 105, + 442, + 506, + 457 + ], + "score": 1.0, + "content": "of Enlighten GAN, DeepLPF, and CUD-NET. User study has tested upon the total of 6 subjects, 4", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 85, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 85, + 456, + 100, + 467 + ], + "score": 1.0, + "content": "287", + "type": "text" + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "deuteranomaly and 2 protanomaly. The subjects were asked to list the ranks of object distinguishability", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 465, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 86, + 468, + 99, + 477 + ], + "score": 1.0, + "content": "288", + "type": "text" + }, + { + "bbox": [ + 105, + 465, + 507, + 478 + ], + "score": 1.0, + "content": "and color harmony of 4-paired-image for each model-blinded item. As the experimental results,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 474, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 100, + 488 + ], + "score": 1.0, + "content": "289", + "type": "text" + }, + { + "bbox": [ + 104, + 474, + 506, + 491 + ], + "score": 1.0, + "content": "the deuteranopia subject ranked the 1-st in the object distinguishability of CUD-NET at an average", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 486, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 85, + 489, + 100, + 500 + ], + "score": 1.0, + "content": "290", + "type": "text" + }, + { + "bbox": [ + 105, + 486, + 506, + 501 + ], + "score": 1.0, + "content": "rank of 1.821, followed by Enlighten-GAN at an average rank of 2.512. Similarly, the protanopia", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 86, + 501, + 99, + 510 + ], + "score": 1.0, + "content": "291", + "type": "text" + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "subject also ranked the 1-st in CUD, followed by Enlighten-GAN, DeepLPF, and input images. 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The evaluation metrics", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 86, + 435, + 100, + 444 + ], + "score": 1.0, + "content": "285", + "type": "text" + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "consist of object distinguishability and color harmony in order of input image I, predicted image", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 442, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 86, + 446, + 100, + 455 + ], + "score": 1.0, + "content": "286", + "type": "text" + }, + { + "bbox": [ + 105, + 442, + 506, + 457 + ], + "score": 1.0, + "content": "of Enlighten GAN, DeepLPF, and CUD-NET. 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ArchitecturesSIM(i,1)SSIM(i,T)PSNR(i,1)PSNR(i,T)SSIM-MAEPSNR·MAE
Cycle-GANZero-DCE0.6300.63413.2813.830.31918.4430
0.9240.88821.9518.670.06613.7300
DeepLPFEnlighten-GAN0.8500.83126.3120.340.12202.0566
0.8200.80821.8519.580.14703.9983
Enlighten-GAN(scaled)0.9660.92124.8621.360.03923.4937
CUD-NET(low bottle-neck feature)CUD-NET0.8970.86627.7721.010.09012.0826
0.9620.92429.5421.190.03121.4760
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sha256:61600f45eccfbf85c91a76cf3a249b3863a8b6a3c50ed7dce79e4cf0627cb1f3 +size 538829 diff --git a/parse/train/Bkxonh4Ywr/Bkxonh4Ywr_span.pdf b/parse/train/Bkxonh4Ywr/Bkxonh4Ywr_span.pdf new file mode 100644 index 0000000000000000000000000000000000000000..649f9b55fa6e1d96d120cc312435953f7463900c --- /dev/null +++ b/parse/train/Bkxonh4Ywr/Bkxonh4Ywr_span.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:918962bce7b95a8a729f695509dcc4abb60111e5d07f8bf4682c1571f33f19af +size 722640 diff --git a/parse/train/BkxthxHYvr/BkxthxHYvr.md b/parse/train/BkxthxHYvr/BkxthxHYvr.md new file mode 100644 index 0000000000000000000000000000000000000000..17241a2c7299d2dac3a3de34070321526d32778a --- /dev/null +++ b/parse/train/BkxthxHYvr/BkxthxHYvr.md @@ -0,0 +1,368 @@ +# CONDITIONAL GENERATION OF MOLECULES FROM DISENTANGLED REPRESENTATIONS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Though machine learning approaches have shown great success in estimating properties of small molecules, the inverse problem of generating molecules with desired properties remains challenging. This difficulty is in part because the set of molecules which have a given property is structurally very diverse. Treating this inverse problem as a conditional distribution estimation task, we draw upon work in learning disentangled representations to learn a conditional distribution over molecules given a desired property, where the molecular structure is encoded in a continuous latent random variable. By including property information as an input factor independent from the structure representation, one can perform conditional molecule generation via a “style transfer” process, in which we explicitly set the property to a desired value at generation time. In contrast to existing approaches, we disentangle the latent factors from the property factors using a regularization term which constrains the generated molecules to have the property provided to the generation network, no matter how the latent factor changes. + +# 1 INTRODUCTION + +Conditional molecule generation is far from being solved. The main challenge is the enormous and discrete nature of the molecules space and the fact that molecule properties are highly sensitive to molecular structure (Kirkpatrick & Ellis, 2004). Approaches to conditional generation are typically two-step, either using a model or genetic algorithm to generate candidates which are later filtered, or learning a continuous embedding of the discrete molecules and optimizing in a real-valued representation space. The former is computationally expensive, the latter performs conditional generation only very obliquely. + +We propose a conditional generative model that produces candidate molecules which targeting a desired property in a single step. This approach builds on work in structured deep generative models (Kingma et al., 2014; Siddharth et al., 2017), which aim to learn a disentangled representation that factors into observed properties we want to control for, and latent factors that account for the remaining features which are either hard to annotate or irrelevant to the properties we wish to optimize. + +We derive a regularizer for supervised variational autoencoders which exploits property information that we provide as supervision, ensuring that produced molecules adhere to target properties they are conditioned on. We demonstrate the ability of our model to perform accurate conditional molecule generation and a sort of “style transfer” on molecules, where a latent representation for a single molecule can have its target properties perturbed independently of its learnt structural characteristics, allowing direct and efficient generation of candidates for local optimization of molecules. + +# 2 BACKGROUND + +Molecule discovery tasks come in two flavors. Global optimization seeks to find molecules that have a particular target property. Local optimization starts from some initial molecule and searches for molecules which have a desired property while not straying too far from the prototype. There is some overlap in methods used in the two approaches. + +# 2.1 DEEP GENERATIVE MODELS FOR MOLECULES + +Virtual screening methods start from a large database of possible molecules and retain the promising ones (Eckert & Bajorath, 2007), as measured by some quality function $f ( \cdot )$ . Machine learning approaches expand on this by dynamically generating additional candidate molecules; Segler et al. (2017) uses a stacked LSTM to produce large numbers of novel molecules which have similar characteristics to an existing database. + +For properties which are expensive to evaluate, generating large sets of candidate molecules is not particularly useful. More sample-efficient global search can be achieved using Bayesian optimization methods, which use a generative model with a latent space that functions as a continuous representation of molecules (Gomez-Bombarelli et al., 2016; Kusner et al., 2017). Optimization is then ´ carried out over this continuous representation space to find candidates which are expected to have the desired property. Local gradient-based search can also be applied on continuous latent spaces to optimize the latent representation with respect to a target property (Jin et al., 2018; Liu et al., 2018). + +A challenge for these latent variable models is to reliably produce valid molecules. Character variational autoencoders (CVAEs) (Gomez-Bombarelli et al., 2016) generate molecules one character ´ at a time, and are prone to syntactic and semantic errors; the grammar-based variational autoencoder (GVAE) (Kusner et al., 2017) and syntax-directed variational autoencoder (SD-VAE) (Dai et al., 2018) instead operate in the space of context-free and attribute grammars, respectively, to ensure syntactic validity. Other work generative models that operates on graph representations (Simonovsky & Komodakis, 2018; De Cao & Kipf, 2018; Jin et al., 2018; You et al., 2018; Liu et al., 2018), largely improving the ability to generate valid molecules. + +Suppose we are given a training set of pairs $\mathcal { D } = \{ ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \} , i = 1 , \ldots , N .$ , where x corresponds to molecules and y represents a value of some properties of the molecule $\mathbf { x }$ . Assume the molecules represent an i.i.d. sample from some unknown distribution $\tilde { p } ( { \bf x } )$ , which assigns high probability to molecules believed to be useful for a given task. Aside from Segler et al. (2017), which has no latent space and thus directly trains via maximum likelihood, these latent variable models are trained by optimizing a standard ELBO objective for variational autoencoders (?). This entails learning a stochastic encoder $q _ { \phi } ( { \bf z } | { \bf x } )$ which maps molecules into a latent space, and a stochastic decoder $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ for reconstructing molecules, by maximizing + +$$ +\mathcal { L } ( \boldsymbol { \theta } , \boldsymbol { \phi } ) = \sum _ { i = 1 } ^ { N } \bigg \{ \mathbb { E } _ { q _ { \boldsymbol { \phi } } ( \mathbf { z } _ { i } | \mathbf { x } _ { i } ) } [ \log p _ { \boldsymbol { \theta } } ( \mathbf { x } _ { i } | \mathbf { z } _ { i } ) ] - D _ { K L } ( q _ { \boldsymbol { \phi } } ( \mathbf { z } _ { i } | \mathbf { x } _ { i } ) | | p ( \mathbf { z } _ { i } ) ) \bigg \} . +$$ + +Notably, the objective is not a function of y: most existing generative models with latent variables do not perform direct conditional generation, and approaches for targeted molecule discovery are bolted on to the learnt model. Some, e.g. Kusner et al. (2017), are trained in an “unsupervised” manner, agnostic to any property which later may need to be optimized. Others, e.g. Gomez-Bombarelli ´ et al. (2016); Liu et al. (2018), train the autoencoder jointly alongside a function to predict y from z, hoping to guide the latent space to be also good for predicting the desired property. A recent exception is Assouel et al. (2018), which learns a deterministic autoencoder where the decoder takes the latent code and the desired property as input, using a mutual information term in training to steer the model towards generating molecules whose target properties match the input. Guimaraes et al. (2017); De Cao & Kipf (2018); You et al. (2018) instead learn generation models optimized towards specific metrics, such as drug-likeliness and solubility; the major downside is that these models must be retrained each time for a new property. In contrast, the autoencoder-based methods can be re-used to optimize towards any particular value of the property. + +# 2.2 STYLE TRANSFER WITH SUPERVISED VAES + +While the latent representations learned through standard VAE models perform well on the task of molecule reconstruction they do not necessarily provide interpretable factorised representations. A disentangled representation gives us additional control on the molecule generation process, allowing us to modify a single property leaving the remaining unaffected (Bengio et al., 2013a). In many cases important variation in the data is easy to annotate. For example in the case of molecule datasets we have access to different functional descriptors of the molecules obtained by chemoinformatics software such as RDKit (Landrum). Particularly useful to us here are supervised methods for learning disentangled representations (Kingma et al., 2014; Siddharth et al., 2017). These are distinct from unsupervised disentangling approaches such as InfoGAN (Chen et al., 2016) or $\beta$ -VAE (Higgins et al., 2017), which encourages the latent factor to learn a disentangled representation by modifying the objective to promote component independence. + +We will learn representations that specifically disentangle molecular properties of interest which we may later want to modify. Kingma et al. (2014) demonstrates how disentangling can be used to take two MNIST images of different digits, written in different styles, and independently change the digit while holding the style constant. An analogous operation on molecules would involve holding the physical structure of a molecule (its “style”) relatively fixed while modifying a salient property. Unlike (say) the style transfer example for the MNIST digits, the conditional distribution of molecules with a particular value of properties might be very diverse; for example, the QED score attempts to measure the drug-likeness of a molecule, and the set of molecules generated at high values of this score would hopefully have high probability on a large, varied set of molecules. An essential challenge here is that the property only provides a very weak signal as to the overall structure of the molecule. To account for this diversity, we model the conditional distribution with a latent variable $\mathbf { z }$ , such that $\begin{array} { r } { p _ { \theta } ( \mathbf { x } | \mathbf { y } ) = \int p _ { \theta } ( \mathbf { x } | \mathbf { z } , \mathbf { y } ) p ( \mathbf { \bar { z } } ) d \mathbf { z } } \end{array}$ . + +Disentangling the latent code $\mathbf { z }$ from the property y enables style transfer. This is done by taking an initial $\mathbf { x }$ , computing the posterior over the latent variable $\mathbf { z }$ , and then generating a new $\mathbf { x } ^ { \prime }$ with the property modified to have a target value $\mathbf { y } ^ { \prime }$ , with $p _ { \theta } ( \mathbf { x } ^ { \prime } | \mathbf { y } ^ { \prime } , \mathbf { x } ) =$ $\begin{array} { r } { \int p _ { \theta } ( \mathbf { x } ^ { \prime } | \mathbf { y } ^ { \prime } , \mathbf { z } ) p _ { \theta } ( \mathbf { z } | \mathbf { x } ) d \mathbf { z } } \end{array}$ . + +![](images/d6fdd5f7e997772d80bd45c81d2d6a587e1202c9b2e03df60b1f8ea2b7c46b4e.jpg) +Figure 1: A demonstration of style transfer + +Concretely, this involves fitting a joint generative model of the form $p _ { \boldsymbol { \theta } } ( \mathbf { x } , \mathbf { y } , \mathbf { z } ) = p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { y } , \mathbf { z } ) p ( \mathbf { y } ) \bar { p ( \mathbf { z } ) }$ , in which $\mathbf { y }$ and $\mathbf { z }$ are independent under the prior, and we assume a unit multivariate normal prior $p ( \mathbf { z } )$ . To infer the latent variable $\mathbf { z }$ we will use a variational distribution $q _ { \phi } ( { \bf z } | { \bf x } )$ , which takes the form of a multivariate normal distribution with parameters a nonlinear function of $\mathbf { x }$ , to approximate the true posterior $p _ { \theta } ( \mathbf { z } | \mathbf { x } , \mathbf { y } )$ . This objective function + +$$ +\mathcal { L } _ { E L B O } ( \theta , \phi ) = \sum _ { i = 1 } ^ { N } \left\{ \mathbb { E } _ { q _ { \phi } ( \mathbf { z } _ { i } | \mathbf { x } _ { i } ) } [ \log p _ { \theta } ( \mathbf { x } _ { i } | \mathbf { y } _ { i } , \mathbf { z } _ { i } ) ] - D _ { K L } ( q _ { \phi } ( \mathbf { z } _ { i } | \mathbf { x } _ { i } ) | | p ( \mathbf { z } _ { i } ) ) \right\} +$$ + +corresponds to learning a supervised VAE (Kingma et al., 2014), and represents a fairly na¨ıve approach to modeling a conditional distribution. + +# 3 CONDITIONAL GENERATION BY DISENTANGLING + +Maximizing this conditional ELBO in Eq (2) will likely yield good reconstructions of molecules from an embedding $\mathbf { z }$ (alongside the true property y), but for properties which only weakly inform the generative model there is nothing to enforce that the variable $\mathbf { y }$ actually directly has an effect on the generative process. Since the value $\mathbf { y }$ is something we know is a derived property of the molecule $\mathbf { x }$ , it is completely possible for all information about y to also be encoded in the representation $\mathbf { z }$ , in which case there is no guarantee that the learnt likelihood $p _ { \theta } ( \mathbf { x } | \mathbf { y } , \mathbf { z } )$ actually takes into account the value of $\mathbf { y }$ — in fact, we know it is possible to fit variational autoencoders where the decoder simply has the form $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ — and we are relying on the utility of $\mathbf { y }$ in reconstructions to see any sort of disentangling effect. + +# 3.1 CONSTRAINED ELBO + +In the case of conditional generation of molecules, we often have access to some oracle function $f$ (possibly non-differentiable) which for any given $\mathbf { x }$ outputs a property estimate $\mathbf { y }$ , for instance, the chemoinformatics software RDKit (Landrum). Since for conditional generation our ultimate goal is to generate a molecule $\mathbf { x }$ for any given target property $\mathbf { y } _ { 0 }$ , which then actually has $f ( \mathbf { x } ) = \bar { \mathbf { y } } _ { 0 }$ , we can reframe the problem by introducing hard constraints on the generated values, i.e. if restricting to values of $\mathbf { y }$ in the training set, + +$$ +\begin{array} { r l } & { \underset { \theta , \phi } { \operatorname* { m a x } } \mathcal { L } _ { E L B O } ( \theta , \phi ) } \\ & { \mathrm { s u b j e c t ~ t o ~ } \mathbb { E } _ { \mathbf { x } \sim p ( \mathbf { x } \mid \mathbf { y } _ { i } ) } [ \mathbb { I } [ f ( \mathbf { x } ) = \mathbf { y } _ { i } ] ] = 1 } \end{array} +$$ + +![](images/11c4f6bc728cef0a1f7156261ab4dd96723193136e4805467ad1b99711c5ae7c.jpg) +Figure 2: Setting and modeling pipeline for conditional generation of molecules, with supervision provided via an external property prediction oracle. Red lines correspond to non-differentiable components, including both a potentially complex sampling process and the property prediction itself. The blue dashed line corresponds to the approximate property predictor, which aims to predict the expected value of the property from a continuous relaxation, marginalized over the sampling process. + +for all $i = 1 , \ldots , N$ . This is an unreasonably hard constraint, unlikely to be satisfied by any distribution other than one which simply places a point mass on the single training $\mathbf { x } _ { i }$ associated with $\mathbf { y } _ { i }$ , but we can relax it by considering that (unlike the molecular space $\mathbf { x }$ ) the property space $\mathbf { y }$ is typically smooth, as many properties are continuous-valued and correspond to a human-interpretable scale. Following Ma et al. (2018) and Hu et al. (2017), we reframe the constraint as a soft penalty on the ELBO, + +$$ +\mathcal { L } ( \theta , \phi ) = \mathcal { L } _ { E L B O } ( \theta , \phi ) - \frac { \lambda _ { 1 } } { 2 } \sum _ { i = 1 } ^ { N } \mathbb { E } _ { \hat { \mathbf { x } } \sim p _ { \theta } ( \mathbf { x } | \mathbf { y } _ { i } ) } \Vert f ( \hat { \mathbf { x } } ) - \mathbf { y } _ { i } \Vert ^ { 2 } +$$ + +so that they are consistent with the property prediction, i.e., as we have an oracle function $f$ which enable us to access the property of any generated data, we can explicitly add a soft constraint to our loss function to provide explicit guidance for the generative model such that $f ( \hat { \mathbf x } ) = \mathbf y$ . This constraint is expected to hold for any pair $\displaystyle ( \mathbf { x } , \mathbf { y } )$ we may happen to come across, not just those in the training data. We also show optimizing the relaxed constraint is equivalent to maximizing mutual information with the target $\mathbf { y } _ { i }$ and generated molecule $\hat { \bf x }$ ; for details see appendix Section 6.1. + +# 3.2 APPROXIMATING THE PROPERTY PREDICTOR + +Introducing the regularizer as in Eq. (3) implicitly guides the reconstruction to take into account the property information, such that the reconstructed data should exhibit properties which match the input properties it is conditioned on. However, existing implementations of $f$ are often non-differentiable or CPU-bound, and $\hat { \bf x }$ are discrete samples from a categorical distribution, all of which means the gradient of the regularizer can’t flow back to the generator. This is outlined in Figure 2. To enable the gradient based methods on GPUs during training and avoid discrete sampling, one approach would be to first fit a differentiable approximation to $f$ , and then use either a Gumbel-softmax relaxation (Jang et al., 2016) or tricks like a “straight-through” estimator (Bengio et al., 2013b) as a continuous approximation for the discrete samples. Instead, we propose bypassing the discrete sampling step entirely and learning a function $f _ { \omega }$ that can map from a learned representation of the molecules directly to molecules property (Hu et al., 2017). + +To do this, we take as input the last hidden layer of the decoder network which parameterizes $p _ { \theta } ( \mathbf { x } | \mathbf { z } , \mathbf { y } )$ , denoting this deterministic transformation as $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \mathbf { y } )$ . For the grammar VAE and the syntax-directed VAE, this last layer $\mathbf { h } = g _ { \boldsymbol { \theta } } ( \mathbf { z } , \mathbf { y } )$ is the output of a recurrent layer that generates logits corresponding to unmasked and unnormalized log probabilities for each character at each position in the string; see Kusner et al. (2017) and Dai et al. (2018) for details on the implementation of the somewhat complex sampling process in the decoder. Ideally, $f _ { \omega }$ would estimate the property distribution obtained by marginalizing out the discrete sampling step, with + +$$ +f _ { \omega } ( \mathbf { h } \equiv g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { 0 } ) ) \approx \mathbb { E } _ { p _ { \theta } ( \mathbf { x } | \mathbf { z } , \mathbf { y } _ { 0 } ) } [ f ( \mathbf { x } ) ] , +$$ + +where we condition on $\mathbf { z }$ , and $\mathbf { y } _ { 0 }$ refers to an arbitrary input target property. + +Assuming the approximation in Eq. (4), we have + +$$ +\begin{array} { r } { \mathbb { E } _ { p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { y } _ { i } ) } \| f ( \hat { \mathbf { x } } ) - \mathbf { y } _ { i } \| ^ { 2 } = \mathbb { E } _ { p ( \mathbf { z } ) } \left[ \mathbb { E } _ { p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { y } _ { i } , \mathbf { z } ) } \| f ( \hat { \mathbf { x } } ) - \mathbf { y } _ { i } \| ^ { 2 } \right] \approx \mathbb { E } _ { p ( \mathbf { z } ) } \| f _ { \omega } ( g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { i } ) ) - \mathbf { y } _ { i } \| ^ { 2 } , } \end{array} +$$ + +an expectation over a real-valued variable which does not depend on any of the parameters we are estimating, meaning we can use a simple path estimate of the gradient with respect to $\theta , \omega$ by exchanging the gradient with the expectation. We thus define a regularization term + +$$ +\mathcal { L } _ { d i s e n t } ( \theta , \omega ) = \frac { \lambda _ { 1 } } { 2 } \sum _ { i = 1 } ^ { N } \mathbb { E } _ { p ( \mathbf { z } ) } \| f _ { \omega } \big ( g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { i } ) \big ) - \mathbf { y } _ { i } \| ^ { 2 } +$$ + +which can be used as a drop-in replacement for the non-differentiable penalty term in Eq. (3), yielding a candidate objective function + +$$ +\mathcal { L } ( \theta , \phi ) \approx \hat { \mathcal { L } } _ { \omega } ( \theta , \phi ) = \mathcal { L } _ { E L B O } ( \theta , \phi ) - \mathcal { L } _ { d i s e n t } ( \theta , \omega ) +$$ + +3.3 LEARNING THE PROPERTY ESTIMATOR JOINTLY WITH GENERATIVE MODEL + +While one could imagine attempting to learn $f _ { \omega }$ jointly with $\phi , \theta$ by direct optimization of Eq. (6), in practice this is very unstable, as values of $g _ { \boldsymbol { \theta } } ( \mathbf { z } , \mathbf { y } _ { i } )$ early in training may correspond to very poor generated molecules $\hat { \mathbf { x } } _ { i }$ which may not have properties at all similar to $\mathbf { y } _ { i }$ . This can be sidestepped by training the property estimator jointly as part of an extended generative model on [x, y]. + +We note that the property estimator $f _ { \omega }$ parameterizes a probability distribution $p _ { \omega } ( f ( \mathbf { x } ) | \mathbf { z } , \mathbf { y } _ { 0 } )$ , where $\mathbf { x } \sim p _ { \theta } ( \mathbf { x } | \mathbf { z } , \mathbf { y } _ { 0 } )$ and f is the oracle function that $f ( \mathbf { x } ) = \mathbf { y }$ . With a Gaussian distribution over the error, we can consider + +$$ +p _ { \omega } ( f ( \mathbf { x } ) \vert \mathbf { z } , \mathbf { y } _ { 0 } ) = \mathcal { N } ( f ( \mathbf { x } ) \vert f _ { \omega } ( g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { 0 } ) ) , \lambda _ { 2 } ^ { - 1 } \mathbf { I } ) +$$ + +for small, fixed $\lambda _ { 2 }$ . Therefore, we propose defining a new ELBO based on a joint autoencoder for $\{ f ( \mathbf { x } _ { i } ) , \mathbf { y } _ { i } ) \}$ , albeit with a factorization such that the input $\mathbf { y } _ { i }$ bypasses the encoder and is passed directly into the decoder, with a joint likelihood + +$$ +p _ { \{ \theta , \omega \} } ( \mathbf { x } _ { i } , f ( \mathbf { x } _ { i } ) | \mathbf { z } _ { i } , \mathbf { y } _ { i } ) = p _ { \omega } ( f ( \mathbf { x } _ { i } ) | \mathbf { z } _ { i } , \mathbf { y } _ { i } ) p _ { \theta } ( \mathbf { x } _ { i } | \mathbf { z } _ { i } , \mathbf { y } _ { i } ) . +$$ + +This yields a joint ELBO for the training set of + +$$ +\mathcal { L } _ { E L B O } ( \omega , \theta , \phi ) = \sum _ { i = 1 } ^ { N } \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) } \left[ \log \frac { p _ { \omega } ( f ( \mathbf { x } _ { i } ) | \mathbf { z } , \mathbf { y } _ { i } ) p _ { \theta } ( \mathbf { x } _ { i } | \mathbf { z } , \mathbf { y } _ { i } ) p ( \mathbf { z } ) } { q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) } \right] . +$$ + +Note that we can rewrite this ELBO as a function of the previous one, with + +$$ +\mathcal { L } _ { E L B O } ( \boldsymbol { \omega } , \boldsymbol { \theta } , \boldsymbol { \phi } ) = \mathcal { L } _ { E L B O } ( \boldsymbol { \theta } , \boldsymbol { \phi } ) - \frac { \lambda _ { 2 } } { 2 } \sum _ { i = 1 } ^ { N } \mathbb { E } _ { \boldsymbol { q } _ { \boldsymbol { \phi } } ( \mathbf { z } | \mathbf { x } _ { i } ) } \| f _ { \boldsymbol { \omega } } \big ( g _ { \boldsymbol { \theta } } ( \mathbf { z } , \mathbf { y } _ { i } ) \big ) - \mathbf { y } _ { i } \| _ { 2 } ^ { 2 } , +$$ + +where we also see that $\mathcal { L } _ { E L B O } ( \omega , \theta , \phi ) \leq \mathcal { L } _ { E L B O } ( \theta , \phi )$ , allowing us to define an objective + +$$ +\hat { \mathcal { L } } ( \omega , \theta , \phi ) = \mathcal { L } _ { E L B O } ( \omega , \theta , \phi ) - \mathcal { L } _ { d i s e n t } ( \theta , \omega ) , +$$ + +which is a lower bound on Eq. (6). Notice the two terms we have added to the original ELBO are quite similar, differing only in choice of distribution: for learning $f _ { \omega }$ , we wish to use values of $\mathbf { z }$ simulated form the approximate posterior $q _ { \phi } ( { \bf z } | { \bf x } )$ , whereas for enforcing a constraint across all possible generations we simulate $\mathbf { z }$ from the prior $p ( \mathbf { z } )$ . + +# 3.4 GRADIENT ESTIMATION + +As the regularizer $\mathcal { L } _ { d i s e n t } ( \theta , \omega )$ encourages disentangling by constraining the molecules generated from $\mathbf { y } _ { i }$ to have property $\mathbf { y } _ { i }$ no matter what value $\mathbf { z }$ takes, we found that it does not necessarily evaluate at meaningful values of $\mathbf { z }$ when sampled randomly from $p ( \mathbf { z } )$ . This roughly corresponds to the notion that not all combinations of “style” and property are physically attainable; ideally for style transfer we would like the generated molecule to stay “close” in structure to the original molecule that we intended to modify. When estimating (gradients of) the soft constraint term $\mathcal { L } _ { d i s e n t } ( \theta , \omega )$ , we found it advantageous to use samples of $\mathbf { z }$ which correspond to encodings of actual data points, as opposed to random samples from the prior. We approximate expectations with respect to $p ( \mathbf { x } )$ by looking at the so-called marginal posterior; we note that + +Table 1: Reconstruction performance and generation quality (Valid, Unique, Novel). + +
ModelQM9ZINC
Reconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE Gómez-Bombarelli et al. (2016)3.6110.30=90.044.60.70·100
GVAE Kusner et al. (2017)96.0060.20=80.9053.707.20=100
SD-VAE Dai et al.(2018)97.8498.4099.2891.9776.2043.50--
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8494.61100
CGD-VAE-1-GRU99.2795.6193.6587.8788.6429.0099.24100
Sup-VAE-3-GRU97.8197.9095.0989.4782.4036.1686.26100
CGD-VAE-3-GRU99.3197,8098.7796.2181.8037.7898.75100
+ +$$ +p ( \mathbf { z } ) = \int p _ { \theta } ( \mathbf { z } | \mathbf { x } ) p _ { \theta } ( \mathbf { x } ) d \mathbf { x } \approx \frac { 1 } { N } \sum _ { j } p _ { \theta } ( \mathbf { z } | \mathbf { x } _ { j } ) \approx \frac { 1 } { N } \sum _ { j } q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { j } ) , +$$ + +where the first approximation uses the empirical data distribution as an approximation to the model marginal $p _ { \theta } ( \mathbf { x } )$ , and the second uses our variational posterior approximation $q _ { \phi } ( { \bf z } | { \bf x } )$ . We define this quantity as $\begin{array} { r } { q ( \mathbf { z } ) = \frac { 1 } { N } \sum _ { j } q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { j } ) } \end{array}$ , a mixture of Gaussians, which we can sample from by drawing random values from our dataset and then drawing from their encoding distributions. + +When we use this in estimating gradients of the soft constraint, we can use samples from the same minibatch, exactly corresponding to a property transfer task. That is, for any particular $y _ { i }$ in the dataset, we can estimate + +$$ +\begin{array} { r } { \mathbb { E } _ { p ( \mathbf { z } ) } \nabla _ { \theta , \omega } \| f _ { \omega } \big ( g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { i } ) \big ) - \mathbf { y } _ { i } \| ^ { 2 } \approx \mathbb { E } _ { q ( \mathbf { z } _ { j } | \mathbf { x } _ { j } ) } \nabla _ { \theta , \omega } \| f _ { \omega } \big ( g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { i } ) \big ) - \mathbf { y } _ { i } \| ^ { 2 } . } \end{array} +$$ + +for any uniformly randomly sampled $j \neq i$ . By sampling $\mathbf { z } _ { j }$ from $q ( \mathbf { z } _ { j } | \mathbf { x } _ { j } )$ where $j \neq i$ , we make sure that all the label information decoder is receiving comes from the actual $\mathbf { y } _ { i }$ that is feed to the decoder and $\mathbf { z } _ { j }$ does not include any information about label. This can be evaluated easily by simply evaluating the penalty term of Eq. (10) twice per minibatch; once as in Eq. (10), and once to approximate $\mathcal { L } _ { d i s e n t } ( \theta , \omega )$ by permuting the properties in the minibatch to be assigned to incorrect molecules. We detail the training algorithm in Section 6.2 of the appendix. + +# 4 EXPERIMENTS + +We experiment with the QM9 dataset (Ramakrishnan et al., 2014), that contains $1 3 4 \mathbf { k }$ molecules with up to 9 heavy atoms, and the ZINC dataset (Sterling & Irwin, 2015) containing $2 5 0 \mathrm { k }$ druglike molecules. Our goal here is two-fold: we would like to understand (1) whether a supervised variational autoencoder is capable of learning suitable conditional distributions over molecules, and (2) to what extent this task is assisted by the additional regularization term corresponding to the soft constraint. + +We represent molecules using the one-hot encoding of their SMILES production rules (Kusner et al., 2017) and add a semantic constraint (Dai et al., 2018) on the decoder network to avoid generating syntactically correct but semantically invalid molecules. We use 80 production rules to describe molecules and set the maximum SMILES sequence length to 100 for the QM9 dataset and 278 for the Zinc dataset. We experiment with the logP property of the molecules (Wildman & Crippen, 1999). We use the same encoder and decoder network structure as Dai et al. (2018) with the only difference that our decoder takes as input the concatenation of $\mathbf { y } , \mathbf { z }$ . We give the details of the architecture in the appendix section 6.2. + +We evaluate the reconstruction accuracy and the quality of the molecules generated by our method, which we denote by CGD-VAE (conditional generation with disentangling) and compare against CVAE (Gomez-Bombarelli et al., 2016), GVAE (Kusner et al., 2017), and SD-VAE (Dai et al., 2018). ´ We explore its conditional generation performance in two settings: controlling only the property value and controlling both the property value and the molecule structure to what can be seen as property transfer. We took the results of CVAE, GVAE from the literature. For SD-VAE we used the authors code with the default values to generate results for QM9 since these were not available for QM9. We also implemented supervised VAE versions of SD-VAE which we denote Sup-VAE-X-GRU $\mathrm { ( X \in \{ 1 , 3 \} }$ , denotes GRU layers) and which can do conditional generation. + +![](images/d83af7d84ad5e4b1f43f6ce0ea2358c91c263cc5eae0206da718a5bc7d37f5ab.jpg) +Figure 3: Conditional generation given the desired $\mathrm { l o g P { = } { - } 0 . 5 7 5 9 }$ , row molecules have a logP within a $15 \%$ range of the desired one. + +![](images/e0f8c116c2f73cfe17b82b2094f1aeb8977e80722189f751d7235bfe6e3e375d.jpg) +Figure 4: Property transfer + +Before proceeding with the experiments we will give some additional details on how we do conditional generation from $p _ { \theta } ( \mathbf { x } | \mathbf { y } _ { 0 } )$ given the target property $\mathbf { y } _ { 0 }$ . Instead of marginalizing over the prior marginal inference distribution mass of the dataset is in the late $p ( z )$ , we mirror the approach taken during training and integrate over an approximation to the $\begin{array} { r } { q _ { \phi } ( \mathbf { z } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { i } ) } \end{array}$ which better characterizes where the large and we do not wish to keep the $N$ entire dataset available at test time, we approximate $q _ { \phi } ( \mathbf { z } )$ with an isotropic Gaussian distribution $\hat { q } _ { \sigma } ( \mathbf { z } ) = \mathcal { N } ( \mathbf { z } | \mathbf { 0 } , \sigma ^ { 2 } \mathbf { I } )$ . We estimate $\sigma$ for each model by Monte Carlo samples from $q _ { \phi } ( \mathbf { z } )$ . For the supervised VAE without the soft constraint regularizer this yields 0.053 for QM9 and 0.118 for ZINC. For our model with the soft constraint we get 0.0354 for QM9 and 0.096 for ZINC. We do conditional generation of $\mathbf { x }$ given $\mathbf { y } _ { 0 }$ by sampling from $\begin{array} { r } { p _ { \theta } ( \mathbf { x } | \mathbf { y } _ { 0 } ) = \int \hat { q } _ { \sigma } ( \mathbf { z } ) p _ { \theta } ( \mathbf { x } | \mathbf { z } , \mathbf { y } _ { 0 } ) d \mathbf { z } } \end{array}$ . + +We evaluate reconstruction performance in terms of the correctly reconstructed molecules on test sets of size 10k for QM9 and $5 \mathrm { k }$ for ZINC, for the latter we used the default test set. We evaluate the generated molecules’ quality by the percentage of valid, unique (i.e. percentage of unique molecules among the generated valid molecules) and novel (i.e. percentage of molecules never seen in the training set among the generated molecules) molecules. We estimate these quantities by sampling 10k (5K for ZINC) $\mathbf { z }$ from the ${ \hat { q } } _ { \sigma } ( \mathbf { z } )$ and coupling each one of them with a logP value, $\mathbf { y }$ , randomly selected from the test set, and we subsequently decode the $\mathbf { z } , \mathbf { y }$ concatenation. We can see that our model has a better reconstruction performance compared to the baselines while in some cases generating slightly less valid molecules table 1. In terms of the three quality measures achieves an excellent performance across all three metrics being always one of the two best performing methods for any metric. + +To visualise how the conditional generation operates we randomly sample from the test set some molecule and obtain its property value $\mathbf { y } _ { 0 }$ . We then draw 50 random samples $\mathbf { z } _ { i }$ from $\hat { q } _ { \sigma } ( \mathbf { z } )$ and decode the $\left[ \mathbf { z } _ { i } , \mathbf { y } _ { 0 } \right]$ vectors. Among the generated valid molecules we compute the percentage of those that have a property value $\mathbf { y } _ { i }$ that is within a $15 \%$ range from the $\mathbf { y } _ { 0 }$ property value. In Figure 3 we present the molecules obtained for a test molecule that had a logP of $- 0 . 5 7 5 9$ . Out of the 50 generated molecules 46 were valid of which we give the five that were within a $15 \%$ range from the $\mathbf { y } _ { 0 }$ value in Figure 3. As we can see we get molecules that are structurally very different from the original one yet they have similar logP value. + +To quantify the quality of the conditional generations we measure the correlation between the property value we obtain by the conditional generation and the property value on which we conditioned the generation. We randomly sample $1 0 0 0 { \textbf { y } }$ values from the test set and $1 0 0 0 \textbf { z }$ values from the approximate learned prior $\hat { q } _ { \sigma } ( \mathbf { z } )$ . We decode each pair, obtain $\hat { \mathbf { x } } \sim p _ { \theta } ( \mathbf { x } | \mathbf { y } , \mathbf { z } )$ , and then measure the correlation of the original y with the $\hat { \mathbf { y } }$ of generated $\hat { \bf x }$ . In Table 2, we give the correlation estimates for our method and the Sup-VAE baselines. As we can see our method has a considerably higher correlation score between the input and the obtained property than Sup-VAE. Conditional generation seems considerably harder for the ZINC dataset for all methods. + +To visualise the style transfer behavior of our model we randomly sample two molecules $\mathbf { x } _ { A } , \mathbf { x } _ { B }$ from the test set. We then sample $\mathbf { z } _ { A }$ from the learned posterior $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { A } )$ . We subsequently decode $\left[ \mathbf { z } _ { A } , \mathbf { y } _ { B } \right]$ , $\mathbf { y } _ { B }$ is the property of $\mathbf { x } _ { B }$ , and get a new molecule $\hat { \mathbf { x } } _ { A B }$ . Ideally, the obtained molecule $\hat { \mathbf { x } } _ { A B }$ should have a property value (logP) close to the target $\mathbf { y } _ { B }$ and be similar to $\mathbf { x } _ { A }$ . In Figure 4 we give one such example. To put the results into context in Figure 8 in appendix, we give the results of a virtual screening method, where we select from the full dataset five molecules which are structurally similar to $\mathbf { x } _ { A }$ and have logP values close to $\mathbf { y } _ { B }$ . As we can see the molecule that our model generates is a new one. + +![](images/ffa61574cd2f40b5b194d3b7515745269b23e87d9fcf063708e5b80200110fb9.jpg) +Figure 5: Style transfer. The $\mathbf { z }$ of the nine real molecules placed in the $\mathbf { X }$ -axis is combined with 11 y property values, sampled in [-4.9, 4.9], the resulting pair is decoded to a molecule. + +To quantify the style transfer performance we proceed in exactly the same manner as we did to quantify the conditional generation performance. However, now instead of sampling $\mathbf { z }$ from the approximate learned prior, ${ \hat { q } } _ { \sigma } ( \mathbf { z } )$ , we first sample some $\mathbf { x }$ from the test set and then we sample $\mathbf { z }$ from the learned posterior $q ( \mathbf { z } | \mathbf { x } )$ . The results are in the second column of Table 2. As we can see the correlation values are now lower than the ones we obtained in the simple conditional generation case. This can be explained by the fact that now we are forcing a specific combination of structure, z comes from a real molecule, and property, which might simply be physically infeasible since the molecule space is discrete and not all combinations are possible. In addition, as it was the case for the conditional generation, style transfer is considerably more difficult for the ZINC dataset. + +We further explore the style transfer and visualize how our model covers the combined space of molecule structure and properties. We sample nine molecules from the QM9 test set, and get their $\mathbf { z }$ encodings. For each such encoding we decode the vectors $[ \mathbf { z } , y ] , y \in [ - 4 . 9 , 4 . 9 ]$ , with the $y$ (logP) interval sampled at 11 points. We give in Figure 5 the resulting valid molecules, each column there corresponds to one of the nine original molecules, the ones surrounded by dotted rectangle, and their decodings with different logP values.For each original molecule we give the generated molecules ordered along the y axis according to the y property that they actually exhibit. The x-axis does not provide an ordering of the original molecules according to $\mathbf { Z }$ , in fact we have ordered the original molecules by their y property. As we can see not all $( \mathbf { z } , y )$ combinations produce a result. These holes can be explained either by the physical infeasibility of the combination and/or a limitation of the learned model. + +We can use conditional generation to control in a fine manner the value of the desired property, to what can be seen as direct property optimization. We visualise the level of control we have on an experiment with a single molecule (with logP is -1.137), which we randomly sample from the test set. We obtain its $\mathbf { z }$ encoding and perform generations with increased logP taking values in 1000 point grid in $\left[ - 1 . 1 3 7 , 4 . 9 \right]$ . We then decode $\left[ \mathbf { z } , \mathbf { y } _ { i } \right]$ and compute the logP value of the generated molecules. Among the 1000 generated molecules only 19 are unique. We get an increase of logP of a very discrete nature, Figure 6. As already discussed not all combinations of structure and properties are possible. The generated molecules themselves are shown in the supplemental material. + +
Modelz~qo(z)z~q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.71850.5005
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.74140.4715
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.38770.0880
Sup-VAE-3-GRU0.35140.1808
CGD-VAE-3-GRU0.39660.1559
+ +Table 2: Correlation between the desired input property and the obtained property . $\mathbf { z } \sim \hat { q } _ { \sigma } ( \mathbf { z } )$ corresponding to conditional generation), and x, $\mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } )$ to property transfer case. + +![](images/dd9ea64a1ba0ee993b06ada6658a78a2896a28fd1475d73046b1a5c7a4f5c11d.jpg) +Figure 6: Property optimization. Given a start molecule (red), we combine its $\mathbf { z }$ with $1 0 0 0 \log \mathrm { P }$ values and decode (blue) + +![](images/6daa2bdd2b6b3d32bf6a2bb30e18d85a323167036361749ae8c33e10b946aa5e.jpg) + +![](images/171721555e584b5336b980f58a15f7e4a7ff8c35dce2ba5cd515c171c0be165b.jpg) +Figure 7: A comparison of simulated logP values and Tanimoto similarity to a target on the ZINC dataset. While the stacked LSTM model has high accuracy in terms of matching the desired property, it would require drawing many samples before finding any close matches to any particular desired prototype. The CGD-VAE-3-GRU model represents a middle ground between a standard VAE model which does not condition on the property, and the stacked LSTM model which does not learn a reusable representation. + +# 4.1 CONDITIONAL LSTM BASELINE + +Finally, we consider a variant of the stacked LSTM model of Segler et al. (2017), with no latent space, where the model is modified to take a target logP value as an additional input at each generation step. This model forms a very strong baseline for many distribution matching tasks (Liu et al., 2018; ?), though as best we are aware this has never been used directly for conditional generation given a target property. We use a modification of the implementation provided by (?) with three layers and default settings, and fit the model by maximum likelihood training on $\begin{array} { r } { p _ { \theta } ( \mathbf { x } | \mathbf { y } ) = \prod _ { t = 1 } ^ { T } p _ { \theta } ( x _ { t } | x _ { 1 : t - 1 } , \mathbf { y } ) } \end{array}$ . + +Training on the ZINC dataset, we find the generated molecules from this model have a very high correlation 0.975 with the target logP value, greatly outperforming any of the latent variable models we consider. This suggests that such a model would be very useful for generating candidates globally, but as the model has no latent variable it is not amenable to style transfer. We observe this in Figure 7, which samples 100 candidate molecules from both the stacked LSTM model and for CGD-VAE-3- GRU, conditioning on the property of one randomly-chosen test set example, while computing the Tanimoto similarity (computed using Morgan fingerprints of radius 2) to a second randomly-chosen test set example, across 200 pairs. The VAE has higher Tanimoto similarities as it can condition on the latent variable of the target molecule, representing a trade-off against the better adherence to the target property value of the unconditioned LSTM. + +# 5 CONCLUSION + +We presented a single step approach for the conditional generation of molecules with desired properties. Our model allows also to condition generation on a prototype molecule with a desired high-level structure. This work thus directly inverts the traditional relationship between molecules and their properties. We found that training the deep generative models conditional on target properties, following a supervised VAE approach, does not appreciably harm the quality of the unconditional generative model as measured by validity, novelty, and uniqueness of samples. Furthermore, we see that the additional act of regularizing the output using an approximate property predictor helps improve both reconstruction accuracy and property correlations in most combinations of tasks and datasets, particularly for the smaller QM9 dataset and for smaller models with fewer RNN layers. We also note that although none of the deep latent variable models are competitive with an LSTM baseline when purely considering generation conditioned on a target property value, the low Tanimoto similarity between randomly sampled candidates and an arbitrary style transfer target makes clear that such a model is not suitable for targeted generation of candidates which are close in structure to a particular prototype. + +In future work, we want to explore how to further improve the correlation between the desired input properties to the decoder, and the properties of the generated molecules. Moreover, we want also to condition on multiple properties; while this is in principle possible in our framework, we do not explore it empirically here. Modifying a single property while constraining the remaining to be close to the original can further aggravate the infeasibility problem, as not all combinations of molecular properties may even be feasible, perhaps requiring learning a dependency structure between multiple properties. + +# REFERENCES + +Rim Assouel, Mohamed Ahmed, Marwin H. Segler, Amir Saffari, and Yoshua Bengio. Defactor: Differentiable edge factorization-based probabilistic graph generation. CoRR, abs/1811.09766, 2018. URL http://arxiv.org/abs/1811.09766. + +Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE transactions on pattern analysis and machine intelligence, 35(8):1798–1828, 2013a. + +Yoshua Bengio, Nicholas Leonard, and Aaron Courville. Estimating or propagating gradients through ´ stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013b. + +Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in neural information processing systems, pp. 2172–2180, 2016. + +Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio. Empirical evaluation of gated recurrent neural networks on sequence modeling. arXiv preprint arXiv:1412.3555, 2014. + +Hanjun Dai, Yingtao Tian, Bo Dai, Steven Skiena, and Le Song. Syntax-directed variational autoencoder for structured data. arXiv preprint arXiv:1802.08786, 2018. + +Nicola De Cao and Thomas Kipf. MolGAN: An implicit generative model for small molecular graphs. May 2018. + +Hanna Eckert and Jurgen Bajorath. Molecular similarity analysis in virtual screening: foundations, ¨ limitations and novel approaches. Drug discovery today, 12(5-6):225–233, 2007. + +Rafael Gomez-Bombarelli, David K. Duvenaud, Jos ´ e Miguel Hern ´ andez-Lobato, Jorge Aguilera- ´ Iparraguirre, Timothy D. Hirzel, Ryan P. Adams, and Alan Aspuru-Guzik. Automatic chemical ´ design using a data-driven continuous representation of molecules. CoRR, abs/1610.02415, 2016. URL http://arxiv.org/abs/1610.02415. + +Gabriel Lima Guimaraes, Benjamin Sanchez-Lengeling, Carlos Outeiral, Pedro Luis Cunha Farias, and Alan Aspuru-Guzik. Objective-Reinforced generative adversarial networks (ORGAN) for ´ sequence generation models. May 2017. + +Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations, 2017. + +Zhiting Hu, Zichao Yang, Xiaodan Liang, Ruslan Salakhutdinov, and Eric P Xing. Toward controlled generation of text. In Proceedings of the 34th International Conference on Machine LearningVolume 70, pp. 1587–1596. JMLR. org, 2017. + +Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. arXiv preprint arXiv:1611.01144, 2016. + +Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Junction tree variational autoencoder for molecular graph generation. arXiv preprint arXiv:1802.04364, 2018. + +Diederik P Kingma, Shakir Mohamed, Danilo Jimenez Rezende, and Max Welling. Semi-supervised learning with deep generative models. In Advances in Neural Information Processing Systems, pp. 3581–3589, 2014. + +Peter Kirkpatrick and Clare Ellis. Chemical space. Nature, 432(7019):823, 2004. + +Matt J Kusner, Brooks Paige, and Jose Miguel Hern ´ andez-Lobato. Grammar variational autoencoder. ´ arXiv preprint arXiv:1703.01925, 2017. + +Greg Landrum. Rdkit: Open-source cheminformatics. URL http://www.rdkit.org. + +Qi Liu, Miltiadis Allamanis, Marc Brockschmidt, and Alexander Gaunt. Constrained graph variational autoencoders for molecule design. In Advances in Neural Information Processing Systems 31, pp. 7806–7815. 2018. + +Tengfei Ma, Jie Chen, and Cao Xiao. Constrained generation of semantically valid graphs via regularizing variational autoencoders. In Advances in Neural Information Processing Systems 31, pp. 7113–7124. 2018. + +Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and O Anatole Von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. Scientific data, 1:140022, 2014. + +Marwin HS Segler, Thierry Kogej, Christian Tyrchan, and Mark P Waller. Generating focused molecule libraries for drug discovery with recurrent neural networks. ACS central science, 4(1): 120–131, 2017. + +N. Siddharth, Brooks Paige, Jan-Willem Van de Meent, Alban Desmaison, Noah Goodman, Pushmeet Kohli, Frank Wood, and Philip Torr. Learning disentangled representations with semi-supervised deep generative models. In Advances in Neural Information Processing Systems, pp. 5925–5935, 2017. + +Martin Simonovsky and Nikos Komodakis. Graphvae: Towards generation of small graphs using variational autoencoders. arXiv preprint arXiv:1802.03480, 2018. + +Teague Sterling and John J Irwin. Zinc 15–ligand discovery for everyone. Journal of chemical information and modeling, 55(11):2324–2337, 2015. + +Scott A Wildman and Gordon M Crippen. Prediction of physicochemical parameters by atomic contributions. Journal of chemical information and computer sciences, 39(5):868–873, 1999. + +Jiaxuan You, Bowen Liu, Rex Ying, Vijay Pande, and Jure Leskovec. Graph convolutional policy network for goal-directed molecular graph generation. In Advances in Neural Information Processing Systems 31, pp. 6412–6422. 2018. + +# 6 APPENDIX + +# 6.1 THE REGULARISER AND ITS RELATION TO THE MUTUAL INFORMATION MAXIMIZATION + +The soft constrain in the loss 3, in fact , is equivalent to a simple maximizing mutual information formulation between generated molecules $\hat { \bf x }$ and the target property $\mathbf { y }$ provided to the generator. Assume the true conditional distribution is $\tilde { p } ( \mathbf { y } | \mathbf { x } )$ : + +$$ +\begin{array} { r l } & { \mathbf { I } ( \mathbf { y } ; \hat { \mathbf { x } } ) = H ( \mathbf { y } ) - H ( \mathbf { y } \vert \hat { \mathbf { x } } ) } \\ & { \qquad = H ( \mathbf { y } ) + \mathbb { E } _ { \hat { \mathbf { x } } \sim p _ { \theta } ( \mathbf { x } \vert \mathbf { y } ) } \mathbb { E } _ { \mathbf { y } ^ { \prime } \sim \tilde { p } ( \mathbf { y } \vert \hat { \mathbf { x } } ) } [ \log \tilde { p } ( \mathbf { y } ^ { \prime } \vert \hat { \mathbf { x } } ) ] } \end{array} +$$ + +We do not know the true distribution $\tilde { p } ( \mathbf { y } | \hat { \mathbf { x } } )$ , however, the RDKit provides an estimation $p ( \mathbf { y } \vert \hat { \mathbf { x } } )$ of the distribution assuming a Gaussian distribution over error: + +$$ +p ( \mathbf { y } \vert \hat { \mathbf { x } } ) = \mathcal { N } ( \mathbf { y } \vert f ( \hat { \mathbf { x } } ) , \lambda _ { 1 } ^ { - 1 } \mathbf { I } ) +$$ + +where $f$ is the molecule property estimator, i.e., RDKit. We have: + +$$ +\begin{array} { l } { { \displaystyle { \bf I } ( { \bf y } ; \hat { \bf x } ) = H ( { \bf y } ) + \mathbb { E } _ { \hat { \bf x } \sim p _ { \theta } ( { \bf x } \mid { \bf y } ) } \mathbb { E } _ { { \bf y } ^ { \prime } \sim \tilde { p } ( { \bf y } \mid \hat { \bf x } ) } [ \log \frac { \tilde { p } ( { \bf y } ^ { \prime } \mid \hat { \bf x } ) } { p ( { \bf y } ^ { \prime } \mid \hat { \bf x } ) } p ( { \bf y } ^ { \prime } \mid \hat { \bf x } ) ] } \ ~ } \\ { { \displaystyle ~ = H ( { \bf y } ) + \mathbb { E } _ { \hat { \bf x } \sim p _ { \theta } ( { \bf x } \mid { \bf y } ) } [ D _ { k l } ( \tilde { p } ( { \bf y } ^ { \prime } \mid \hat { \bf x } ) \vert \vert p ( { \bf y } ^ { \prime } \vert \hat { \bf x } ) ) + \mathbb { E } _ { { \bf y } ^ { \prime } \sim \tilde { p } ( { \bf y } \mid \hat { \bf x } ) } \log p ( { \bf y } ^ { \prime } \mid \hat { \bf x } ) ] } \ ~ } \\ { { \displaystyle ~ \geq H ( { \bf y } ) + \mathbb { E } _ { \hat { \bf x } \sim p _ { \theta } ( { \bf x } \mid { \bf y } ) } [ \mathbb { E } _ { { \bf y } ^ { \prime } \sim \tilde { p } ( { \bf y } \mid \hat { \bf x } ) } \log p ( { \bf y } ^ { \prime } \mid \hat { \bf x } ) ] } } \end{array} +$$ + +By following the Lemma 5.1 given in Chen et al. (2016), we have + +$$ +\mathbf { I } ( \mathbf { y } ; \hat { \mathbf { x } } ) \geq H ( \mathbf { y } ) + \mathbb { E } _ { \mathbf { y } \sim p ( \mathbf { y } ) , \hat { \mathbf { x } } \sim p _ { \theta } ( \mathbf { x } | \mathbf { y } ) } [ \log p ( \mathbf { y } | \hat { \mathbf { x } } ) ] +$$ + +As $H ( \mathbf { y } )$ is constant, minimizing $\mathbb { E } _ { \hat { \mathbf { x } } \sim p _ { \theta } ( \mathbf { x } | \mathbf { y } _ { i } ) } \| f ( \hat { \mathbf { x } } ) - \mathbf { y } _ { i } \| ^ { 2 }$ is equivalent to maximizing $\mathbf { I } ( \mathbf { y } ; \hat { \mathbf { x } } )$ under the assumption that $p ( \mathbf { y } \vert \hat { \mathbf { x } } )$ is close to $\tilde { p } ( \mathbf { y } | \hat { \mathbf { x } } )$ + +# 6.2 ARCHITECTURE AND TRAINING PROCEDURE DESCRIPTION + +We use the same encoder and decoder network structure as Dai et al. (2018) with the only difference that our decoder takes as input the concatenation of $\mathbf { y } , \mathbf { z }$ . As GRU layers become computationally expensive when the sequences length increase, we also examined the model using less layer GRU. To be precise, the decoder in Dai et al. (2018) takes the from of a dense hidden layer with ReLU activation followed by three layers GRU Chung et al. (2014). We tried two different settings of decoder: in the first setting, we feed the concatenation of $\mathbf { y } , \mathbf { z }$ to dense layer then apply one layer GRU, in the second setting, to enhance the effect of $\mathbf { y }$ in the decoder, we feed y not only to the dense layer but also to each layer of GRU. Furthermore, we set the dimension of the latent representation to 56. For the oracle function estimator $f _ { w }$ , we use the same network architecture as the encoder (there is no parameter sharing) and we add one more fully connected layer followed by a Tanh transformation. + +To speed up convergence, we initialize the $f _ { w }$ from a pre-training, where we train $f _ { w }$ on the welltrained (maximum 500 epochs with early stopping) supervised VAE’s decoder output to predict the molecules property value. We also initialize the parameters of the encoder/decoder networks with the partially trained supervised VAE model (after 40 epochs for QM9, 100 epochs for ZINC). We do not update $\omega$ and $\phi , \theta$ simultaneously, instead we do an alternate optimization. We update $\omega$ continuously for five epochs while holding $\phi , \theta$ and do the same for updating $\phi , \theta$ . We set the hyper-parameter value $\lambda _ { 1 }$ to 50 and $\lambda _ { 2 }$ to 1. The mini-batch size is set to 300 for QM9 and 100 for ZINC. We use ADAM optimizer with learning rate 0.0001 and pytorch lr-schedular on the validation loss. The general training algorithm is described in below algorithm block1. In our experiment, to train $f _ { \omega }$ , we skipped the second term in step 7, which means we only train $f _ { \omega }$ on the training data but not the newly generated molecules obtained by permuting the property. The reason for this is that, during the training, we found that it is easy for the model to learn to reconstruct but hard to conditionally generate the molecules with given properties while we have no guidance of what the molecules should look like. Further more, some combination of $\mathbf { z }$ and $\mathbf { y }$ are physically not feasible. In this case, when the conditional generation is not good enough yet during the training, we end up fitting $f _ { \omega }$ on the miss represented molecules representations and it makes the optimization harder. + +# Algorithm 1 Training algorithm + +1: Initialize $p _ { \theta } ( \mathbf { x } | \mathbf { z } , \mathbf { y } )$ , $q _ { \phi } ( { \bf z } | { \bf x } )$ , $f _ { w }$ +2: for i=1,2, . . . , N (maximum epoch number) do +3: for ${ \bf j } = 1 , 2 , \ldots , { \bf L }$ , sample a minibatch $D = ( { \bf X } , { \bf Y } ) = \{ { \bf x } _ { i } , { \bf y } _ { i } ) \} _ { i = 1 } ^ { M }$ of M samples do +4: randomly permute the property set $\mathbf { Y }$ to obtain $\mathbf { Y } ^ { \ast }$ and define a label permuted mini +batch $D ^ { * } = ( \mathbf { X } , \mathbf { \bar { Y } } ^ { * } ) = \{ \mathbf { x } _ { i } , \mathbf { y } _ { i } ^ { * } \} _ { i = 1 } ^ { \bar { M } }$ +5: $\begin{array} { l } { { \displaystyle \theta ^ { j } = \theta ^ { j - 1 } - \gamma \big ( - \nabla _ { \theta } \mathcal { L } _ { E L B O } \big ( \theta , \phi , \omega \big ) + \frac { \lambda _ { 1 } } { 2 } \sum _ { \substack { ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \in D ^ { * } } } \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } _ { i } ) } \nabla _ { \theta } \| f _ { \omega } \big ( g _ { \theta } \big ( \mathbf { z } , \mathbf { y } _ { i } \big ) \big ) - \mathbf { y } _ { i } \| ^ { 2 } \big ) } } \\ { { \displaystyle \phi ^ { j } = \phi ^ { j - 1 } - \gamma \big ( - \nabla _ { \phi } \mathcal { L } _ { E L B O } \big ( \theta , \phi , \omega \big ) \big ) } } \\ { { \displaystyle w ^ { j } = w ^ { j - 1 } - \gamma \big ( \frac { \lambda _ { 2 } } { 2 } \sum _ { \substack { ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \in D } } \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } _ { i } ) } \nabla _ { \omega } \| f _ { \omega } \big ( g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { i } ) \big ) - \mathbf { y } _ { i } \| ^ { 2 } + \frac { \lambda _ { 1 } } { 2 } \sum _ { \substack { ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \in D ^ { * } } } \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } _ { i } ) } \nabla _ { \omega } \| f _ { \omega } \big ( g _ { \theta } ( \mathbf { z } , \mathbf { y } _ { i } ) \big ) - \mathbf { y } _ { i } \| ^ { 2 } \big ) } } \end{array}$ +6: +7: + +# 6.3 USE A FIXED PRE-TRAINED PROPERTY PREDICTION FUNCTION + +We also investigate the case where we train the property prediction function $f _ { \omega }$ on the well trained supervised VAE output and keep it fixed during the training of the main model. We give the performance in tables 3, 4. Using a fixed $f _ { \omega }$ , in terms of reconstruction and generation performance, delivers mixed results 3. However, in terms of conditional generation 4, it does perform better than the baselines but worse compared to the case where we also update $f _ { \omega }$ during the learning. + +Table 3: Sup-VAE-1-GRU /Sup-VAE-3-GRU: supervised version of SD-VAE model where y is been feed to only the first layer/all layer decoder. CGD-VAE: our model, conditional generation with disentangling. The result for CVAE and GVAE are taken from the literature, ”-” refers that those measures are not reported. The rest of baseline result is obtained by rerunning (SD-VAE) and editing the original code (Sup-VAE). + +
QM9ZINC
ModelReconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE3.6110.30190.0044.600.70-100
GVAE96.0060.20=80.9053.707.20100
SD-VAE97.8498.4099.2891.9776.2043.50-、
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8495.61100
CGD-VAE-1-GRU98.9695.0392.7789.7467.4617.3882.44100
Sup-VAE-3-GRU97.8197.995.0989.4782.4036.1686.26100
CGD-VAE-3-GRU96.993.898.2989.5581.8037.7898.70100
+ +
Modelz~qq(z)z ~ q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.63310.3835
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.66650.4507
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.29810.0866
Sup-VAE-3-GRU0.36380.1818
CGD-VAE-3-GRU0.37650.1310
+ +Table 4: Correlation between the desired input property and the obtained property when z is sampled from the approximate learned prior, $\mathbf { z } \sim \hat { q } _ { \sigma } ( \mathbf { z } )$ , (conditional generation), and when $\mathbf { z }$ is sampled from the learned posterior given some x, $\mathbf { z } \sim q ( \mathbf { z } | \mathbf { x } )$ , (property transfer). + +# 6.4 VIRTUAL SCREENING + +The figure 8 displays the results of a virtual screening method, where we select from the full dataset five molecules which are structurally similar to $\mathbf { x } _ { A }$ in figure 4 in section 4 and have logP values close to $\mathbf { y } _ { B }$ . As we can see the molecule that our model generates is a new one. + +![](images/a7e485939ef9875fdba8293f1eb15bcfd2c0bfca822f4dd0ac966cbb93a44c64.jpg) +Figure 8: Molecules selected with virtual screening over the full dataset in a manner that they are structurally similar to the A molecule of figure 4 while they have a logP value which is close to the logP value of the B molecule also of figure 4 + +# 6.5 METRICS AS FUNCTION OF $\mathbf { y } - \mathbf { y } ^ { \prime }$ + +The validity and novelty were mainly used to assess purely the generative model performance. However, it is also interesting to see if the model capable of generating valid and novel molecules if we start from an existing molecules and drift away from the original label, i.e., if we get $\mathbf { z }$ from $q ( \mathbf { z } | \mathbf { x } )$ , and then compare different values from $p ( \mathbf { \bar { x } } | \mathbf { z } , \mathbf { y } ^ { \prime } )$ as $\mathbf { y } ^ { \prime }$ moves far from y. We randomly sample a molecule $\mathbf { x }$ whose LogP is y from the test set, then sample $1 0 \textbf { z }$ from $q ( \mathbf { z } | \mathbf { x } )$ . For each such $\mathbf { z }$ we couple it with a $\mathbf { y } ^ { \prime }$ that is different that $\mathbf { y }$ , and sample 10 molecules from $p ( \mathbf { x } | \mathbf { z } , \mathbf { y } ^ { \prime } )$ . Eventually, for each such $\mathbf { y } ^ { \prime }$ , starting from original molecule $\mathbf { x }$ , we generated 100 molecules, and we report the validity, uniqueness and novelty as a function of $\mathbf { y } - \mathbf { y } ^ { \prime }$ . The figure 9 displays result of repeating above process for 20 randomly sampled $\displaystyle ( \mathbf { x } , \mathbf { y } )$ along 100 grid points for $\mathbf { y } - \mathbf { y } ^ { \prime }$ . The result confirms that on a big data set, conditional generative models uniqueness, validity and novelty performance is not affected by the size of the modification done on the property. However, on a small dataset, uniqueness is not affected. As expected, novelty increases as the properties modification size increases and validity drops slightly as the property modification size increase. + +![](images/05561149d0652633c285e17687e4cc57ccdd333927040429da49e149095794c1.jpg) +Figure 9: CGD-VAE-3-GRU model validity, novelty, uniqueness performance on property transfer task as a function of $\mathbf { y } - \mathbf { y } ^ { \prime }$ on QM9 dataset. + +![](images/7e913f7fbf7e5ee958020cae4f83630081d3c4db545c8f072beab85a15a4684d.jpg) +Figure 10: CGD-VAE-3-GRU model validity, novelty, uniqueness performance on property transfer task as a function of $\mathbf { y } - \mathbf { y } ^ { \prime }$ on ZINC dataset. + +With our model, during the generation, we observe that sampling from the approximated marginal posterior improves generation performance when compered to sampling from the prior. Here we investigate if this findings holds for other baseline models or not. We explored the behavior of baselines when the $\mathbf { z }$ is sampled from the approximate marginal posterior, we observe that for CVAE and GVAE the validity did not change (as the ${ \hat { q } } ( \mathbf { z } )$ and $p ( \mathbf { z } )$ are essentially identical); for the SD-VAE the validity increases (Table 5). + +SDVAE +Table 5: Baseline model performance on ZINC + +
Valid%Unique %Novel %
z ~qσ(z)56.0580.69100
z ~p(z)43.5082.26100
+ +# 6.7 MOLECULE PROPERTY OPTIMIZATION + +Figure 11 is the visualization of the generated molecules from property optimization task, given in figure 6 section 4. The molecules are generated by increasing the logP of a given molecule, i.e. we hold $\mathbf { z }$ fixed and increase the y value. Among the all generated molecules, 19 of them are unique. + +![](images/cc585b7043d323b6f84b9809b8adc19b851ad1196e606dc71b64fabbf987c156.jpg) +Figure 11: Molecules generated with an increasing logP \ No newline at end of file diff --git a/parse/train/BkxthxHYvr/BkxthxHYvr_content_list.json b/parse/train/BkxthxHYvr/BkxthxHYvr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..75ce77ac9b71efbeb95d121c9c254c90a5f09ca3 --- /dev/null +++ b/parse/train/BkxthxHYvr/BkxthxHYvr_content_list.json @@ -0,0 +1,1859 @@ +[ + { + "type": "text", + "text": "CONDITIONAL GENERATION OF MOLECULES FROM DISENTANGLED REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 101, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Though machine learning approaches have shown great success in estimating properties of small molecules, the inverse problem of generating molecules with desired properties remains challenging. This difficulty is in part because the set of molecules which have a given property is structurally very diverse. Treating this inverse problem as a conditional distribution estimation task, we draw upon work in learning disentangled representations to learn a conditional distribution over molecules given a desired property, where the molecular structure is encoded in a continuous latent random variable. By including property information as an input factor independent from the structure representation, one can perform conditional molecule generation via a “style transfer” process, in which we explicitly set the property to a desired value at generation time. In contrast to existing approaches, we disentangle the latent factors from the property factors using a regularization term which constrains the generated molecules to have the property provided to the generation network, no matter how the latent factor changes. ", + "bbox": [ + 233, + 270, + 766, + 463 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 497, + 336, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Conditional molecule generation is far from being solved. The main challenge is the enormous and discrete nature of the molecules space and the fact that molecule properties are highly sensitive to molecular structure (Kirkpatrick & Ellis, 2004). Approaches to conditional generation are typically two-step, either using a model or genetic algorithm to generate candidates which are later filtered, or learning a continuous embedding of the discrete molecules and optimizing in a real-valued representation space. The former is computationally expensive, the latter performs conditional generation only very obliquely. ", + "bbox": [ + 174, + 530, + 825, + 628 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose a conditional generative model that produces candidate molecules which targeting a desired property in a single step. This approach builds on work in structured deep generative models (Kingma et al., 2014; Siddharth et al., 2017), which aim to learn a disentangled representation that factors into observed properties we want to control for, and latent factors that account for the remaining features which are either hard to annotate or irrelevant to the properties we wish to optimize. ", + "bbox": [ + 174, + 636, + 825, + 718 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We derive a regularizer for supervised variational autoencoders which exploits property information that we provide as supervision, ensuring that produced molecules adhere to target properties they are conditioned on. We demonstrate the ability of our model to perform accurate conditional molecule generation and a sort of “style transfer” on molecules, where a latent representation for a single molecule can have its target properties perturbed independently of its learnt structural characteristics, allowing direct and efficient generation of candidates for local optimization of molecules. ", + "bbox": [ + 174, + 726, + 825, + 809 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 833, + 326, + 849 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Molecule discovery tasks come in two flavors. Global optimization seeks to find molecules that have a particular target property. Local optimization starts from some initial molecule and searches for molecules which have a desired property while not straying too far from the prototype. There is some overlap in methods used in the two approaches. ", + "bbox": [ + 174, + 867, + 823, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2.1 DEEP GENERATIVE MODELS FOR MOLECULES ", + "text_level": 1, + "bbox": [ + 178, + 104, + 527, + 117 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Virtual screening methods start from a large database of possible molecules and retain the promising ones (Eckert & Bajorath, 2007), as measured by some quality function $f ( \\cdot )$ . Machine learning approaches expand on this by dynamically generating additional candidate molecules; Segler et al. (2017) uses a stacked LSTM to produce large numbers of novel molecules which have similar characteristics to an existing database. ", + "bbox": [ + 174, + 128, + 825, + 199 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "For properties which are expensive to evaluate, generating large sets of candidate molecules is not particularly useful. More sample-efficient global search can be achieved using Bayesian optimization methods, which use a generative model with a latent space that functions as a continuous representation of molecules (Gomez-Bombarelli et al., 2016; Kusner et al., 2017). Optimization is then ´ carried out over this continuous representation space to find candidates which are expected to have the desired property. Local gradient-based search can also be applied on continuous latent spaces to optimize the latent representation with respect to a target property (Jin et al., 2018; Liu et al., 2018). ", + "bbox": [ + 174, + 207, + 825, + 304 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A challenge for these latent variable models is to reliably produce valid molecules. Character variational autoencoders (CVAEs) (Gomez-Bombarelli et al., 2016) generate molecules one character ´ at a time, and are prone to syntactic and semantic errors; the grammar-based variational autoencoder (GVAE) (Kusner et al., 2017) and syntax-directed variational autoencoder (SD-VAE) (Dai et al., 2018) instead operate in the space of context-free and attribute grammars, respectively, to ensure syntactic validity. Other work generative models that operates on graph representations (Simonovsky & Komodakis, 2018; De Cao & Kipf, 2018; Jin et al., 2018; You et al., 2018; Liu et al., 2018), largely improving the ability to generate valid molecules. ", + "bbox": [ + 173, + 310, + 825, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Suppose we are given a training set of pairs $\\mathcal { D } = \\{ ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\} , i = 1 , \\ldots , N .$ , where x corresponds to molecules and y represents a value of some properties of the molecule $\\mathbf { x }$ . Assume the molecules represent an i.i.d. sample from some unknown distribution $\\tilde { p } ( { \\bf x } )$ , which assigns high probability to molecules believed to be useful for a given task. Aside from Segler et al. (2017), which has no latent space and thus directly trains via maximum likelihood, these latent variable models are trained by optimizing a standard ELBO objective for variational autoencoders (?). This entails learning a stochastic encoder $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ which maps molecules into a latent space, and a stochastic decoder $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ for reconstructing molecules, by maximizing ", + "bbox": [ + 173, + 428, + 825, + 541 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/515df8a88edf91025448b21438c14e0e0054dae72cef08ba7e1526e1e611922b.jpg", + "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) = \\sum _ { i = 1 } ^ { N } \\bigg \\{ \\mathbb { E } _ { q _ { \\boldsymbol { \\phi } } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) } [ \\log p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { i } | \\mathbf { z } _ { i } ) ] - D _ { K L } ( q _ { \\boldsymbol { \\phi } } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) | | p ( \\mathbf { z } _ { i } ) ) \\bigg \\} .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 544, + 730, + 587 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Notably, the objective is not a function of y: most existing generative models with latent variables do not perform direct conditional generation, and approaches for targeted molecule discovery are bolted on to the learnt model. Some, e.g. Kusner et al. (2017), are trained in an “unsupervised” manner, agnostic to any property which later may need to be optimized. Others, e.g. Gomez-Bombarelli ´ et al. (2016); Liu et al. (2018), train the autoencoder jointly alongside a function to predict y from z, hoping to guide the latent space to be also good for predicting the desired property. A recent exception is Assouel et al. (2018), which learns a deterministic autoencoder where the decoder takes the latent code and the desired property as input, using a mutual information term in training to steer the model towards generating molecules whose target properties match the input. Guimaraes et al. (2017); De Cao & Kipf (2018); You et al. (2018) instead learn generation models optimized towards specific metrics, such as drug-likeliness and solubility; the major downside is that these models must be retrained each time for a new property. In contrast, the autoencoder-based methods can be re-used to optimize towards any particular value of the property. ", + "bbox": [ + 173, + 589, + 825, + 770 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 STYLE TRANSFER WITH SUPERVISED VAES ", + "text_level": 1, + "bbox": [ + 174, + 786, + 516, + 800 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While the latent representations learned through standard VAE models perform well on the task of molecule reconstruction they do not necessarily provide interpretable factorised representations. A disentangled representation gives us additional control on the molecule generation process, allowing us to modify a single property leaving the remaining unaffected (Bengio et al., 2013a). In many cases important variation in the data is easy to annotate. For example in the case of molecule datasets we have access to different functional descriptors of the molecules obtained by chemoinformatics software such as RDKit (Landrum). Particularly useful to us here are supervised methods for learning disentangled representations (Kingma et al., 2014; Siddharth et al., 2017). These are distinct from unsupervised disentangling approaches such as InfoGAN (Chen et al., 2016) or $\\beta$ -VAE (Higgins et al., 2017), which encourages the latent factor to learn a disentangled representation by modifying the objective to promote component independence. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We will learn representations that specifically disentangle molecular properties of interest which we may later want to modify. Kingma et al. (2014) demonstrates how disentangling can be used to take two MNIST images of different digits, written in different styles, and independently change the digit while holding the style constant. An analogous operation on molecules would involve holding the physical structure of a molecule (its “style”) relatively fixed while modifying a salient property. Unlike (say) the style transfer example for the MNIST digits, the conditional distribution of molecules with a particular value of properties might be very diverse; for example, the QED score attempts to measure the drug-likeness of a molecule, and the set of molecules generated at high values of this score would hopefully have high probability on a large, varied set of molecules. An essential challenge here is that the property only provides a very weak signal as to the overall structure of the molecule. To account for this diversity, we model the conditional distribution with a latent variable $\\mathbf { z }$ , such that $\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } ) = \\int p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } ) p ( \\mathbf { \\bar { z } } ) d \\mathbf { z } } \\end{array}$ . ", + "bbox": [ + 173, + 152, + 826, + 320 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Disentangling the latent code $\\mathbf { z }$ from the property y enables style transfer. This is done by taking an initial $\\mathbf { x }$ , computing the posterior over the latent variable $\\mathbf { z }$ , and then generating a new $\\mathbf { x } ^ { \\prime }$ with the property modified to have a target value $\\mathbf { y } ^ { \\prime }$ , with $p _ { \\theta } ( \\mathbf { x } ^ { \\prime } | \\mathbf { y } ^ { \\prime } , \\mathbf { x } ) =$ $\\begin{array} { r } { \\int p _ { \\theta } ( \\mathbf { x } ^ { \\prime } | \\mathbf { y } ^ { \\prime } , \\mathbf { z } ) p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } ) d \\mathbf { z } } \\end{array}$ . ", + "bbox": [ + 174, + 325, + 614, + 397 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/d6fdd5f7e997772d80bd45c81d2d6a587e1202c9b2e03df60b1f8ea2b7c46b4e.jpg", + "image_caption": [ + "Figure 1: A demonstration of style transfer " + ], + "image_footnote": [], + "bbox": [ + 632, + 353, + 823, + 441 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Concretely, this involves fitting a joint generative model of the form $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } , \\mathbf { y } , \\mathbf { z } ) = p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { y } , \\mathbf { z } ) p ( \\mathbf { y } ) \\bar { p ( \\mathbf { z } ) }$ , in which $\\mathbf { y }$ and $\\mathbf { z }$ are independent under the prior, and we assume a unit multivariate normal prior $p ( \\mathbf { z } )$ . To infer the latent variable $\\mathbf { z }$ we will use a variational distribution $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ , which takes the form of a multivariate normal distribution with parameters a nonlinear function of $\\mathbf { x }$ , to approximate the true posterior $p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } , \\mathbf { y } )$ . This objective function ", + "bbox": [ + 173, + 402, + 616, + 501 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/19adcf923cd42af20f25625075fd5a5e386342b2aa0bc26bcb53f0f419d51259.jpg", + "text": "$$\n\\mathcal { L } _ { E L B O } ( \\theta , \\phi ) = \\sum _ { i = 1 } ^ { N } \\left\\{ \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) } [ \\log p _ { \\theta } ( \\mathbf { x } _ { i } | \\mathbf { y } _ { i } , \\mathbf { z } _ { i } ) ] - D _ { K L } ( q _ { \\phi } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) | | p ( \\mathbf { z } _ { i } ) ) \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 503, + 759, + 546 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "corresponds to learning a supervised VAE (Kingma et al., 2014), and represents a fairly na¨ıve approach to modeling a conditional distribution. ", + "bbox": [ + 173, + 546, + 826, + 575 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 CONDITIONAL GENERATION BY DISENTANGLING ", + "text_level": 1, + "bbox": [ + 174, + 594, + 611, + 611 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Maximizing this conditional ELBO in Eq (2) will likely yield good reconstructions of molecules from an embedding $\\mathbf { z }$ (alongside the true property y), but for properties which only weakly inform the generative model there is nothing to enforce that the variable $\\mathbf { y }$ actually directly has an effect on the generative process. Since the value $\\mathbf { y }$ is something we know is a derived property of the molecule $\\mathbf { x }$ , it is completely possible for all information about y to also be encoded in the representation $\\mathbf { z }$ , in which case there is no guarantee that the learnt likelihood $p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } , \\mathbf { z } )$ actually takes into account the value of $\\mathbf { y }$ — in fact, we know it is possible to fit variational autoencoders where the decoder simply has the form $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ — and we are relying on the utility of $\\mathbf { y }$ in reconstructions to see any sort of disentangling effect. ", + "bbox": [ + 173, + 623, + 825, + 751 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 CONSTRAINED ELBO ", + "text_level": 1, + "bbox": [ + 176, + 766, + 369, + 781 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the case of conditional generation of molecules, we often have access to some oracle function $f$ (possibly non-differentiable) which for any given $\\mathbf { x }$ outputs a property estimate $\\mathbf { y }$ , for instance, the chemoinformatics software RDKit (Landrum). Since for conditional generation our ultimate goal is to generate a molecule $\\mathbf { x }$ for any given target property $\\mathbf { y } _ { 0 }$ , which then actually has $f ( \\mathbf { x } ) = \\bar { \\mathbf { y } } _ { 0 }$ , we can reframe the problem by introducing hard constraints on the generated values, i.e. if restricting to values of $\\mathbf { y }$ in the training set, ", + "bbox": [ + 173, + 791, + 825, + 876 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9d3d6f8ab2be0b3d36c74b23d206b7d8770334fdea9b7551ea7a51f7f2b7b768.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\theta , \\phi } { \\operatorname* { m a x } } \\mathcal { L } _ { E L B O } ( \\theta , \\phi ) } \\\\ & { \\mathrm { s u b j e c t ~ t o ~ } \\mathbb { E } _ { \\mathbf { x } \\sim p ( \\mathbf { x } \\mid \\mathbf { y } _ { i } ) } [ \\mathbb { I } [ f ( \\mathbf { x } ) = \\mathbf { y } _ { i } ] ] = 1 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 877, + 632, + 922 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/11c4f6bc728cef0a1f7156261ab4dd96723193136e4805467ad1b99711c5ae7c.jpg", + "image_caption": [ + "Figure 2: Setting and modeling pipeline for conditional generation of molecules, with supervision provided via an external property prediction oracle. Red lines correspond to non-differentiable components, including both a potentially complex sampling process and the property prediction itself. The blue dashed line corresponds to the approximate property predictor, which aims to predict the expected value of the property from a continuous relaxation, marginalized over the sampling process. " + ], + "image_footnote": [], + "bbox": [ + 178, + 99, + 823, + 242 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "for all $i = 1 , \\ldots , N$ . This is an unreasonably hard constraint, unlikely to be satisfied by any distribution other than one which simply places a point mass on the single training $\\mathbf { x } _ { i }$ associated with $\\mathbf { y } _ { i }$ , but we can relax it by considering that (unlike the molecular space $\\mathbf { x }$ ) the property space $\\mathbf { y }$ is typically smooth, as many properties are continuous-valued and correspond to a human-interpretable scale. Following Ma et al. (2018) and Hu et al. (2017), we reframe the constraint as a soft penalty on the ELBO, ", + "bbox": [ + 173, + 349, + 825, + 435 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5fab4abe27c3d00185148805fef678d755dee2fa1728393bf860fe90413109a5.jpg", + "text": "$$\n\\mathcal { L } ( \\theta , \\phi ) = \\mathcal { L } _ { E L B O } ( \\theta , \\phi ) - \\frac { \\lambda _ { 1 } } { 2 } \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { i } ) } \\Vert f ( \\hat { \\mathbf { x } } ) - \\mathbf { y } _ { i } \\Vert ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 297, + 441, + 700, + 484 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "so that they are consistent with the property prediction, i.e., as we have an oracle function $f$ which enable us to access the property of any generated data, we can explicitly add a soft constraint to our loss function to provide explicit guidance for the generative model such that $f ( \\hat { \\mathbf x } ) = \\mathbf y$ . This constraint is expected to hold for any pair $\\displaystyle ( \\mathbf { x } , \\mathbf { y } )$ we may happen to come across, not just those in the training data. We also show optimizing the relaxed constraint is equivalent to maximizing mutual information with the target $\\mathbf { y } _ { i }$ and generated molecule $\\hat { \\bf x }$ ; for details see appendix Section 6.1. ", + "bbox": [ + 173, + 491, + 825, + 575 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 APPROXIMATING THE PROPERTY PREDICTOR", + "text_level": 1, + "bbox": [ + 174, + 594, + 526, + 608 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Introducing the regularizer as in Eq. (3) implicitly guides the reconstruction to take into account the property information, such that the reconstructed data should exhibit properties which match the input properties it is conditioned on. However, existing implementations of $f$ are often non-differentiable or CPU-bound, and $\\hat { \\bf x }$ are discrete samples from a categorical distribution, all of which means the gradient of the regularizer can’t flow back to the generator. This is outlined in Figure 2. To enable the gradient based methods on GPUs during training and avoid discrete sampling, one approach would be to first fit a differentiable approximation to $f$ , and then use either a Gumbel-softmax relaxation (Jang et al., 2016) or tricks like a “straight-through” estimator (Bengio et al., 2013b) as a continuous approximation for the discrete samples. Instead, we propose bypassing the discrete sampling step entirely and learning a function $f _ { \\omega }$ that can map from a learned representation of the molecules directly to molecules property (Hu et al., 2017). ", + "bbox": [ + 173, + 618, + 825, + 773 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To do this, we take as input the last hidden layer of the decoder network which parameterizes $p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } )$ , denoting this deterministic transformation as $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } )$ . For the grammar VAE and the syntax-directed VAE, this last layer $\\mathbf { h } = g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } )$ is the output of a recurrent layer that generates logits corresponding to unmasked and unnormalized log probabilities for each character at each position in the string; see Kusner et al. (2017) and Dai et al. (2018) for details on the implementation of the somewhat complex sampling process in the decoder. Ideally, $f _ { \\omega }$ would estimate the property distribution obtained by marginalizing out the discrete sampling step, with ", + "bbox": [ + 173, + 779, + 825, + 877 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5ed239fd633ddf54d4b7a1e159489adf8dd099640a506d32c340e273e987628c.jpg", + "text": "$$\nf _ { \\omega } ( \\mathbf { h } \\equiv g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { 0 } ) ) \\approx \\mathbb { E } _ { p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } _ { 0 } ) } [ f ( \\mathbf { x } ) ] ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 885, + 630, + 904 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where we condition on $\\mathbf { z }$ , and $\\mathbf { y } _ { 0 }$ refers to an arbitrary input target property. ", + "bbox": [ + 174, + 909, + 666, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assuming the approximation in Eq. (4), we have ", + "bbox": [ + 174, + 103, + 491, + 118 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2c1d061a373428519f20b25a02e5965063a9736bc1fdea5a507d64bfcef34e5d.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { y } _ { i } ) } \\| f ( \\hat { \\mathbf { x } } ) - \\mathbf { y } _ { i } \\| ^ { 2 } = \\mathbb { E } _ { p ( \\mathbf { z } ) } \\left[ \\mathbb { E } _ { p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { y } _ { i } , \\mathbf { z } ) } \\| f ( \\hat { \\mathbf { x } } ) - \\mathbf { y } _ { i } \\| ^ { 2 } \\right] \\approx \\mathbb { E } _ { p ( \\mathbf { z } ) } \\| f _ { \\omega } ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) ) - \\mathbf { y } _ { i } \\| ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 199, + 122, + 797, + 142 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "an expectation over a real-valued variable which does not depend on any of the parameters we are estimating, meaning we can use a simple path estimate of the gradient with respect to $\\theta , \\omega$ by exchanging the gradient with the expectation. We thus define a regularization term ", + "bbox": [ + 174, + 147, + 825, + 189 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e46441682ad2adc01467e4ee612ea4f93bd7a475626f79c4ad3a6bd5d8ff42cb.jpg", + "text": "$$\n\\mathcal { L } _ { d i s e n t } ( \\theta , \\omega ) = \\frac { \\lambda _ { 1 } } { 2 } \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { p ( \\mathbf { z } ) } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 325, + 195, + 671, + 238 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "which can be used as a drop-in replacement for the non-differentiable penalty term in Eq. (3), yielding a candidate objective function ", + "bbox": [ + 171, + 242, + 823, + 272 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a9bc09229d3d3410c05de845a0f50287eeb9aa745a3439d6933f023664f4608d.jpg", + "text": "$$\n\\mathcal { L } ( \\theta , \\phi ) \\approx \\hat { \\mathcal { L } } _ { \\omega } ( \\theta , \\phi ) = \\mathcal { L } _ { E L B O } ( \\theta , \\phi ) - \\mathcal { L } _ { d i s e n t } ( \\theta , \\omega )\n$$", + "text_format": "latex", + "bbox": [ + 323, + 277, + 674, + 297 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 LEARNING THE PROPERTY ESTIMATOR JOINTLY WITH GENERATIVE MODEL", + "bbox": [ + 174, + 313, + 732, + 327 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "While one could imagine attempting to learn $f _ { \\omega }$ jointly with $\\phi , \\theta$ by direct optimization of Eq. (6), in practice this is very unstable, as values of $g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } _ { i } )$ early in training may correspond to very poor generated molecules $\\hat { \\mathbf { x } } _ { i }$ which may not have properties at all similar to $\\mathbf { y } _ { i }$ . This can be sidestepped by training the property estimator jointly as part of an extended generative model on [x, y]. ", + "bbox": [ + 174, + 337, + 826, + 395 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We note that the property estimator $f _ { \\omega }$ parameterizes a probability distribution $p _ { \\omega } ( f ( \\mathbf { x } ) | \\mathbf { z } , \\mathbf { y } _ { 0 } )$ , where $\\mathbf { x } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } _ { 0 } )$ and f is the oracle function that $f ( \\mathbf { x } ) = \\mathbf { y }$ . With a Gaussian distribution over the error, we can consider ", + "bbox": [ + 174, + 401, + 823, + 443 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f3b9079cadea7a9df088ce28fe8d191b08b86758e85349a90b14eca103e27037.jpg", + "text": "$$\np _ { \\omega } ( f ( \\mathbf { x } ) \\vert \\mathbf { z } , \\mathbf { y } _ { 0 } ) = \\mathcal { N } ( f ( \\mathbf { x } ) \\vert f _ { \\omega } ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { 0 } ) ) , \\lambda _ { 2 } ^ { - 1 } \\mathbf { I } )\n$$", + "text_format": "latex", + "bbox": [ + 336, + 448, + 661, + 467 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "for small, fixed $\\lambda _ { 2 }$ . Therefore, we propose defining a new ELBO based on a joint autoencoder for $\\{ f ( \\mathbf { x } _ { i } ) , \\mathbf { y } _ { i } ) \\}$ , albeit with a factorization such that the input $\\mathbf { y } _ { i }$ bypasses the encoder and is passed directly into the decoder, with a joint likelihood ", + "bbox": [ + 174, + 472, + 825, + 515 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/cfa05656a242d1dc2ac18e6c7ffd00b4885814eb837aa125aff4e1676ad39d6f.jpg", + "text": "$$\np _ { \\{ \\theta , \\omega \\} } ( \\mathbf { x } _ { i } , f ( \\mathbf { x } _ { i } ) | \\mathbf { z } _ { i } , \\mathbf { y } _ { i } ) = p _ { \\omega } ( f ( \\mathbf { x } _ { i } ) | \\mathbf { z } _ { i } , \\mathbf { y } _ { i } ) p _ { \\theta } ( \\mathbf { x } _ { i } | \\mathbf { z } _ { i } , \\mathbf { y } _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 520, + 689, + 539 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This yields a joint ELBO for the training set of ", + "bbox": [ + 174, + 542, + 483, + 558 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4e5ecb79e686fed417c19f489830af7e9b3aee9af952354f5749e92d20256fd4.jpg", + "text": "$$\n\\mathcal { L } _ { E L B O } ( \\omega , \\theta , \\phi ) = \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\left[ \\log \\frac { p _ { \\omega } ( f ( \\mathbf { x } _ { i } ) | \\mathbf { z } , \\mathbf { y } _ { i } ) p _ { \\theta } ( \\mathbf { x } _ { i } | \\mathbf { z } , \\mathbf { y } _ { i } ) p ( \\mathbf { z } ) } { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 256, + 563, + 741, + 607 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that we can rewrite this ELBO as a function of the previous one, with ", + "bbox": [ + 173, + 611, + 665, + 627 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d42c69718c2bfc4ea19cb72b2db20cf4880d9a13e3412b0b80eaf9b780f40e24.jpg", + "text": "$$\n\\mathcal { L } _ { E L B O } ( \\boldsymbol { \\omega } , \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) = \\mathcal { L } _ { E L B O } ( \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) - \\frac { \\lambda _ { 2 } } { 2 } \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { \\boldsymbol { q } _ { \\boldsymbol { \\phi } } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\| f _ { \\boldsymbol { \\omega } } \\big ( g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| _ { 2 } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 246, + 632, + 750, + 676 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where we also see that $\\mathcal { L } _ { E L B O } ( \\omega , \\theta , \\phi ) \\leq \\mathcal { L } _ { E L B O } ( \\theta , \\phi )$ , allowing us to define an objective ", + "bbox": [ + 171, + 680, + 772, + 698 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/69eb19c7108316d23bb2e4f1035e85db9beaa690c0bc3cb5bbba0a710a5cceaa.jpg", + "text": "$$\n\\hat { \\mathcal { L } } ( \\omega , \\theta , \\phi ) = \\mathcal { L } _ { E L B O } ( \\omega , \\theta , \\phi ) - \\mathcal { L } _ { d i s e n t } ( \\theta , \\omega ) ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 703, + 653, + 722 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "which is a lower bound on Eq. (6). Notice the two terms we have added to the original ELBO are quite similar, differing only in choice of distribution: for learning $f _ { \\omega }$ , we wish to use values of $\\mathbf { z }$ simulated form the approximate posterior $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ , whereas for enforcing a constraint across all possible generations we simulate $\\mathbf { z }$ from the prior $p ( \\mathbf { z } )$ . ", + "bbox": [ + 173, + 727, + 825, + 785 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 GRADIENT ESTIMATION ", + "text_level": 1, + "bbox": [ + 174, + 800, + 380, + 814 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As the regularizer $\\mathcal { L } _ { d i s e n t } ( \\theta , \\omega )$ encourages disentangling by constraining the molecules generated from $\\mathbf { y } _ { i }$ to have property $\\mathbf { y } _ { i }$ no matter what value $\\mathbf { z }$ takes, we found that it does not necessarily evaluate at meaningful values of $\\mathbf { z }$ when sampled randomly from $p ( \\mathbf { z } )$ . This roughly corresponds to the notion that not all combinations of “style” and property are physically attainable; ideally for style transfer we would like the generated molecule to stay “close” in structure to the original molecule that we intended to modify. When estimating (gradients of) the soft constraint term $\\mathcal { L } _ { d i s e n t } ( \\theta , \\omega )$ , we found it advantageous to use samples of $\\mathbf { z }$ which correspond to encodings of actual data points, as opposed to random samples from the prior. We approximate expectations with respect to $p ( \\mathbf { x } )$ by looking at the so-called marginal posterior; we note that ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/576dcae7e42a624b914823986f1145cc9ba49a93363af77cd37a03e58a6d5a38.jpg", + "table_caption": [ + "Table 1: Reconstruction performance and generation quality (Valid, Unique, Novel). " + ], + "table_footnote": [], + "table_body": "
ModelQM9ZINC
Reconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE Gómez-Bombarelli et al. (2016)3.6110.30=90.044.60.70·100
GVAE Kusner et al. (2017)96.0060.20=80.9053.707.20=100
SD-VAE Dai et al.(2018)97.8498.4099.2891.9776.2043.50--
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8494.61100
CGD-VAE-1-GRU99.2795.6193.6587.8788.6429.0099.24100
Sup-VAE-3-GRU97.8197.9095.0989.4782.4036.1686.26100
CGD-VAE-3-GRU99.3197,8098.7796.2181.8037.7898.75100
", + "bbox": [ + 173, + 101, + 825, + 196 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 246, + 825, + 275 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/a4545b144594a1bd9864b22a482b9947207e768c0be2785b7d44bf6b765493d9.jpg", + "text": "$$\np ( \\mathbf { z } ) = \\int p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } ) p _ { \\theta } ( \\mathbf { x } ) d \\mathbf { x } \\approx \\frac { 1 } { N } \\sum _ { j } p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } _ { j } ) \\approx \\frac { 1 } { N } \\sum _ { j } q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { j } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 279, + 279, + 718, + 318 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where the first approximation uses the empirical data distribution as an approximation to the model marginal $p _ { \\theta } ( \\mathbf { x } )$ , and the second uses our variational posterior approximation $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ . We define this quantity as $\\begin{array} { r } { q ( \\mathbf { z } ) = \\frac { 1 } { N } \\sum _ { j } q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { j } ) } \\end{array}$ , a mixture of Gaussians, which we can sample from by drawing random values from our dataset and then drawing from their encoding distributions. ", + "bbox": [ + 174, + 320, + 825, + 381 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "When we use this in estimating gradients of the soft constraint, we can use samples from the same minibatch, exactly corresponding to a property transfer task. That is, for any particular $y _ { i }$ in the dataset, we can estimate ", + "bbox": [ + 173, + 387, + 825, + 429 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/cf0c9755535c174c4171a2f8795fd29f320b8e84873c9b4523b27e793e43e0b7.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { p ( \\mathbf { z } ) } \\nabla _ { \\theta , \\omega } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } \\approx \\mathbb { E } _ { q ( \\mathbf { z } _ { j } | \\mathbf { x } _ { j } ) } \\nabla _ { \\theta , \\omega } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 431, + 741, + 452 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "for any uniformly randomly sampled $j \\neq i$ . By sampling $\\mathbf { z } _ { j }$ from $q ( \\mathbf { z } _ { j } | \\mathbf { x } _ { j } )$ where $j \\neq i$ , we make sure that all the label information decoder is receiving comes from the actual $\\mathbf { y } _ { i }$ that is feed to the decoder and $\\mathbf { z } _ { j }$ does not include any information about label. This can be evaluated easily by simply evaluating the penalty term of Eq. (10) twice per minibatch; once as in Eq. (10), and once to approximate $\\mathcal { L } _ { d i s e n t } ( \\theta , \\omega )$ by permuting the properties in the minibatch to be assigned to incorrect molecules. We detail the training algorithm in Section 6.2 of the appendix. ", + "bbox": [ + 174, + 455, + 825, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 559, + 326, + 574 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We experiment with the QM9 dataset (Ramakrishnan et al., 2014), that contains $1 3 4 \\mathbf { k }$ molecules with up to 9 heavy atoms, and the ZINC dataset (Sterling & Irwin, 2015) containing $2 5 0 \\mathrm { k }$ druglike molecules. Our goal here is two-fold: we would like to understand (1) whether a supervised variational autoencoder is capable of learning suitable conditional distributions over molecules, and (2) to what extent this task is assisted by the additional regularization term corresponding to the soft constraint. ", + "bbox": [ + 173, + 589, + 825, + 672 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We represent molecules using the one-hot encoding of their SMILES production rules (Kusner et al., 2017) and add a semantic constraint (Dai et al., 2018) on the decoder network to avoid generating syntactically correct but semantically invalid molecules. We use 80 production rules to describe molecules and set the maximum SMILES sequence length to 100 for the QM9 dataset and 278 for the Zinc dataset. We experiment with the logP property of the molecules (Wildman & Crippen, 1999). We use the same encoder and decoder network structure as Dai et al. (2018) with the only difference that our decoder takes as input the concatenation of $\\mathbf { y } , \\mathbf { z }$ . We give the details of the architecture in the appendix section 6.2. ", + "bbox": [ + 173, + 680, + 825, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate the reconstruction accuracy and the quality of the molecules generated by our method, which we denote by CGD-VAE (conditional generation with disentangling) and compare against CVAE (Gomez-Bombarelli et al., 2016), GVAE (Kusner et al., 2017), and SD-VAE (Dai et al., 2018). ´ We explore its conditional generation performance in two settings: controlling only the property value and controlling both the property value and the molecule structure to what can be seen as property transfer. We took the results of CVAE, GVAE from the literature. For SD-VAE we used the authors code with the default values to generate results for QM9 since these were not available for QM9. We also implemented supervised VAE versions of SD-VAE which we denote Sup-VAE-X-GRU $\\mathrm { ( X \\in \\{ 1 , 3 \\} }$ , denotes GRU layers) and which can do conditional generation. ", + "bbox": [ + 173, + 797, + 826, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/d83af7d84ad5e4b1f43f6ce0ea2358c91c263cc5eae0206da718a5bc7d37f5ab.jpg", + "image_caption": [ + "Figure 3: Conditional generation given the desired $\\mathrm { l o g P { = } { - } 0 . 5 7 5 9 }$ , row molecules have a logP within a $15 \\%$ range of the desired one. " + ], + "image_footnote": [], + "bbox": [ + 179, + 108, + 524, + 207 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e0f8c116c2f73cfe17b82b2094f1aeb8977e80722189f751d7235bfe6e3e375d.jpg", + "image_caption": [ + "Figure 4: Property transfer " + ], + "image_footnote": [], + "bbox": [ + 552, + 99, + 779, + 226 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Before proceeding with the experiments we will give some additional details on how we do conditional generation from $p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { 0 } )$ given the target property $\\mathbf { y } _ { 0 }$ . Instead of marginalizing over the prior marginal inference distribution mass of the dataset is in the late $p ( z )$ , we mirror the approach taken during training and integrate over an approximation to the $\\begin{array} { r } { q _ { \\phi } ( \\mathbf { z } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\end{array}$ which better characterizes where the large and we do not wish to keep the $N$ entire dataset available at test time, we approximate $q _ { \\phi } ( \\mathbf { z } )$ with an isotropic Gaussian distribution $\\hat { q } _ { \\sigma } ( \\mathbf { z } ) = \\mathcal { N } ( \\mathbf { z } | \\mathbf { 0 } , \\sigma ^ { 2 } \\mathbf { I } )$ . We estimate $\\sigma$ for each model by Monte Carlo samples from $q _ { \\phi } ( \\mathbf { z } )$ . For the supervised VAE without the soft constraint regularizer this yields 0.053 for QM9 and 0.118 for ZINC. For our model with the soft constraint we get 0.0354 for QM9 and 0.096 for ZINC. We do conditional generation of $\\mathbf { x }$ given $\\mathbf { y } _ { 0 }$ by sampling from $\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { 0 } ) = \\int \\hat { q } _ { \\sigma } ( \\mathbf { z } ) p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } _ { 0 } ) d \\mathbf { z } } \\end{array}$ . ", + "bbox": [ + 173, + 292, + 825, + 438 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate reconstruction performance in terms of the correctly reconstructed molecules on test sets of size 10k for QM9 and $5 \\mathrm { k }$ for ZINC, for the latter we used the default test set. We evaluate the generated molecules’ quality by the percentage of valid, unique (i.e. percentage of unique molecules among the generated valid molecules) and novel (i.e. percentage of molecules never seen in the training set among the generated molecules) molecules. We estimate these quantities by sampling 10k (5K for ZINC) $\\mathbf { z }$ from the ${ \\hat { q } } _ { \\sigma } ( \\mathbf { z } )$ and coupling each one of them with a logP value, $\\mathbf { y }$ , randomly selected from the test set, and we subsequently decode the $\\mathbf { z } , \\mathbf { y }$ concatenation. We can see that our model has a better reconstruction performance compared to the baselines while in some cases generating slightly less valid molecules table 1. In terms of the three quality measures achieves an excellent performance across all three metrics being always one of the two best performing methods for any metric. ", + "bbox": [ + 173, + 444, + 825, + 597 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To visualise how the conditional generation operates we randomly sample from the test set some molecule and obtain its property value $\\mathbf { y } _ { 0 }$ . We then draw 50 random samples $\\mathbf { z } _ { i }$ from $\\hat { q } _ { \\sigma } ( \\mathbf { z } )$ and decode the $\\left[ \\mathbf { z } _ { i } , \\mathbf { y } _ { 0 } \\right]$ vectors. Among the generated valid molecules we compute the percentage of those that have a property value $\\mathbf { y } _ { i }$ that is within a $15 \\%$ range from the $\\mathbf { y } _ { 0 }$ property value. In Figure 3 we present the molecules obtained for a test molecule that had a logP of $- 0 . 5 7 5 9$ . Out of the 50 generated molecules 46 were valid of which we give the five that were within a $15 \\%$ range from the $\\mathbf { y } _ { 0 }$ value in Figure 3. As we can see we get molecules that are structurally very different from the original one yet they have similar logP value. ", + "bbox": [ + 173, + 603, + 825, + 715 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To quantify the quality of the conditional generations we measure the correlation between the property value we obtain by the conditional generation and the property value on which we conditioned the generation. We randomly sample $1 0 0 0 { \\textbf { y } }$ values from the test set and $1 0 0 0 \\textbf { z }$ values from the approximate learned prior $\\hat { q } _ { \\sigma } ( \\mathbf { z } )$ . We decode each pair, obtain $\\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } , \\mathbf { z } )$ , and then measure the correlation of the original y with the $\\hat { \\mathbf { y } }$ of generated $\\hat { \\bf x }$ . In Table 2, we give the correlation estimates for our method and the Sup-VAE baselines. As we can see our method has a considerably higher correlation score between the input and the obtained property than Sup-VAE. Conditional generation seems considerably harder for the ZINC dataset for all methods. ", + "bbox": [ + 174, + 722, + 825, + 833 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To visualise the style transfer behavior of our model we randomly sample two molecules $\\mathbf { x } _ { A } , \\mathbf { x } _ { B }$ from the test set. We then sample $\\mathbf { z } _ { A }$ from the learned posterior $q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { A } )$ . We subsequently decode $\\left[ \\mathbf { z } _ { A } , \\mathbf { y } _ { B } \\right]$ , $\\mathbf { y } _ { B }$ is the property of $\\mathbf { x } _ { B }$ , and get a new molecule $\\hat { \\mathbf { x } } _ { A B }$ . Ideally, the obtained molecule $\\hat { \\mathbf { x } } _ { A B }$ should have a property value (logP) close to the target $\\mathbf { y } _ { B }$ and be similar to $\\mathbf { x } _ { A }$ . In Figure 4 we give one such example. To put the results into context in Figure 8 in appendix, we give the results of a virtual screening method, where we select from the full dataset five molecules which are structurally similar to $\\mathbf { x } _ { A }$ and have logP values close to $\\mathbf { y } _ { B }$ . As we can see the molecule that our model generates is a new one. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/ffa61574cd2f40b5b194d3b7515745269b23e87d9fcf063708e5b80200110fb9.jpg", + "image_caption": [ + "Figure 5: Style transfer. The $\\mathbf { z }$ of the nine real molecules placed in the $\\mathbf { X }$ -axis is combined with 11 y property values, sampled in [-4.9, 4.9], the resulting pair is decoded to a molecule. " + ], + "image_footnote": [], + "bbox": [ + 209, + 83, + 776, + 397 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 486, + 823, + 513 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To quantify the style transfer performance we proceed in exactly the same manner as we did to quantify the conditional generation performance. However, now instead of sampling $\\mathbf { z }$ from the approximate learned prior, ${ \\hat { q } } _ { \\sigma } ( \\mathbf { z } )$ , we first sample some $\\mathbf { x }$ from the test set and then we sample $\\mathbf { z }$ from the learned posterior $q ( \\mathbf { z } | \\mathbf { x } )$ . The results are in the second column of Table 2. As we can see the correlation values are now lower than the ones we obtained in the simple conditional generation case. This can be explained by the fact that now we are forcing a specific combination of structure, z comes from a real molecule, and property, which might simply be physically infeasible since the molecule space is discrete and not all combinations are possible. In addition, as it was the case for the conditional generation, style transfer is considerably more difficult for the ZINC dataset. ", + "bbox": [ + 174, + 520, + 825, + 646 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We further explore the style transfer and visualize how our model covers the combined space of molecule structure and properties. We sample nine molecules from the QM9 test set, and get their $\\mathbf { z }$ encodings. For each such encoding we decode the vectors $[ \\mathbf { z } , y ] , y \\in [ - 4 . 9 , 4 . 9 ]$ , with the $y$ (logP) interval sampled at 11 points. We give in Figure 5 the resulting valid molecules, each column there corresponds to one of the nine original molecules, the ones surrounded by dotted rectangle, and their decodings with different logP values.For each original molecule we give the generated molecules ordered along the y axis according to the y property that they actually exhibit. The x-axis does not provide an ordering of the original molecules according to $\\mathbf { Z }$ , in fact we have ordered the original molecules by their y property. As we can see not all $( \\mathbf { z } , y )$ combinations produce a result. These holes can be explained either by the physical infeasibility of the combination and/or a limitation of the learned model. ", + "bbox": [ + 174, + 652, + 825, + 805 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We can use conditional generation to control in a fine manner the value of the desired property, to what can be seen as direct property optimization. We visualise the level of control we have on an experiment with a single molecule (with logP is -1.137), which we randomly sample from the test set. We obtain its $\\mathbf { z }$ encoding and perform generations with increased logP taking values in 1000 point grid in $\\left[ - 1 . 1 3 7 , 4 . 9 \\right]$ . We then decode $\\left[ \\mathbf { z } , \\mathbf { y } _ { i } \\right]$ and compute the logP value of the generated molecules. Among the 1000 generated molecules only 19 are unique. We get an increase of logP of a very discrete nature, Figure 6. As already discussed not all combinations of structure and properties are possible. The generated molecules themselves are shown in the supplemental material. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/316bb7a4e909e85ec0724b2b6aa986623202551cad1f77165f46d291f086ea62.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Modelz~qo(z)z~q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.71850.5005
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.74140.4715
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.38770.0880
Sup-VAE-3-GRU0.35140.1808
CGD-VAE-3-GRU0.39660.1559
", + "bbox": [ + 173, + 101, + 468, + 213 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 2: Correlation between the desired input property and the obtained property . $\\mathbf { z } \\sim \\hat { q } _ { \\sigma } ( \\mathbf { z } )$ corresponding to conditional generation), and x, $\\mathbf { z } \\sim q ( \\mathbf { z } | \\mathbf { x } )$ to property transfer case. ", + "bbox": [ + 173, + 223, + 470, + 291 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/dd9ea64a1ba0ee993b06ada6658a78a2896a28fd1475d73046b1a5c7a4f5c11d.jpg", + "image_caption": [ + "Figure 6: Property optimization. Given a start molecule (red), we combine its $\\mathbf { z }$ with $1 0 0 0 \\log \\mathrm { P }$ values and decode (blue) " + ], + "image_footnote": [], + "bbox": [ + 529, + 93, + 782, + 232 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/6daa2bdd2b6b3d32bf6a2bb30e18d85a323167036361749ae8c33e10b946aa5e.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 534, + 290, + 756, + 412 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/171721555e584b5336b980f58a15f7e4a7ff8c35dce2ba5cd515c171c0be165b.jpg", + "image_caption": [ + "Figure 7: A comparison of simulated logP values and Tanimoto similarity to a target on the ZINC dataset. While the stacked LSTM model has high accuracy in terms of matching the desired property, it would require drawing many samples before finding any close matches to any particular desired prototype. The CGD-VAE-3-GRU model represents a middle ground between a standard VAE model which does not condition on the property, and the stacked LSTM model which does not learn a reusable representation. " + ], + "image_footnote": [], + "bbox": [ + 236, + 290, + 472, + 412 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.1 CONDITIONAL LSTM BASELINE ", + "text_level": 1, + "bbox": [ + 176, + 539, + 437, + 553 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, we consider a variant of the stacked LSTM model of Segler et al. (2017), with no latent space, where the model is modified to take a target logP value as an additional input at each generation step. This model forms a very strong baseline for many distribution matching tasks (Liu et al., 2018; ?), though as best we are aware this has never been used directly for conditional generation given a target property. We use a modification of the implementation provided by (?) with three layers and default settings, and fit the model by maximum likelihood training on $\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( x _ { t } | x _ { 1 : t - 1 } , \\mathbf { y } ) } \\end{array}$ . ", + "bbox": [ + 174, + 565, + 826, + 652 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Training on the ZINC dataset, we find the generated molecules from this model have a very high correlation 0.975 with the target logP value, greatly outperforming any of the latent variable models we consider. This suggests that such a model would be very useful for generating candidates globally, but as the model has no latent variable it is not amenable to style transfer. We observe this in Figure 7, which samples 100 candidate molecules from both the stacked LSTM model and for CGD-VAE-3- GRU, conditioning on the property of one randomly-chosen test set example, while computing the Tanimoto similarity (computed using Morgan fingerprints of radius 2) to a second randomly-chosen test set example, across 200 pairs. The VAE has higher Tanimoto similarities as it can condition on the latent variable of the target molecule, representing a trade-off against the better adherence to the target property value of the unconditioned LSTM. ", + "bbox": [ + 173, + 659, + 825, + 797 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 820, + 318, + 837 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We presented a single step approach for the conditional generation of molecules with desired properties. Our model allows also to condition generation on a prototype molecule with a desired high-level structure. This work thus directly inverts the traditional relationship between molecules and their properties. We found that training the deep generative models conditional on target properties, following a supervised VAE approach, does not appreciably harm the quality of the unconditional generative model as measured by validity, novelty, and uniqueness of samples. Furthermore, we see that the additional act of regularizing the output using an approximate property predictor helps improve both reconstruction accuracy and property correlations in most combinations of tasks and datasets, particularly for the smaller QM9 dataset and for smaller models with fewer RNN layers. We also note that although none of the deep latent variable models are competitive with an LSTM baseline when purely considering generation conditioned on a target property value, the low Tanimoto similarity between randomly sampled candidates and an arbitrary style transfer target makes clear that such a model is not suitable for targeted generation of candidates which are close in structure to a particular prototype. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In future work, we want to explore how to further improve the correlation between the desired input properties to the decoder, and the properties of the generated molecules. Moreover, we want also to condition on multiple properties; while this is in principle possible in our framework, we do not explore it empirically here. Modifying a single property while constraining the remaining to be close to the original can further aggravate the infeasibility problem, as not all combinations of molecular properties may even be feasible, perhaps requiring learning a dependency structure between multiple properties. ", + "bbox": [ + 174, + 236, + 825, + 333 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 356, + 285, + 371 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Rim Assouel, Mohamed Ahmed, Marwin H. Segler, Amir Saffari, and Yoshua Bengio. Defactor: Differentiable edge factorization-based probabilistic graph generation. CoRR, abs/1811.09766, 2018. URL http://arxiv.org/abs/1811.09766. ", + "bbox": [ + 178, + 380, + 823, + 422 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. 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", + "bbox": [ + 173, + 738, + 823, + 781 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "6 APPENDIX ", + "text_level": 1, + "bbox": [ + 173, + 102, + 294, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "6.1 THE REGULARISER AND ITS RELATION TO THE MUTUAL INFORMATION MAXIMIZATION ", + "text_level": 1, + "bbox": [ + 171, + 136, + 816, + 150 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The soft constrain in the loss 3, in fact , is equivalent to a simple maximizing mutual information formulation between generated molecules $\\hat { \\bf x }$ and the target property $\\mathbf { y }$ provided to the generator. Assume the true conditional distribution is $\\tilde { p } ( \\mathbf { y } | \\mathbf { x } )$ : ", + "bbox": [ + 174, + 161, + 826, + 204 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/f3def3078a70acd5de4fbedc7647e95f7ee8cf236f0ac3b8770154a6f4a8731b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { I } ( \\mathbf { y } ; \\hat { \\mathbf { x } } ) = H ( \\mathbf { y } ) - H ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) } \\\\ & { \\qquad = H ( \\mathbf { y } ) + \\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } \\vert \\mathbf { y } ) } \\mathbb { E } _ { \\mathbf { y } ^ { \\prime } \\sim \\tilde { p } ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) } [ \\log \\tilde { p } ( \\mathbf { y } ^ { \\prime } \\vert \\hat { \\mathbf { x } } ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 212, + 678, + 252 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We do not know the true distribution $\\tilde { p } ( \\mathbf { y } | \\hat { \\mathbf { x } } )$ , however, the RDKit provides an estimation $p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } )$ of the distribution assuming a Gaussian distribution over error: ", + "bbox": [ + 173, + 260, + 825, + 289 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/289e50d0f7798a72284b200d021c3baab2f2a34670611452c4a211a23b8db034.jpg", + "text": "$$\np ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) = \\mathcal { N } ( \\mathbf { y } \\vert f ( \\hat { \\mathbf { x } } ) , \\lambda _ { 1 } ^ { - 1 } \\mathbf { I } )\n$$", + "text_format": "latex", + "bbox": [ + 405, + 295, + 593, + 315 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where $f$ is the molecule property estimator, i.e., RDKit. We have: ", + "bbox": [ + 173, + 323, + 606, + 338 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/9313d409114746a7bb4d13520d49b98c88c18a40042204504361c3ccec2df0c1.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\bf I } ( { \\bf y } ; \\hat { \\bf x } ) = H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } [ \\log \\frac { \\tilde { p } ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) } { p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) } p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } \\ ~ } \\\\ { { \\displaystyle ~ = H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } [ D _ { k l } ( \\tilde { p } ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) \\vert \\vert p ( { \\bf y } ^ { \\prime } \\vert \\hat { \\bf x } ) ) + \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } \\log p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } \\ ~ } \\\\ { { \\displaystyle ~ \\geq H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } [ \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } \\log p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 233, + 345, + 763, + 422 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "By following the Lemma 5.1 given in Chen et al. (2016), we have ", + "bbox": [ + 173, + 428, + 606, + 443 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/7f4de000aa662d02a9dc5ba4efae62b0270f7b9250580018179096fe3e0471b3.jpg", + "text": "$$\n\\mathbf { I } ( \\mathbf { y } ; \\hat { \\mathbf { x } } ) \\geq H ( \\mathbf { y } ) + \\mathbb { E } _ { \\mathbf { y } \\sim p ( \\mathbf { y } ) , \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } ) } [ \\log p ( \\mathbf { y } | \\hat { \\mathbf { x } } ) ]\n$$", + "text_format": "latex", + "bbox": [ + 333, + 452, + 663, + 469 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "As $H ( \\mathbf { y } )$ is constant, minimizing $\\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { i } ) } \\| f ( \\hat { \\mathbf { x } } ) - \\mathbf { y } _ { i } \\| ^ { 2 }$ is equivalent to maximizing $\\mathbf { I } ( \\mathbf { y } ; \\hat { \\mathbf { x } } )$ under the assumption that $p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } )$ is close to $\\tilde { p } ( \\mathbf { y } | \\hat { \\mathbf { x } } )$ ", + "bbox": [ + 176, + 478, + 823, + 511 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "6.2 ARCHITECTURE AND TRAINING PROCEDURE DESCRIPTION ", + "text_level": 1, + "bbox": [ + 173, + 529, + 622, + 544 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We use the same encoder and decoder network structure as Dai et al. (2018) with the only difference that our decoder takes as input the concatenation of $\\mathbf { y } , \\mathbf { z }$ . As GRU layers become computationally expensive when the sequences length increase, we also examined the model using less layer GRU. To be precise, the decoder in Dai et al. (2018) takes the from of a dense hidden layer with ReLU activation followed by three layers GRU Chung et al. (2014). We tried two different settings of decoder: in the first setting, we feed the concatenation of $\\mathbf { y } , \\mathbf { z }$ to dense layer then apply one layer GRU, in the second setting, to enhance the effect of $\\mathbf { y }$ in the decoder, we feed y not only to the dense layer but also to each layer of GRU. Furthermore, we set the dimension of the latent representation to 56. For the oracle function estimator $f _ { w }$ , we use the same network architecture as the encoder (there is no parameter sharing) and we add one more fully connected layer followed by a Tanh transformation. ", + "bbox": [ + 173, + 555, + 825, + 695 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "To speed up convergence, we initialize the $f _ { w }$ from a pre-training, where we train $f _ { w }$ on the welltrained (maximum 500 epochs with early stopping) supervised VAE’s decoder output to predict the molecules property value. We also initialize the parameters of the encoder/decoder networks with the partially trained supervised VAE model (after 40 epochs for QM9, 100 epochs for ZINC). We do not update $\\omega$ and $\\phi , \\theta$ simultaneously, instead we do an alternate optimization. We update $\\omega$ continuously for five epochs while holding $\\phi , \\theta$ and do the same for updating $\\phi , \\theta$ . We set the hyper-parameter value $\\lambda _ { 1 }$ to 50 and $\\lambda _ { 2 }$ to 1. The mini-batch size is set to 300 for QM9 and 100 for ZINC. We use ADAM optimizer with learning rate 0.0001 and pytorch lr-schedular on the validation loss. The general training algorithm is described in below algorithm block1. In our experiment, to train $f _ { \\omega }$ , we skipped the second term in step 7, which means we only train $f _ { \\omega }$ on the training data but not the newly generated molecules obtained by permuting the property. The reason for this is that, during the training, we found that it is easy for the model to learn to reconstruct but hard to conditionally generate the molecules with given properties while we have no guidance of what the molecules should look like. Further more, some combination of $\\mathbf { z }$ and $\\mathbf { y }$ are physically not feasible. In this case, when the conditional generation is not good enough yet during the training, we end up fitting $f _ { \\omega }$ on the miss represented molecules representations and it makes the optimization harder. ", + "bbox": [ + 173, + 702, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1 Training algorithm ", + "text_level": 1, + "bbox": [ + 174, + 103, + 388, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1: Initialize $p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } )$ , $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ , $f _ { w }$ \n2: for i=1,2, . . . , N (maximum epoch number) do \n3: for ${ \\bf j } = 1 , 2 , \\ldots , { \\bf L }$ , sample a minibatch $D = ( { \\bf X } , { \\bf Y } ) = \\{ { \\bf x } _ { i } , { \\bf y } _ { i } ) \\} _ { i = 1 } ^ { M }$ of M samples do \n4: randomly permute the property set $\\mathbf { Y }$ to obtain $\\mathbf { Y } ^ { \\ast }$ and define a label permuted mini \nbatch $D ^ { * } = ( \\mathbf { X } , \\mathbf { \\bar { Y } } ^ { * } ) = \\{ \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ^ { * } \\} _ { i = 1 } ^ { \\bar { M } }$ \n5: $\\begin{array} { l } { { \\displaystyle \\theta ^ { j } = \\theta ^ { j - 1 } - \\gamma \\big ( - \\nabla _ { \\theta } \\mathcal { L } _ { E L B O } \\big ( \\theta , \\phi , \\omega \\big ) + \\frac { \\lambda _ { 1 } } { 2 } \\sum _ { \\substack { ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\in D ^ { * } } } \\mathbb { E } _ { q ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\nabla _ { \\theta } \\| f _ { \\omega } \\big ( g _ { \\theta } \\big ( \\mathbf { z } , \\mathbf { y } _ { i } \\big ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } \\big ) } } \\\\ { { \\displaystyle \\phi ^ { j } = \\phi ^ { j - 1 } - \\gamma \\big ( - \\nabla _ { \\phi } \\mathcal { L } _ { E L B O } \\big ( \\theta , \\phi , \\omega \\big ) \\big ) } } \\\\ { { \\displaystyle w ^ { j } = w ^ { j - 1 } - \\gamma \\big ( \\frac { \\lambda _ { 2 } } { 2 } \\sum _ { \\substack { ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\in D } } \\mathbb { E } _ { q ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\nabla _ { \\omega } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } + \\frac { \\lambda _ { 1 } } { 2 } \\sum _ { \\substack { ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\in D ^ { * } } } \\mathbb { E } _ { q ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\nabla _ { \\omega } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } \\big ) } } \\end{array}$ \n6: \n7: ", + "bbox": [ + 178, + 122, + 836, + 295 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "6.3 USE A FIXED PRE-TRAINED PROPERTY PREDICTION FUNCTION", + "text_level": 1, + "bbox": [ + 174, + 340, + 645, + 354 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We also investigate the case where we train the property prediction function $f _ { \\omega }$ on the well trained supervised VAE output and keep it fixed during the training of the main model. We give the performance in tables 3, 4. Using a fixed $f _ { \\omega }$ , in terms of reconstruction and generation performance, delivers mixed results 3. However, in terms of conditional generation 4, it does perform better than the baselines but worse compared to the case where we also update $f _ { \\omega }$ during the learning. ", + "bbox": [ + 173, + 373, + 826, + 444 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/87d4bd18b4c8c2fd59ecb849572f398ef0279aae15024add5804f3b0c07dbd2e.jpg", + "table_caption": [ + "Table 3: Sup-VAE-1-GRU /Sup-VAE-3-GRU: supervised version of SD-VAE model where y is been feed to only the first layer/all layer decoder. CGD-VAE: our model, conditional generation with disentangling. The result for CVAE and GVAE are taken from the literature, ”-” refers that those measures are not reported. The rest of baseline result is obtained by rerunning (SD-VAE) and editing the original code (Sup-VAE). " + ], + "table_footnote": [], + "table_body": "
QM9ZINC
ModelReconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE3.6110.30190.0044.600.70-100
GVAE96.0060.20=80.9053.707.20100
SD-VAE97.8498.4099.2891.9776.2043.50-、
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8495.61100
CGD-VAE-1-GRU98.9695.0392.7789.7467.4617.3882.44100
Sup-VAE-3-GRU97.8197.995.0989.4782.4036.1686.26100
CGD-VAE-3-GRU96.993.898.2989.5581.8037.7898.70100
", + "bbox": [ + 173, + 476, + 825, + 585 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/6fbf171757b6258aa6a8fe1e8a68d65348c503e12eeb358625899a33b54024f0.jpg", + "table_caption": [], + "table_footnote": [ + "Table 4: Correlation between the desired input property and the obtained property when z is sampled from the approximate learned prior, $\\mathbf { z } \\sim \\hat { q } _ { \\sigma } ( \\mathbf { z } )$ , (conditional generation), and when $\\mathbf { z }$ is sampled from the learned posterior given some x, $\\mathbf { z } \\sim q ( \\mathbf { z } | \\mathbf { x } )$ , (property transfer). " + ], + "table_body": "
Modelz~qq(z)z ~ q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.63310.3835
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.66650.4507
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.29810.0866
Sup-VAE-3-GRU0.36380.1818
CGD-VAE-3-GRU0.37650.1310
", + "bbox": [ + 320, + 731, + 678, + 867 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "6.4 VIRTUAL SCREENING ", + "text_level": 1, + "bbox": [ + 174, + 103, + 362, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The figure 8 displays the results of a virtual screening method, where we select from the full dataset five molecules which are structurally similar to $\\mathbf { x } _ { A }$ in figure 4 in section 4 and have logP values close to $\\mathbf { y } _ { B }$ . As we can see the molecule that our model generates is a new one. ", + "bbox": [ + 173, + 128, + 825, + 172 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/a7e485939ef9875fdba8293f1eb15bcfd2c0bfca822f4dd0ac966cbb93a44c64.jpg", + "image_caption": [ + "Figure 8: Molecules selected with virtual screening over the full dataset in a manner that they are structurally similar to the A molecule of figure 4 while they have a logP value which is close to the logP value of the B molecule also of figure 4 " + ], + "image_footnote": [], + "bbox": [ + 178, + 196, + 763, + 280 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "6.5 METRICS AS FUNCTION OF $\\mathbf { y } - \\mathbf { y } ^ { \\prime }$ ", + "text_level": 1, + "bbox": [ + 176, + 366, + 449, + 380 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The validity and novelty were mainly used to assess purely the generative model performance. However, it is also interesting to see if the model capable of generating valid and novel molecules if we start from an existing molecules and drift away from the original label, i.e., if we get $\\mathbf { z }$ from $q ( \\mathbf { z } | \\mathbf { x } )$ , and then compare different values from $p ( \\mathbf { \\bar { x } } | \\mathbf { z } , \\mathbf { y } ^ { \\prime } )$ as $\\mathbf { y } ^ { \\prime }$ moves far from y. We randomly sample a molecule $\\mathbf { x }$ whose LogP is y from the test set, then sample $1 0 \\textbf { z }$ from $q ( \\mathbf { z } | \\mathbf { x } )$ . For each such $\\mathbf { z }$ we couple it with a $\\mathbf { y } ^ { \\prime }$ that is different that $\\mathbf { y }$ , and sample 10 molecules from $p ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } ^ { \\prime } )$ . Eventually, for each such $\\mathbf { y } ^ { \\prime }$ , starting from original molecule $\\mathbf { x }$ , we generated 100 molecules, and we report the validity, uniqueness and novelty as a function of $\\mathbf { y } - \\mathbf { y } ^ { \\prime }$ . The figure 9 displays result of repeating above process for 20 randomly sampled $\\displaystyle ( \\mathbf { x } , \\mathbf { y } )$ along 100 grid points for $\\mathbf { y } - \\mathbf { y } ^ { \\prime }$ . The result confirms that on a big data set, conditional generative models uniqueness, validity and novelty performance is not affected by the size of the modification done on the property. However, on a small dataset, uniqueness is not affected. As expected, novelty increases as the properties modification size increases and validity drops slightly as the property modification size increase. ", + "bbox": [ + 173, + 392, + 826, + 571 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/05561149d0652633c285e17687e4cc57ccdd333927040429da49e149095794c1.jpg", + "image_caption": [ + "Figure 9: CGD-VAE-3-GRU model validity, novelty, uniqueness performance on property transfer task as a function of $\\mathbf { y } - \\mathbf { y } ^ { \\prime }$ on QM9 dataset. " + ], + "image_footnote": [], + "bbox": [ + 179, + 588, + 496, + 761 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/7e913f7fbf7e5ee958020cae4f83630081d3c4db545c8f072beab85a15a4684d.jpg", + "image_caption": [ + "Figure 10: CGD-VAE-3-GRU model validity, novelty, uniqueness performance on property transfer task as a function of $\\mathbf { y } - \\mathbf { y } ^ { \\prime }$ on ZINC dataset. " + ], + "image_footnote": [], + "bbox": [ + 511, + 588, + 821, + 761 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "With our model, during the generation, we observe that sampling from the approximated marginal posterior improves generation performance when compered to sampling from the prior. Here we investigate if this findings holds for other baseline models or not. We explored the behavior of baselines when the $\\mathbf { z }$ is sampled from the approximate marginal posterior, we observe that for CVAE and GVAE the validity did not change (as the ${ \\hat { q } } ( \\mathbf { z } )$ and $p ( \\mathbf { z } )$ are essentially identical); for the SD-VAE the validity increases (Table 5). ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/c3de2f63686b84ac41b690d9ca173237b6801eee4ea9eb9834c33d5c24250bc9.jpg", + "table_caption": [ + "SDVAE ", + "Table 5: Baseline model performance on ZINC " + ], + "table_footnote": [], + "table_body": "
Valid%Unique %Novel %
z ~qσ(z)56.0580.69100
z ~p(z)43.5082.26100
", + "bbox": [ + 348, + 188, + 648, + 236 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "6.7 MOLECULE PROPERTY OPTIMIZATION", + "text_level": 1, + "bbox": [ + 176, + 286, + 477, + 301 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Figure 11 is the visualization of the generated molecules from property optimization task, given in figure 6 section 4. The molecules are generated by increasing the logP of a given molecule, i.e. we hold $\\mathbf { z }$ fixed and increase the y value. Among the all generated molecules, 19 of them are unique. ", + "bbox": [ + 174, + 313, + 825, + 356 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/cc585b7043d323b6f84b9809b8adc19b851ad1196e606dc71b64fabbf987c156.jpg", + "image_caption": [ + "Figure 11: Molecules generated with an increasing logP " + ], + "image_footnote": [], + "bbox": [ + 313, + 368, + 681, + 550 + ], + "page_idx": 14 + } +] \ No newline at end of file diff --git a/parse/train/BkxthxHYvr/BkxthxHYvr_middle.json b/parse/train/BkxthxHYvr/BkxthxHYvr_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..5c86f50a56d0c454d8a9c035b20ebb68e36ef136 --- /dev/null +++ b/parse/train/BkxthxHYvr/BkxthxHYvr_middle.json @@ -0,0 +1,44803 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 99 + ], + "score": 1.0, + "content": "CONDITIONAL GENERATION OF MOLECULES FROM", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 102, + 365, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 365, + 118 + ], + "score": 1.0, + "content": "DISENTANGLED REPRESENTATIONS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 214, + 469, + 367 + ], + "lines": [ + { + "bbox": [ + 141, + 213, + 469, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 469, + 227 + ], + "score": 1.0, + "content": "Though machine learning approaches have shown great success in estimating", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 225, + 470, + 237 + ], + "spans": [ + { + "bbox": [ + 141, + 225, + 470, + 237 + ], + "score": 1.0, + "content": "properties of small molecules, the inverse problem of generating molecules with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 236, + 470, + 248 + ], + "spans": [ + { + "bbox": [ + 141, + 236, + 470, + 248 + ], + "score": 1.0, + "content": "desired properties remains challenging. 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This difficulty is in part because the set of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 246, + 470, + 260 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 470, + 260 + ], + "score": 1.0, + "content": "molecules which have a given property is structurally very diverse. Treating this", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 258, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 470, + 270 + ], + "score": 1.0, + "content": "inverse problem as a conditional distribution estimation task, we draw upon work", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 470, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 470, + 281 + ], + "score": 1.0, + "content": "in learning disentangled representations to learn a conditional distribution over", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 280, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 470, + 291 + ], + "score": 1.0, + "content": "molecules given a desired property, where the molecular structure is encoded in a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "score": 1.0, + "content": "continuous latent random variable. By including property information as an input", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 470, + 315 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 470, + 315 + ], + "score": 1.0, + "content": "factor independent from the structure representation, one can perform conditional", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "score": 1.0, + "content": "molecule generation via a “style transfer” process, in which we explicitly set the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "score": 1.0, + "content": "property to a desired value at generation time. In contrast to existing approaches,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 469, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 469, + 347 + ], + "score": 1.0, + "content": "we disentangle the latent factors from the property factors using a regularization", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 345, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 469, + 358 + ], + "score": 1.0, + "content": "term which constrains the generated molecules to have the property provided to the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 356, + 387, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 356, + 387, + 369 + ], + "score": 1.0, + "content": "generation network, no matter how the latent factor changes.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5, + "bbox_fs": [ + 141, + 213, + 470, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 394, + 206, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 208, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 208, + 410 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "Conditional molecule generation is far from being solved. The main challenge is the enormous and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "discrete nature of the molecules space and the fact that molecule properties are highly sensitive to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "molecular structure (Kirkpatrick & Ellis, 2004). Approaches to conditional generation are typically", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "score": 1.0, + "content": "two-step, either using a model or genetic algorithm to generate candidates which are later filtered,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "or learning a continuous embedding of the discrete molecules and optimizing in a real-valued", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "representation space. 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Machine learning", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "score": 1.0, + "content": "approaches expand on this by dynamically generating additional candidate molecules; Segler et al.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 149 + ], + "score": 1.0, + "content": "(2017) uses a stacked LSTM to produce large numbers of novel molecules which have similar", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 261, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 261, + 159 + ], + "score": 1.0, + "content": "characteristics to an existing database.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "For properties which are expensive to evaluate, generating large sets of candidate molecules is not", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "particularly useful. More sample-efficient global search can be achieved using Bayesian optimization", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 507, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 507, + 199 + ], + "score": 1.0, + "content": "methods, which use a generative model with a latent space that functions as a continuous repre-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "sentation of molecules (Gomez-Bombarelli et al., 2016; Kusner et al., 2017). Optimization is then ´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "carried out over this continuous representation space to find candidates which are expected to have", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "the desired property. Local gradient-based search can also be applied on continuous latent spaces to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "optimize the latent representation with respect to a target property (Jin et al., 2018; Liu et al., 2018).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 246, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "A challenge for these latent variable models is to reliably produce valid molecules. Character", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "variational autoencoders (CVAEs) (Gomez-Bombarelli et al., 2016) generate molecules one character ´", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "at a time, and are prone to syntactic and semantic errors; the grammar-based variational autoencoder", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 279, + 507, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 507, + 293 + ], + "score": 1.0, + "content": "(GVAE) (Kusner et al., 2017) and syntax-directed variational autoencoder (SD-VAE) (Dai et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "2018) instead operate in the space of context-free and attribute grammars, respectively, to ensure", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "syntactic validity. Other work generative models that operates on graph representations (Simonovsky", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "& Komodakis, 2018; De Cao & Kipf, 2018; Jin et al., 2018; You et al., 2018; Liu et al., 2018), largely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 307, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 307, + 335 + ], + "score": 1.0, + "content": "improving the ability to generate valid molecules.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 284, + 353 + ], + "score": 1.0, + "content": "Suppose we are given a training set of pairs", + "type": "text" + }, + { + "bbox": [ + 284, + 340, + 405, + 352 + ], + "score": 0.89, + "content": "\\mathcal { D } = \\{ ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\} , i = 1 , \\ldots , N .", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 340, + 506, + 353 + ], + "score": 1.0, + "content": ", where x corresponds to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 397, + 363 + ], + "score": 1.0, + "content": "molecules and y represents a value of some properties of the molecule", + "type": "text" + }, + { + "bbox": [ + 398, + 353, + 405, + 361 + ], + "score": 0.27, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 351, + 506, + 363 + ], + "score": 1.0, + "content": ". Assume the molecules", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 344, + 375 + ], + "score": 1.0, + "content": "represent an i.i.d. sample from some unknown distribution", + "type": "text" + }, + { + "bbox": [ + 345, + 362, + 364, + 374 + ], + "score": 0.91, + "content": "\\tilde { p } ( { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 363, + 505, + 375 + ], + "score": 1.0, + "content": ", which assigns high probability to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "molecules believed to be useful for a given task. Aside from Segler et al. (2017), which has no", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "latent space and thus directly trains via maximum likelihood, these latent variable models are trained", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "by optimizing a standard ELBO objective for variational autoencoders (?). This entails learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 192, + 418 + ], + "score": 1.0, + "content": "a stochastic encoder", + "type": "text" + }, + { + "bbox": [ + 192, + 406, + 225, + 419 + ], + "score": 0.92, + "content": "q _ { \\phi } ( { \\bf z } | { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 406, + 506, + 418 + ], + "score": 1.0, + "content": "which maps molecules into a latent space, and a stochastic decoder", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 416, + 322, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 138, + 429 + ], + "score": 0.93, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 416, + 322, + 430 + ], + "score": 1.0, + "content": "for reconstructing molecules, by maximizing", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 431, + 447, + 465 + ], + "lines": [ + { + "bbox": [ + 164, + 431, + 447, + 465 + ], + "spans": [ + { + "bbox": [ + 164, + 431, + 447, + 465 + ], + "score": 0.94, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) = \\sum _ { i = 1 } ^ { N } \\bigg \\{ \\mathbb { E } _ { q _ { \\boldsymbol { \\phi } } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) } [ \\log p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { i } | \\mathbf { z } _ { i } ) ] - D _ { K L } ( q _ { \\boldsymbol { \\phi } } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) | | p ( \\mathbf { z } _ { i } ) ) \\bigg \\} .", + "type": "interline_equation", + "image_path": "515df8a88edf91025448b21438c14e0e0054dae72cef08ba7e1526e1e611922b.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 164, + 431, + 447, + 442.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 164, + 442.3333333333333, + 447, + 453.66666666666663 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 164, + 453.66666666666663, + 447, + 464.99999999999994 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "score": 1.0, + "content": "Notably, the objective is not a function of y: most existing generative models with latent variables do", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "not perform direct conditional generation, and approaches for targeted molecule discovery are bolted", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "on to the learnt model. 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Guimaraes et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "(2017); De Cao & Kipf (2018); You et al. (2018) instead learn generation models optimized towards", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "specific metrics, such as drug-likeliness and solubility; the major downside is that these models must", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "be retrained each time for a new property. 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A", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "disentangled representation gives us additional control on the molecule generation process, allowing", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "us to modify a single property leaving the remaining unaffected (Bengio et al., 2013a). In many cases", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "score": 1.0, + "content": "important variation in the data is easy to annotate. For example in the case of molecule datasets", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "we have access to different functional descriptors of the molecules obtained by chemoinformatics", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "software such as RDKit (Landrum). Particularly useful to us here are supervised methods for learning", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "disentangled representations (Kingma et al., 2014; Siddharth et al., 2017). 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Machine learning", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 137 + ], + "score": 1.0, + "content": "approaches expand on this by dynamically generating additional candidate molecules; Segler et al.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 149 + ], + "score": 1.0, + "content": "(2017) uses a stacked LSTM to produce large numbers of novel molecules which have similar", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 261, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 261, + 159 + ], + "score": 1.0, + "content": "characteristics to an existing database.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 102, + 506, + 159 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "For properties which are expensive to evaluate, generating large sets of candidate molecules is not", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "particularly useful. More sample-efficient global search can be achieved using Bayesian optimization", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 507, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 507, + 199 + ], + "score": 1.0, + "content": "methods, which use a generative model with a latent space that functions as a continuous repre-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "sentation of molecules (Gomez-Bombarelli et al., 2016; Kusner et al., 2017). Optimization is then ´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "carried out over this continuous representation space to find candidates which are expected to have", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "the desired property. Local gradient-based search can also be applied on continuous latent spaces to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "optimize the latent representation with respect to a target property (Jin et al., 2018; Liu et al., 2018).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 163, + 507, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 246, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "A challenge for these latent variable models is to reliably produce valid molecules. Character", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "variational autoencoders (CVAEs) (Gomez-Bombarelli et al., 2016) generate molecules one character ´", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "at a time, and are prone to syntactic and semantic errors; the grammar-based variational autoencoder", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 279, + 507, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 507, + 293 + ], + "score": 1.0, + "content": "(GVAE) (Kusner et al., 2017) and syntax-directed variational autoencoder (SD-VAE) (Dai et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 304 + ], + "score": 1.0, + "content": "2018) instead operate in the space of context-free and attribute grammars, respectively, to ensure", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 314 + ], + "score": 1.0, + "content": "syntactic validity. Other work generative models that operates on graph representations (Simonovsky", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "& Komodakis, 2018; De Cao & Kipf, 2018; Jin et al., 2018; You et al., 2018; Liu et al., 2018), largely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 307, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 307, + 335 + ], + "score": 1.0, + "content": "improving the ability to generate valid molecules.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 247, + 507, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 284, + 353 + ], + "score": 1.0, + "content": "Suppose we are given a training set of pairs", + "type": "text" + }, + { + "bbox": [ + 284, + 340, + 405, + 352 + ], + "score": 0.89, + "content": "\\mathcal { D } = \\{ ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\} , i = 1 , \\ldots , N .", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 340, + 506, + 353 + ], + "score": 1.0, + "content": ", where x corresponds to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 397, + 363 + ], + "score": 1.0, + "content": "molecules and y represents a value of some properties of the molecule", + "type": "text" + }, + { + "bbox": [ + 398, + 353, + 405, + 361 + ], + "score": 0.27, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 351, + 506, + 363 + ], + "score": 1.0, + "content": ". Assume the molecules", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 344, + 375 + ], + "score": 1.0, + "content": "represent an i.i.d. sample from some unknown distribution", + "type": "text" + }, + { + "bbox": [ + 345, + 362, + 364, + 374 + ], + "score": 0.91, + "content": "\\tilde { p } ( { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 363, + 505, + 375 + ], + "score": 1.0, + "content": ", which assigns high probability to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 506, + 385 + ], + "score": 1.0, + "content": "molecules believed to be useful for a given task. Aside from Segler et al. (2017), which has no", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "latent space and thus directly trains via maximum likelihood, these latent variable models are trained", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "by optimizing a standard ELBO objective for variational autoencoders (?). This entails learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 192, + 418 + ], + "score": 1.0, + "content": "a stochastic encoder", + "type": "text" + }, + { + "bbox": [ + 192, + 406, + 225, + 419 + ], + "score": 0.92, + "content": "q _ { \\phi } ( { \\bf z } | { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 406, + 506, + 418 + ], + "score": 1.0, + "content": "which maps molecules into a latent space, and a stochastic decoder", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 416, + 322, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 138, + 429 + ], + "score": 0.93, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 416, + 322, + 430 + ], + "score": 1.0, + "content": "for reconstructing molecules, by maximizing", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 340, + 506, + 430 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 431, + 447, + 465 + ], + "lines": [ + { + "bbox": [ + 164, + 431, + 447, + 465 + ], + "spans": [ + { + "bbox": [ + 164, + 431, + 447, + 465 + ], + "score": 0.94, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) = \\sum _ { i = 1 } ^ { N } \\bigg \\{ \\mathbb { E } _ { q _ { \\boldsymbol { \\phi } } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) } [ \\log p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { i } | \\mathbf { z } _ { i } ) ] - D _ { K L } ( q _ { \\boldsymbol { \\phi } } ( \\mathbf { z } _ { i } | \\mathbf { x } _ { i } ) | | p ( \\mathbf { z } _ { i } ) ) \\bigg \\} .", + "type": "interline_equation", + "image_path": "515df8a88edf91025448b21438c14e0e0054dae72cef08ba7e1526e1e611922b.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 164, + 431, + 447, + 442.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 164, + 442.3333333333333, + 447, + 453.66666666666663 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 164, + 453.66666666666663, + 447, + 464.99999999999994 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "score": 1.0, + "content": "Notably, the objective is not a function of y: most existing generative models with latent variables do", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "not perform direct conditional generation, and approaches for targeted molecule discovery are bolted", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "on to the learnt model. Some, e.g. Kusner et al. (2017), are trained in an “unsupervised” manner,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "agnostic to any property which later may need to be optimized. Others, e.g. Gomez-Bombarelli ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "et al. (2016); Liu et al. (2018), train the autoencoder jointly alongside a function to predict y from", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "z, hoping to guide the latent space to be also good for predicting the desired property. A recent", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "exception is Assouel et al. (2018), which learns a deterministic autoencoder where the decoder takes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "the latent code and the desired property as input, using a mutual information term in training to steer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 554, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 507, + 568 + ], + "score": 1.0, + "content": "the model towards generating molecules whose target properties match the input. Guimaraes et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "(2017); De Cao & Kipf (2018); You et al. (2018) instead learn generation models optimized towards", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "specific metrics, such as drug-likeliness and solubility; the major downside is that these models must", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "be retrained each time for a new property. In contrast, the autoencoder-based methods can be re-used", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 599, + 333, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 333, + 612 + ], + "score": 1.0, + "content": "to optimize towards any particular value of the property.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 467, + 507, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 623, + 316, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 317, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 317, + 636 + ], + "score": 1.0, + "content": "2.2 STYLE TRANSFER WITH SUPERVISED VAES", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "While the latent representations learned through standard VAE models perform well on the task of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "molecule reconstruction they do not necessarily provide interpretable factorised representations. A", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "disentangled representation gives us additional control on the molecule generation process, allowing", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "us to modify a single property leaving the remaining unaffected (Bengio et al., 2013a). In many cases", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 504, + 699 + ], + "score": 1.0, + "content": "important variation in the data is easy to annotate. For example in the case of molecule datasets", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "we have access to different functional descriptors of the molecules obtained by chemoinformatics", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "software such as RDKit (Landrum). Particularly useful to us here are supervised methods for learning", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "disentangled representations (Kingma et al., 2014; Siddharth et al., 2017). 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An analogous operation on molecules would involve holding", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 507, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 507, + 179 + ], + "score": 1.0, + "content": "the physical structure of a molecule (its “style”) relatively fixed while modifying a salient property.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "Unlike (say) the style transfer example for the MNIST digits, the conditional distribution of molecules", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "with a particular value of properties might be very diverse; for example, the QED score attempts", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "to measure the drug-likeness of a molecule, and the set of molecules generated at high values of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "this score would hopefully have high probability on a large, varied set of molecules. 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This is an unreasonably hard constraint, unlikely to be satisfied by any", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 430, + 303 + ], + "score": 1.0, + "content": "distribution other than one which simply places a point mass on the single training", + "type": "text" + }, + { + "bbox": [ + 430, + 291, + 441, + 301 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "associated with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 117, + 312 + ], + "score": 0.83, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 300, + 396, + 314 + ], + "score": 1.0, + "content": ", but we can relax it by considering that (unlike the molecular space", + "type": "text" + }, + { + "bbox": [ + 396, + 303, + 404, + 311 + ], + "score": 0.28, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 300, + 486, + 314 + ], + "score": 1.0, + "content": ") the property space", + "type": "text" + }, + { + "bbox": [ + 487, + 303, + 494, + 312 + ], + "score": 0.34, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "typically smooth, as many properties are continuous-valued and correspond to a human-interpretable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "scale. Following Ma et al. (2018) and Hu et al. 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This", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 272, + 436 + ], + "score": 1.0, + "content": "constraint is expected to hold for any pair", + "type": "text" + }, + { + "bbox": [ + 272, + 423, + 297, + 434 + ], + "score": 0.9, + "content": "\\displaystyle ( \\mathbf { x } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "we may happen to come across, not just those in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "training data. We also show optimizing the relaxed constraint is equivalent to maximizing mutual", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 444, + 483, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 215, + 457 + ], + "score": 1.0, + "content": "information with the target", + "type": "text" + }, + { + "bbox": [ + 216, + 446, + 226, + 456 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 444, + 324, + 457 + ], + "score": 1.0, + "content": "and generated molecule", + "type": "text" + }, + { + "bbox": [ + 325, + 445, + 333, + 455 + ], + "score": 0.63, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 444, + 483, + 457 + ], + "score": 1.0, + "content": "; for details see appendix Section 6.1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 107, + 471, + 322, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 322, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 322, + 483 + ], + "score": 1.0, + "content": "3.2 APPROXIMATING THE PROPERTY PREDICTOR", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "Introducing the regularizer as in Eq. (3) implicitly guides the reconstruction to take into account the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "property information, such that the reconstructed data should exhibit properties which match the input", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 514, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 385, + 525 + ], + "score": 1.0, + "content": "properties it is conditioned on. However, existing implementations of", + "type": "text" + }, + { + "bbox": [ + 385, + 514, + 392, + 525 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 514, + 506, + 525 + ], + "score": 1.0, + "content": "are often non-differentiable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 524, + 504, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 188, + 536 + ], + "score": 1.0, + "content": "or CPU-bound, and", + "type": "text" + }, + { + "bbox": [ + 189, + 524, + 196, + 534 + ], + "score": 0.68, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 525, + 504, + 536 + ], + "score": 1.0, + "content": "are discrete samples from a categorical distribution, all of which means the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "gradient of the regularizer can’t flow back to the generator. This is outlined in Figure 2. To enable the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "gradient based methods on GPUs during training and avoid discrete sampling, one approach would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 294, + 569 + ], + "score": 1.0, + "content": "be to first fit a differentiable approximation to", + "type": "text" + }, + { + "bbox": [ + 294, + 558, + 301, + 568 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 557, + 505, + 569 + ], + "score": 1.0, + "content": ", and then use either a Gumbel-softmax relaxation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "(Jang et al., 2016) or tricks like a “straight-through” estimator (Bengio et al., 2013b) as a continuous", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 578, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 593 + ], + "score": 1.0, + "content": "approximation for the discrete samples. Instead, we propose bypassing the discrete sampling step", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 239, + 603 + ], + "score": 1.0, + "content": "entirely and learning a function", + "type": "text" + }, + { + "bbox": [ + 240, + 590, + 252, + 601 + ], + "score": 0.87, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "that can map from a learned representation of the molecules", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 601, + 299, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 299, + 613 + ], + "score": 1.0, + "content": "directly to molecules property (Hu et al., 2017).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "To do this, we take as input the last hidden layer of the decoder network which parameterizes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 149, + 641 + ], + "score": 0.92, + "content": "p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 628, + 339, + 641 + ], + "score": 1.0, + "content": ", denoting this deterministic transformation as", + "type": "text" + }, + { + "bbox": [ + 339, + 629, + 373, + 641 + ], + "score": 0.93, + "content": "g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 628, + 506, + 641 + ], + "score": 1.0, + "content": ". 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Red lines correspond to non-differentiable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "components, including both a potentially complex sampling process and the property prediction itself.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 247 + ], + "score": 1.0, + "content": "The blue dashed line corresponds to the approximate property predictor, which aims to predict the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "expected value of the property from a continuous relaxation, marginalized over the sampling process.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 277, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 136, + 292 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 136, + 279, + 195, + 290 + ], + "score": 0.92, + "content": "i = 1 , \\ldots , N", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 277, + 505, + 292 + ], + "score": 1.0, + "content": ". This is an unreasonably hard constraint, unlikely to be satisfied by any", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 430, + 303 + ], + "score": 1.0, + "content": "distribution other than one which simply places a point mass on the single training", + "type": "text" + }, + { + "bbox": [ + 430, + 291, + 441, + 301 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "associated with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 300, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 117, + 312 + ], + "score": 0.83, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 300, + 396, + 314 + ], + "score": 1.0, + "content": ", but we can relax it by considering that (unlike the molecular space", + "type": "text" + }, + { + "bbox": [ + 396, + 303, + 404, + 311 + ], + "score": 0.28, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 300, + 486, + 314 + ], + "score": 1.0, + "content": ") the property space", + "type": "text" + }, + { + "bbox": [ + 487, + 303, + 494, + 312 + ], + "score": 0.34, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 300, + 506, + 314 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "typically smooth, as many properties are continuous-valued and correspond to a human-interpretable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "scale. Following Ma et al. (2018) and Hu et al. 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This", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 272, + 436 + ], + "score": 1.0, + "content": "constraint is expected to hold for any pair", + "type": "text" + }, + { + "bbox": [ + 272, + 423, + 297, + 434 + ], + "score": 0.9, + "content": "\\displaystyle ( \\mathbf { x } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "we may happen to come across, not just those in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "training data. We also show optimizing the relaxed constraint is equivalent to maximizing mutual", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 444, + 483, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 215, + 457 + ], + "score": 1.0, + "content": "information with the target", + "type": "text" + }, + { + "bbox": [ + 216, + 446, + 226, + 456 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 444, + 324, + 457 + ], + "score": 1.0, + "content": "and generated molecule", + "type": "text" + }, + { + "bbox": [ + 325, + 445, + 333, + 455 + ], + "score": 0.63, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 444, + 483, + 457 + ], + "score": 1.0, + "content": "; for details see appendix Section 6.1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 390, + 506, + 457 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 471, + 322, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 322, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 322, + 483 + ], + "score": 1.0, + "content": "3.2 APPROXIMATING THE PROPERTY PREDICTOR", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "Introducing the regularizer as in Eq. (3) implicitly guides the reconstruction to take into account the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "property information, such that the reconstructed data should exhibit properties which match the input", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 514, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 385, + 525 + ], + "score": 1.0, + "content": "properties it is conditioned on. However, existing implementations of", + "type": "text" + }, + { + "bbox": [ + 385, + 514, + 392, + 525 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 514, + 506, + 525 + ], + "score": 1.0, + "content": "are often non-differentiable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 524, + 504, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 188, + 536 + ], + "score": 1.0, + "content": "or CPU-bound, and", + "type": "text" + }, + { + "bbox": [ + 189, + 524, + 196, + 534 + ], + "score": 0.68, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 525, + 504, + 536 + ], + "score": 1.0, + "content": "are discrete samples from a categorical distribution, all of which means the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "gradient of the regularizer can’t flow back to the generator. This is outlined in Figure 2. To enable the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "gradient based methods on GPUs during training and avoid discrete sampling, one approach would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 294, + 569 + ], + "score": 1.0, + "content": "be to first fit a differentiable approximation to", + "type": "text" + }, + { + "bbox": [ + 294, + 558, + 301, + 568 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 557, + 505, + 569 + ], + "score": 1.0, + "content": ", and then use either a Gumbel-softmax relaxation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "(Jang et al., 2016) or tricks like a “straight-through” estimator (Bengio et al., 2013b) as a continuous", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 578, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 593 + ], + "score": 1.0, + "content": "approximation for the discrete samples. Instead, we propose bypassing the discrete sampling step", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 239, + 603 + ], + "score": 1.0, + "content": "entirely and learning a function", + "type": "text" + }, + { + "bbox": [ + 240, + 590, + 252, + 601 + ], + "score": 0.87, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "that can map from a learned representation of the molecules", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 601, + 299, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 299, + 613 + ], + "score": 1.0, + "content": "directly to molecules property (Hu et al., 2017).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 491, + 506, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "To do this, we take as input the last hidden layer of the decoder network which parameterizes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 149, + 641 + ], + "score": 0.92, + "content": "p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 628, + 339, + 641 + ], + "score": 1.0, + "content": ", denoting this deterministic transformation as", + "type": "text" + }, + { + "bbox": [ + 339, + 629, + 373, + 641 + ], + "score": 0.93, + "content": "g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 628, + 506, + 641 + ], + "score": 1.0, + "content": ". 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(3), yielding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 228, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 228, + 217 + ], + "score": 1.0, + "content": "a candidate objective function", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 220, + 413, + 236 + ], + "lines": [ + { + "bbox": [ + 198, + 220, + 413, + 236 + ], + "spans": [ + { + "bbox": [ + 198, + 220, + 413, + 236 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\theta , \\phi ) \\approx \\hat { \\mathcal { L } } _ { \\omega } ( \\theta , \\phi ) = \\mathcal { L } _ { E L B O } ( \\theta , \\phi ) - \\mathcal { L } _ { d i s e n t } ( \\theta , \\omega )", + "type": "interline_equation", + "image_path": "a9bc09229d3d3410c05de845a0f50287eeb9aa745a3439d6933f023664f4608d.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 198, + 220, + 413, + 236 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 448, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 450, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 450, + 259 + ], + "score": 1.0, + "content": "3.3 LEARNING THE PROPERTY ESTIMATOR JOINTLY WITH GENERATIVE MODEL", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 267, + 506, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 284, + 282 + ], + "score": 1.0, + "content": "While one could imagine attempting to learn", + "type": "text" + }, + { + "bbox": [ + 284, + 269, + 295, + 280 + ], + "score": 0.89, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 267, + 345, + 282 + ], + "score": 1.0, + "content": "jointly with", + "type": "text" + }, + { + "bbox": [ + 345, + 268, + 362, + 280 + ], + "score": 0.9, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 267, + 506, + 282 + ], + "score": 1.0, + "content": "by direct optimization of Eq. (6), in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 279, + 293 + ], + "score": 1.0, + "content": "practice this is very unstable, as values of", + "type": "text" + }, + { + "bbox": [ + 279, + 280, + 315, + 291 + ], + "score": 0.92, + "content": "g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "early in training may correspond to very poor", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 191, + 303 + ], + "score": 1.0, + "content": "generated molecules", + "type": "text" + }, + { + "bbox": [ + 191, + 291, + 202, + 301 + ], + "score": 0.88, + "content": "\\hat { \\mathbf { x } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 290, + 392, + 303 + ], + "score": 1.0, + "content": "which may not have properties at all similar to", + "type": "text" + }, + { + "bbox": [ + 392, + 292, + 402, + 302 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 290, + 506, + 303 + ], + "score": 1.0, + "content": ". This can be sidestepped", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 473, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 473, + 315 + ], + "score": 1.0, + "content": "by training the property estimator jointly as part of an extended generative model on [x, y].", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 245, + 331 + ], + "score": 1.0, + "content": "We note that the property estimator", + "type": "text" + }, + { + "bbox": [ + 245, + 319, + 256, + 330 + ], + "score": 0.9, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 317, + 413, + 331 + ], + "score": 1.0, + "content": "parameterizes a probability distribution", + "type": "text" + }, + { + "bbox": [ + 413, + 318, + 475, + 330 + ], + "score": 0.92, + "content": "p _ { \\omega } ( f ( \\mathbf { x } ) | \\mathbf { z } , \\mathbf { y } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 317, + 506, + 331 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 173, + 341 + ], + "score": 0.93, + "content": "\\mathbf { x } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 329, + 305, + 342 + ], + "score": 1.0, + "content": "and f is the oracle function that", + "type": "text" + }, + { + "bbox": [ + 305, + 329, + 346, + 341 + ], + "score": 0.93, + "content": "f ( \\mathbf { x } ) = \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 329, + 505, + 342 + ], + "score": 1.0, + "content": ". With a Gaussian distribution over the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 341, + 197, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 197, + 351 + ], + "score": 1.0, + "content": "error, we can consider", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 355, + 405, + 370 + ], + "lines": [ + { + "bbox": [ + 206, + 355, + 405, + 370 + ], + "spans": [ + { + "bbox": [ + 206, + 355, + 405, + 370 + ], + "score": 0.9, + "content": "p _ { \\omega } ( f ( \\mathbf { x } ) \\vert \\mathbf { z } , \\mathbf { y } _ { 0 } ) = \\mathcal { N } ( f ( \\mathbf { x } ) \\vert f _ { \\omega } ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { 0 } ) ) , \\lambda _ { 2 } ^ { - 1 } \\mathbf { I } )", + "type": "interline_equation", + "image_path": "f3b9079cadea7a9df088ce28fe8d191b08b86758e85349a90b14eca103e27037.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 206, + 355, + 405, + 370 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 170, + 387 + ], + "score": 1.0, + "content": "for small, fixed", + "type": "text" + }, + { + "bbox": [ + 171, + 375, + 182, + 385 + ], + "score": 0.87, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 373, + 506, + 387 + ], + "score": 1.0, + "content": ". 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\\frac { \\lambda _ { 2 } } { 2 } \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { \\boldsymbol { q } _ { \\boldsymbol { \\phi } } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\| f _ { \\boldsymbol { \\omega } } \\big ( g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| _ { 2 } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "d42c69718c2bfc4ea19cb72b2db20cf4880d9a13e3412b0b80eaf9b780f40e24.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 151, + 501, + 459, + 512.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 151, + 512.6666666666666, + 459, + 524.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 151, + 524.3333333333333, + 459, + 535.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 539, + 473, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 473, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 198, + 554 + ], + "score": 1.0, + "content": "where we also see that", + "type": "text" + }, + { + "bbox": [ + 198, + 540, + 333, + 552 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { E L B O } ( \\omega , \\theta , \\phi ) \\leq \\mathcal { L } _ { E L B O } ( \\theta , \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 539, + 473, + 554 + ], + "score": 1.0, + "content": ", allowing us to define an objective", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 557, + 400, + 572 + ], + "lines": [ + { + "bbox": [ + 210, + 557, + 400, + 572 + ], + "spans": [ + { + "bbox": [ + 210, + 557, + 400, + 572 + ], + "score": 0.89, + "content": "\\hat { \\mathcal { L } } ( \\omega , \\theta , \\phi ) = \\mathcal { L } _ { E L B O } ( \\omega , \\theta , \\phi ) - 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We thus define a regularization term", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 116, + 506, + 151 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 155, + 411, + 189 + ], + "lines": [ + { + "bbox": [ + 199, + 155, + 411, + 189 + ], + "spans": [ + { + "bbox": [ + 199, + 155, + 411, + 189 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { d i s e n t } ( \\theta , \\omega ) = \\frac { \\lambda _ { 1 } } { 2 } \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { p ( \\mathbf { z } ) } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 }", + "type": "interline_equation", + "image_path": "e46441682ad2adc01467e4ee612ea4f93bd7a475626f79c4ad3a6bd5d8ff42cb.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 155, + 411, + 172.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 199, + 172.0, + 411, + 189.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 504, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 207 + ], + "score": 1.0, + "content": "which can be used as a drop-in replacement for the non-differentiable penalty term in Eq. (3), yielding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 228, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 228, + 217 + ], + "score": 1.0, + "content": "a candidate objective function", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 191, + 505, + 217 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 220, + 413, + 236 + ], + "lines": [ + { + "bbox": [ + 198, + 220, + 413, + 236 + ], + "spans": [ + { + "bbox": [ + 198, + 220, + 413, + 236 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\theta , \\phi ) \\approx \\hat { \\mathcal { L } } _ { \\omega } ( \\theta , \\phi ) = \\mathcal { L } _ { E L B O } ( \\theta , \\phi ) - \\mathcal { L } _ { d i s e n t } ( \\theta , \\omega )", + "type": "interline_equation", + "image_path": "a9bc09229d3d3410c05de845a0f50287eeb9aa745a3439d6933f023664f4608d.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 198, + 220, + 413, + 236 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 448, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 450, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 450, + 259 + ], + "score": 1.0, + "content": "3.3 LEARNING THE PROPERTY ESTIMATOR JOINTLY WITH GENERATIVE MODEL", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 248, + 450, + 259 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 267, + 506, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 284, + 282 + ], + "score": 1.0, + "content": "While one could imagine attempting to learn", + "type": "text" + }, + { + "bbox": [ + 284, + 269, + 295, + 280 + ], + "score": 0.89, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 267, + 345, + 282 + ], + "score": 1.0, + "content": "jointly with", + "type": "text" + }, + { + "bbox": [ + 345, + 268, + 362, + 280 + ], + "score": 0.9, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 267, + 506, + 282 + ], + "score": 1.0, + "content": "by direct optimization of Eq. (6), in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 279, + 293 + ], + "score": 1.0, + "content": "practice this is very unstable, as values of", + "type": "text" + }, + { + "bbox": [ + 279, + 280, + 315, + 291 + ], + "score": 0.92, + "content": "g _ { \\boldsymbol { \\theta } } ( \\mathbf { z } , \\mathbf { y } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "early in training may correspond to very poor", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 191, + 303 + ], + "score": 1.0, + "content": "generated molecules", + "type": "text" + }, + { + "bbox": [ + 191, + 291, + 202, + 301 + ], + "score": 0.88, + "content": "\\hat { \\mathbf { x } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 290, + 392, + 303 + ], + "score": 1.0, + "content": "which may not have properties at all similar to", + "type": "text" + }, + { + "bbox": [ + 392, + 292, + 402, + 302 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 290, + 506, + 303 + ], + "score": 1.0, + "content": ". This can be sidestepped", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 473, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 473, + 315 + ], + "score": 1.0, + "content": "by training the property estimator jointly as part of an extended generative model on [x, y].", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 267, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 245, + 331 + ], + "score": 1.0, + "content": "We note that the property estimator", + "type": "text" + }, + { + "bbox": [ + 245, + 319, + 256, + 330 + ], + "score": 0.9, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 317, + 413, + 331 + ], + "score": 1.0, + "content": "parameterizes a probability distribution", + "type": "text" + }, + { + "bbox": [ + 413, + 318, + 475, + 330 + ], + "score": 0.92, + "content": "p _ { \\omega } ( f ( \\mathbf { x } ) | \\mathbf { z } , \\mathbf { y } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 317, + 506, + 331 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 173, + 341 + ], + "score": 0.93, + "content": "\\mathbf { x } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 329, + 305, + 342 + ], + "score": 1.0, + "content": "and f is the oracle function that", + "type": "text" + }, + { + "bbox": [ + 305, + 329, + 346, + 341 + ], + "score": 0.93, + "content": "f ( \\mathbf { x } ) = \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 329, + 505, + 342 + ], + "score": 1.0, + "content": ". With a Gaussian distribution over the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 341, + 197, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 197, + 351 + ], + "score": 1.0, + "content": "error, we can consider", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 317, + 506, + 351 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 355, + 405, + 370 + ], + "lines": [ + { + "bbox": [ + 206, + 355, + 405, + 370 + ], + "spans": [ + { + "bbox": [ + 206, + 355, + 405, + 370 + ], + "score": 0.9, + "content": "p _ { \\omega } ( f ( \\mathbf { x } ) \\vert \\mathbf { z } , \\mathbf { y } _ { 0 } ) = \\mathcal { N } ( f ( \\mathbf { x } ) \\vert f _ { \\omega } ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { 0 } ) ) , \\lambda _ { 2 } ^ { - 1 } \\mathbf { I } )", + "type": "interline_equation", + "image_path": "f3b9079cadea7a9df088ce28fe8d191b08b86758e85349a90b14eca103e27037.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 206, + 355, + 405, + 370 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 170, + 387 + ], + "score": 1.0, + "content": "for small, fixed", + "type": "text" + }, + { + "bbox": [ + 171, + 375, + 182, + 385 + ], + "score": 0.87, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 373, + 506, + 387 + ], + "score": 1.0, + "content": ". 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ModelQM9ZINC
Reconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE Gómez-Bombarelli et al. (2016)3.6110.30=90.044.60.70·100
GVAE Kusner et al. (2017)96.0060.20=80.9053.707.20=100
SD-VAE Dai et al.(2018)97.8498.4099.2891.9776.2043.50--
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8494.61100
CGD-VAE-1-GRU99.2795.6193.6587.8788.6429.0099.24100
Sup-VAE-3-GRU97.8197.9095.0989.4782.4036.1686.26100
CGD-VAE-3-GRU99.3197,8098.7796.2181.8037.7898.75100
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Our goal here is two-fold: we would like to understand (1) whether a supervised", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "variational autoencoder is capable of learning suitable conditional distributions over molecules, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "(2) to what extent this task is assisted by the additional regularization term corresponding to the soft", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 523, + 150, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 150, + 534 + ], + "score": 1.0, + "content": "constraint.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "We represent molecules using the one-hot encoding of their SMILES production rules (Kusner et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "2017) and add a semantic constraint (Dai et al., 2018) on the decoder network to avoid generating", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "syntactically correct but semantically invalid molecules. We use 80 production rules to describe", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "molecules and set the maximum SMILES sequence length to 100 for the QM9 dataset and 278 for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "the Zinc dataset. We experiment with the logP property of the molecules (Wildman & Crippen, 1999).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "We use the same encoder and decoder network structure as Dai et al. 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We give the details of the architecture in the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 617, + 193, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 193, + 628 + ], + "score": 1.0, + "content": "appendix section 6.2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "We evaluate the reconstruction accuracy and the quality of the molecules generated by our method,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "which we denote by CGD-VAE (conditional generation with disentangling) and compare against", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "CVAE (Gomez-Bombarelli et al., 2016), GVAE (Kusner et al., 2017), and SD-VAE (Dai et al., 2018). ´", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "We explore its conditional generation performance in two settings: controlling only the property value", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "and controlling both the property value and the molecule structure to what can be seen as property", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "transfer. 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ModelQM9ZINC
Reconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE Gómez-Bombarelli et al. (2016)3.6110.30=90.044.60.70·100
GVAE Kusner et al. (2017)96.0060.20=80.9053.707.20=100
SD-VAE Dai et al.(2018)97.8498.4099.2891.9776.2043.50--
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8494.61100
CGD-VAE-1-GRU99.2795.6193.6587.8788.6429.0099.24100
Sup-VAE-3-GRU97.8197.9095.0989.4782.4036.1686.26100
CGD-VAE-3-GRU99.3197,8098.7796.2181.8037.7898.75100
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This can be evaluated easily by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "simply evaluating the penalty term of Eq. (10) twice per minibatch; once as in Eq. (10), and once to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 404, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 159, + 418 + ], + "score": 1.0, + "content": "approximate", + "type": "text" + }, + { + "bbox": [ + 159, + 405, + 214, + 417 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { d i s e n t } ( \\theta , \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 404, + 506, + 418 + ], + "score": 1.0, + "content": "by permuting the properties in the minibatch to be assigned to incorrect", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 416, + 406, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 406, + 428 + ], + "score": 1.0, + "content": "molecules. We detail the training algorithm in Section 6.2 of the appendix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 361, + 506, + 428 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 443, + 200, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 201, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 201, + 457 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 438, + 479 + ], + "score": 1.0, + "content": "We experiment with the QM9 dataset (Ramakrishnan et al., 2014), that contains", + "type": "text" + }, + { + "bbox": [ + 438, + 468, + 460, + 478 + ], + "score": 0.38, + "content": "1 3 4 \\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "molecules", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 476, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 476, + 458, + 493 + ], + "score": 1.0, + "content": "with up to 9 heavy atoms, and the ZINC dataset (Sterling & Irwin, 2015) containing", + "type": "text" + }, + { + "bbox": [ + 459, + 479, + 481, + 489 + ], + "score": 0.47, + "content": "2 5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 476, + 507, + 493 + ], + "score": 1.0, + "content": "drug-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "like molecules. Our goal here is two-fold: we would like to understand (1) whether a supervised", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "variational autoencoder is capable of learning suitable conditional distributions over molecules, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "(2) to what extent this task is assisted by the additional regularization term corresponding to the soft", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 523, + 150, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 150, + 534 + ], + "score": 1.0, + "content": "constraint.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 468, + 507, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "We represent molecules using the one-hot encoding of their SMILES production rules (Kusner et al.,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "2017) and add a semantic constraint (Dai et al., 2018) on the decoder network to avoid generating", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "syntactically correct but semantically invalid molecules. We use 80 production rules to describe", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "molecules and set the maximum SMILES sequence length to 100 for the QM9 dataset and 278 for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "the Zinc dataset. We experiment with the logP property of the molecules (Wildman & Crippen, 1999).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "We use the same encoder and decoder network structure as Dai et al. (2018) with the only difference", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 309, + 617 + ], + "score": 1.0, + "content": "that our decoder takes as input the concatenation of", + "type": "text" + }, + { + "bbox": [ + 309, + 606, + 327, + 616 + ], + "score": 0.48, + "content": "\\mathbf { y } , \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ". We give the details of the architecture in the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 617, + 193, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 193, + 628 + ], + "score": 1.0, + "content": "appendix section 6.2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 540, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "We evaluate the reconstruction accuracy and the quality of the molecules generated by our method,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "which we denote by CGD-VAE (conditional generation with disentangling) and compare against", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "CVAE (Gomez-Bombarelli et al., 2016), GVAE (Kusner et al., 2017), and SD-VAE (Dai et al., 2018). ´", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "We explore its conditional generation performance in two settings: controlling only the property value", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "and controlling both the property value and the molecule structure to what can be seen as property", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "transfer. We took the results of CVAE, GVAE from the literature. 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Instead of marginalizing over the prior", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 101, + 255, + 510, + 292 + ], + "spans": [ + { + "bbox": [ + 101, + 260, + 237, + 292 + ], + "score": 1.0, + "content": "marginal inference distribution mass of the dataset is in the late", + "type": "text" + }, + { + "bbox": [ + 106, + 255, + 125, + 267 + ], + "score": 0.9, + "content": "p ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 255, + 506, + 268 + ], + "score": 1.0, + "content": ", we mirror the approach taken during training and integrate over an approximation to the", + "type": "text" + }, + { + "bbox": [ + 237, + 266, + 349, + 281 + ], + "score": 0.93, + "content": "\\begin{array} { r } { q _ { \\phi } ( \\mathbf { z } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 260, + 510, + 292 + ], + "score": 1.0, + "content": "which better characterizes where the large and we do not wish to keep the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 330, + 280, + 340, + 289 + ], + "spans": [ + { + "bbox": [ + 330, + 280, + 340, + 289 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 319, + 303 + ], + "score": 1.0, + "content": "entire dataset available at test time, we approximate", + "type": "text" + }, + { + "bbox": [ + 320, + 290, + 343, + 303 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "with an isotropic Gaussian distribution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 196, + 314 + ], + "score": 0.93, + "content": "\\hat { q } _ { \\sigma } ( \\mathbf { z } ) = \\mathcal { N } ( \\mathbf { z } | \\mathbf { 0 } , \\sigma ^ { 2 } \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 301, + 256, + 315 + ], + "score": 1.0, + "content": ". We estimate", + "type": "text" + }, + { + "bbox": [ + 256, + 304, + 264, + 312 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 301, + 458, + 315 + ], + "score": 1.0, + "content": "for each model by Monte Carlo samples from", + "type": "text" + }, + { + "bbox": [ + 459, + 302, + 482, + 314 + ], + "score": 0.91, + "content": "q _ { \\phi } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 301, + 506, + 315 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "the supervised VAE without the soft constraint regularizer this yields 0.053 for QM9 and 0.118 for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "ZINC. For our model with the soft constraint we get 0.0354 for QM9 and 0.096 for ZINC. We do", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 334, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 209, + 348 + ], + "score": 1.0, + "content": "conditional generation of", + "type": "text" + }, + { + "bbox": [ + 209, + 336, + 217, + 345 + ], + "score": 0.66, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 334, + 241, + 348 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 242, + 336, + 254, + 347 + ], + "score": 0.84, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 334, + 328, + 348 + ], + "score": 1.0, + "content": "by sampling from", + "type": "text" + }, + { + "bbox": [ + 328, + 335, + 467, + 348 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { 0 } ) = \\int \\hat { q } _ { \\sigma } ( \\mathbf { z } ) p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } _ { 0 } ) d \\mathbf { z } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 334, + 470, + 348 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 364 + ], + "score": 1.0, + "content": "We evaluate reconstruction performance in terms of the correctly reconstructed molecules on test", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 223, + 375 + ], + "score": 1.0, + "content": "sets of size 10k for QM9 and", + "type": "text" + }, + { + "bbox": [ + 223, + 363, + 235, + 373 + ], + "score": 0.29, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "for ZINC, for the latter we used the default test set. We evaluate the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "generated molecules’ quality by the percentage of valid, unique (i.e. percentage of unique molecules", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "among the generated valid molecules) and novel (i.e. percentage of molecules never seen in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 395, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 410 + ], + "score": 1.0, + "content": "training set among the generated molecules) molecules. We estimate these quantities by sampling", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 184, + 420 + ], + "score": 1.0, + "content": "10k (5K for ZINC)", + "type": "text" + }, + { + "bbox": [ + 184, + 408, + 191, + 417 + ], + "score": 0.28, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 406, + 228, + 420 + ], + "score": 1.0, + "content": "from the", + "type": "text" + }, + { + "bbox": [ + 228, + 407, + 252, + 419 + ], + "score": 0.92, + "content": "{ \\hat { q } } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 406, + 453, + 420 + ], + "score": 1.0, + "content": "and coupling each one of them with a logP value,", + "type": "text" + }, + { + "bbox": [ + 453, + 409, + 461, + 418 + ], + "score": 0.32, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ", randomly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 353, + 430 + ], + "score": 1.0, + "content": "selected from the test set, and we subsequently decode the", + "type": "text" + }, + { + "bbox": [ + 353, + 419, + 371, + 429 + ], + "score": 0.8, + "content": "\\mathbf { z } , \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "concatenation. We can see that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "our model has a better reconstruction performance compared to the baselines while in some cases", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "generating slightly less valid molecules table 1. In terms of the three quality measures achieves an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "excellent performance across all three metrics being always one of the two best performing methods", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 462, + 167, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 167, + 474 + ], + "score": 1.0, + "content": "for any metric.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "To visualise how the conditional generation operates we randomly sample from the test set some", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 267, + 502 + ], + "score": 1.0, + "content": "molecule and obtain its property value", + "type": "text" + }, + { + "bbox": [ + 267, + 491, + 279, + 501 + ], + "score": 0.84, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 489, + 427, + 502 + ], + "score": 1.0, + "content": ". We then draw 50 random samples", + "type": "text" + }, + { + "bbox": [ + 428, + 491, + 438, + 501 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 489, + 462, + 502 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 462, + 489, + 486, + 501 + ], + "score": 0.92, + "content": "\\hat { q } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 153, + 513 + ], + "score": 1.0, + "content": "decode the", + "type": "text" + }, + { + "bbox": [ + 154, + 500, + 183, + 513 + ], + "score": 0.93, + "content": "\\left[ \\mathbf { z } _ { i } , \\mathbf { y } _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "vectors. Among the generated valid molecules we compute the percentage of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 232, + 524 + ], + "score": 1.0, + "content": "those that have a property value", + "type": "text" + }, + { + "bbox": [ + 232, + 513, + 243, + 523 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 510, + 304, + 524 + ], + "score": 1.0, + "content": "that is within a", + "type": "text" + }, + { + "bbox": [ + 304, + 511, + 324, + 522 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 510, + 384, + 524 + ], + "score": 1.0, + "content": "range from the", + "type": "text" + }, + { + "bbox": [ + 384, + 513, + 396, + 523 + ], + "score": 0.85, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "property value. In Figure 3", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 405, + 534 + ], + "score": 1.0, + "content": "we present the molecules obtained for a test molecule that had a logP of", + "type": "text" + }, + { + "bbox": [ + 406, + 523, + 442, + 533 + ], + "score": 0.65, + "content": "- 0 . 5 7 5 9", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 523, + 505, + 534 + ], + "score": 1.0, + "content": ". Out of the 50", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 423, + 546 + ], + "score": 1.0, + "content": "generated molecules 46 were valid of which we give the five that were within a", + "type": "text" + }, + { + "bbox": [ + 424, + 533, + 443, + 544 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "range from the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 118, + 556 + ], + "score": 0.79, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "value in Figure 3. As we can see we get molecules that are structurally very different from the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 289, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 289, + 568 + ], + "score": 1.0, + "content": "original one yet they have similar logP value.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "To quantify the quality of the conditional generations we measure the correlation between the property", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "value we obtain by the conditional generation and the property value on which we conditioned the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 251, + 606 + ], + "score": 1.0, + "content": "generation. We randomly sample", + "type": "text" + }, + { + "bbox": [ + 251, + 594, + 282, + 605 + ], + "score": 0.66, + "content": "1 0 0 0 { \\textbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 594, + 405, + 606 + ], + "score": 1.0, + "content": "values from the test set and", + "type": "text" + }, + { + "bbox": [ + 405, + 594, + 435, + 604 + ], + "score": 0.78, + "content": "1 0 0 0 \\textbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "values from the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 211, + 617 + ], + "score": 1.0, + "content": "approximate learned prior", + "type": "text" + }, + { + "bbox": [ + 212, + 605, + 235, + 617 + ], + "score": 0.92, + "content": "\\hat { q } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 605, + 353, + 617 + ], + "score": 1.0, + "content": ". We decode each pair, obtain", + "type": "text" + }, + { + "bbox": [ + 353, + 605, + 415, + 617 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 605, + 506, + 617 + ], + "score": 1.0, + "content": ", and then measure the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 254, + 629 + ], + "score": 1.0, + "content": "correlation of the original y with the", + "type": "text" + }, + { + "bbox": [ + 254, + 616, + 262, + 627 + ], + "score": 0.83, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 615, + 315, + 629 + ], + "score": 1.0, + "content": "of generated", + "type": "text" + }, + { + "bbox": [ + 315, + 616, + 323, + 626 + ], + "score": 0.42, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 615, + 506, + 629 + ], + "score": 1.0, + "content": ". In Table 2, we give the correlation estimates", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "for our method and the Sup-VAE baselines. As we can see our method has a considerably higher", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "correlation score between the input and the obtained property than Sup-VAE. Conditional generation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 365, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 365, + 660 + ], + "score": 1.0, + "content": "seems considerably harder for the ZINC dataset for all methods.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 504, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 472, + 680 + ], + "score": 1.0, + "content": "To visualise the style transfer behavior of our model we randomly sample two molecules", + "type": "text" + }, + { + "bbox": [ + 473, + 668, + 504, + 677 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { A } , \\mathbf { x } _ { B }", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 241, + 690 + ], + "score": 1.0, + "content": "from the test set. We then sample", + "type": "text" + }, + { + "bbox": [ + 241, + 678, + 254, + 688 + ], + "score": 0.87, + "content": "\\mathbf { z } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 677, + 361, + 690 + ], + "score": 1.0, + "content": "from the learned posterior", + "type": "text" + }, + { + "bbox": [ + 361, + 677, + 400, + 689 + ], + "score": 0.92, + "content": "q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { A } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 677, + 505, + 690 + ], + "score": 1.0, + "content": ". We subsequently decode", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 107, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 142, + 699 + ], + "score": 0.88, + "content": "\\left[ \\mathbf { z } _ { A } , \\mathbf { y } _ { B } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 687, + 146, + 700 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 146, + 689, + 160, + 699 + ], + "score": 0.61, + "content": "\\mathbf { y } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 687, + 235, + 700 + ], + "score": 1.0, + "content": "is the property of", + "type": "text" + }, + { + "bbox": [ + 236, + 690, + 249, + 699 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 687, + 353, + 700 + ], + "score": 1.0, + "content": ", and get a new molecule", + "type": "text" + }, + { + "bbox": [ + 353, + 689, + 373, + 699 + ], + "score": 0.89, + "content": "\\hat { \\mathbf { x } } _ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 687, + 505, + 700 + ], + "score": 1.0, + "content": ". Ideally, the obtained molecule", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 127, + 710 + ], + "score": 0.89, + "content": "\\hat { \\mathbf { x } } _ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 698, + 342, + 712 + ], + "score": 1.0, + "content": "should have a property value (logP) close to the target", + "type": "text" + }, + { + "bbox": [ + 343, + 700, + 357, + 710 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 698, + 426, + 712 + ], + "score": 1.0, + "content": "and be similar to", + "type": "text" + }, + { + "bbox": [ + 426, + 700, + 440, + 710 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". In Figure 4 we", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "give one such example. To put the results into context in Figure 8 in appendix, we give the results of a", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "virtual screening method, where we select from the full dataset five molecules which are structurally", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 86, + 321, + 164 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 86, + 321, + 164 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 86, + 321, + 164 + ], + "spans": [ + { + "bbox": [ + 110, + 86, + 321, + 164 + ], + "score": 0.942, + "type": "image", + "image_path": "d83af7d84ad5e4b1f43f6ce0ea2358c91c263cc5eae0206da718a5bc7d37f5ab.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 110, + 86, + 321, + 99.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 99.0, + 321, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 112.0, + 321, + 125.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 110, + 125.0, + 321, + 138.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 110, + 138.0, + 321, + 151.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 110, + 151.0, + 321, + 164.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 174, + 303, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 173, + 303, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 303, + 186 + ], + "score": 1.0, + "content": "Figure 3: Conditional generation given the de-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 184, + 303, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 129, + 198 + ], + "score": 1.0, + "content": "sired", + "type": "text" + }, + { + "bbox": [ + 129, + 185, + 186, + 195 + ], + "score": 0.34, + "content": "\\mathrm { l o g P { = } { - } 0 . 5 7 5 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 184, + 303, + 198 + ], + "score": 1.0, + "content": ", row molecules have a logP", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 195, + 263, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 141, + 208 + ], + "score": 1.0, + "content": "within a", + "type": "text" + }, + { + "bbox": [ + 141, + 196, + 160, + 207 + ], + "score": 0.88, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 195, + 263, + 208 + ], + "score": 1.0, + "content": "range of the desired one.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + } + ], + "index": 5.75 + }, + { + "type": "image", + "bbox": [ + 338, + 79, + 477, + 179 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 338, + 79, + 477, + 179 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 338, + 79, + 477, + 179 + ], + "spans": [ + { + "bbox": [ + 338, + 79, + 477, + 179 + ], + "score": 0.963, + "type": "image", + "image_path": "e0f8c116c2f73cfe17b82b2094f1aeb8977e80722189f751d7235bfe6e3e375d.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 338, + 79, + 477, + 129.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 338, + 129.0, + 477, + 179.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 353, + 193, + 461, + 204 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 351, + 191, + 462, + 207 + ], + "spans": [ + { + "bbox": [ + 351, + 191, + 462, + 207 + ], + "score": 1.0, + "content": "Figure 4: Property transfer", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + } + ], + "index": 8.75 + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 232, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 504, + 244 + ], + "score": 1.0, + "content": "Before proceeding with the experiments we will give some additional details on how we do conditional", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 176, + 257 + ], + "score": 1.0, + "content": "generation from", + "type": "text" + }, + { + "bbox": [ + 176, + 244, + 214, + 256 + ], + "score": 0.93, + "content": "p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 244, + 321, + 257 + ], + "score": 1.0, + "content": "given the target property", + "type": "text" + }, + { + "bbox": [ + 322, + 246, + 334, + 256 + ], + "score": 0.83, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 244, + 506, + 257 + ], + "score": 1.0, + "content": ". Instead of marginalizing over the prior", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 101, + 255, + 510, + 292 + ], + "spans": [ + { + "bbox": [ + 101, + 260, + 237, + 292 + ], + "score": 1.0, + "content": "marginal inference distribution mass of the dataset is in the late", + "type": "text" + }, + { + "bbox": [ + 106, + 255, + 125, + 267 + ], + "score": 0.9, + "content": "p ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 255, + 506, + 268 + ], + "score": 1.0, + "content": ", we mirror the approach taken during training and integrate over an approximation to the", + "type": "text" + }, + { + "bbox": [ + 237, + 266, + 349, + 281 + ], + "score": 0.93, + "content": "\\begin{array} { r } { q _ { \\phi } ( \\mathbf { z } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 260, + 510, + 292 + ], + "score": 1.0, + "content": "which better characterizes where the large and we do not wish to keep the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 330, + 280, + 340, + 289 + ], + "spans": [ + { + "bbox": [ + 330, + 280, + 340, + 289 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 319, + 303 + ], + "score": 1.0, + "content": "entire dataset available at test time, we approximate", + "type": "text" + }, + { + "bbox": [ + 320, + 290, + 343, + 303 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "with an isotropic Gaussian distribution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 196, + 314 + ], + "score": 0.93, + "content": "\\hat { q } _ { \\sigma } ( \\mathbf { z } ) = \\mathcal { N } ( \\mathbf { z } | \\mathbf { 0 } , \\sigma ^ { 2 } \\mathbf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 301, + 256, + 315 + ], + "score": 1.0, + "content": ". We estimate", + "type": "text" + }, + { + "bbox": [ + 256, + 304, + 264, + 312 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 301, + 458, + 315 + ], + "score": 1.0, + "content": "for each model by Monte Carlo samples from", + "type": "text" + }, + { + "bbox": [ + 459, + 302, + 482, + 314 + ], + "score": 0.91, + "content": "q _ { \\phi } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 301, + 506, + 315 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "the supervised VAE without the soft constraint regularizer this yields 0.053 for QM9 and 0.118 for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "ZINC. For our model with the soft constraint we get 0.0354 for QM9 and 0.096 for ZINC. We do", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 334, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 209, + 348 + ], + "score": 1.0, + "content": "conditional generation of", + "type": "text" + }, + { + "bbox": [ + 209, + 336, + 217, + 345 + ], + "score": 0.66, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 334, + 241, + 348 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 242, + 336, + 254, + 347 + ], + "score": 0.84, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 334, + 328, + 348 + ], + "score": 1.0, + "content": "by sampling from", + "type": "text" + }, + { + "bbox": [ + 328, + 335, + 467, + 348 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { 0 } ) = \\int \\hat { q } _ { \\sigma } ( \\mathbf { z } ) p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } _ { 0 } ) d \\mathbf { z } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 334, + 470, + 348 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 101, + 232, + 510, + 348 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 364 + ], + "score": 1.0, + "content": "We evaluate reconstruction performance in terms of the correctly reconstructed molecules on test", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 223, + 375 + ], + "score": 1.0, + "content": "sets of size 10k for QM9 and", + "type": "text" + }, + { + "bbox": [ + 223, + 363, + 235, + 373 + ], + "score": 0.29, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "for ZINC, for the latter we used the default test set. We evaluate the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "generated molecules’ quality by the percentage of valid, unique (i.e. percentage of unique molecules", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "among the generated valid molecules) and novel (i.e. percentage of molecules never seen in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 395, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 410 + ], + "score": 1.0, + "content": "training set among the generated molecules) molecules. We estimate these quantities by sampling", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 184, + 420 + ], + "score": 1.0, + "content": "10k (5K for ZINC)", + "type": "text" + }, + { + "bbox": [ + 184, + 408, + 191, + 417 + ], + "score": 0.28, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 406, + 228, + 420 + ], + "score": 1.0, + "content": "from the", + "type": "text" + }, + { + "bbox": [ + 228, + 407, + 252, + 419 + ], + "score": 0.92, + "content": "{ \\hat { q } } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 406, + 453, + 420 + ], + "score": 1.0, + "content": "and coupling each one of them with a logP value,", + "type": "text" + }, + { + "bbox": [ + 453, + 409, + 461, + 418 + ], + "score": 0.32, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ", randomly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 353, + 430 + ], + "score": 1.0, + "content": "selected from the test set, and we subsequently decode the", + "type": "text" + }, + { + "bbox": [ + 353, + 419, + 371, + 429 + ], + "score": 0.8, + "content": "\\mathbf { z } , \\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "concatenation. We can see that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "our model has a better reconstruction performance compared to the baselines while in some cases", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "generating slightly less valid molecules table 1. In terms of the three quality measures achieves an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "excellent performance across all three metrics being always one of the two best performing methods", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 462, + 167, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 167, + 474 + ], + "score": 1.0, + "content": "for any metric.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 352, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "To visualise how the conditional generation operates we randomly sample from the test set some", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 267, + 502 + ], + "score": 1.0, + "content": "molecule and obtain its property value", + "type": "text" + }, + { + "bbox": [ + 267, + 491, + 279, + 501 + ], + "score": 0.84, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 489, + 427, + 502 + ], + "score": 1.0, + "content": ". We then draw 50 random samples", + "type": "text" + }, + { + "bbox": [ + 428, + 491, + 438, + 501 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 489, + 462, + 502 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 462, + 489, + 486, + 501 + ], + "score": 0.92, + "content": "\\hat { q } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 153, + 513 + ], + "score": 1.0, + "content": "decode the", + "type": "text" + }, + { + "bbox": [ + 154, + 500, + 183, + 513 + ], + "score": 0.93, + "content": "\\left[ \\mathbf { z } _ { i } , \\mathbf { y } _ { 0 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "vectors. Among the generated valid molecules we compute the percentage of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 232, + 524 + ], + "score": 1.0, + "content": "those that have a property value", + "type": "text" + }, + { + "bbox": [ + 232, + 513, + 243, + 523 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 510, + 304, + 524 + ], + "score": 1.0, + "content": "that is within a", + "type": "text" + }, + { + "bbox": [ + 304, + 511, + 324, + 522 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 510, + 384, + 524 + ], + "score": 1.0, + "content": "range from the", + "type": "text" + }, + { + "bbox": [ + 384, + 513, + 396, + 523 + ], + "score": 0.85, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "property value. In Figure 3", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 405, + 534 + ], + "score": 1.0, + "content": "we present the molecules obtained for a test molecule that had a logP of", + "type": "text" + }, + { + "bbox": [ + 406, + 523, + 442, + 533 + ], + "score": 0.65, + "content": "- 0 . 5 7 5 9", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 523, + 505, + 534 + ], + "score": 1.0, + "content": ". Out of the 50", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 423, + 546 + ], + "score": 1.0, + "content": "generated molecules 46 were valid of which we give the five that were within a", + "type": "text" + }, + { + "bbox": [ + 424, + 533, + 443, + 544 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "range from the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 118, + 556 + ], + "score": 0.79, + "content": "\\mathbf { y } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "value in Figure 3. As we can see we get molecules that are structurally very different from the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 289, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 289, + 568 + ], + "score": 1.0, + "content": "original one yet they have similar logP value.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 478, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "To quantify the quality of the conditional generations we measure the correlation between the property", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "value we obtain by the conditional generation and the property value on which we conditioned the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 251, + 606 + ], + "score": 1.0, + "content": "generation. We randomly sample", + "type": "text" + }, + { + "bbox": [ + 251, + 594, + 282, + 605 + ], + "score": 0.66, + "content": "1 0 0 0 { \\textbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 594, + 405, + 606 + ], + "score": 1.0, + "content": "values from the test set and", + "type": "text" + }, + { + "bbox": [ + 405, + 594, + 435, + 604 + ], + "score": 0.78, + "content": "1 0 0 0 \\textbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "values from the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 211, + 617 + ], + "score": 1.0, + "content": "approximate learned prior", + "type": "text" + }, + { + "bbox": [ + 212, + 605, + 235, + 617 + ], + "score": 0.92, + "content": "\\hat { q } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 605, + 353, + 617 + ], + "score": 1.0, + "content": ". We decode each pair, obtain", + "type": "text" + }, + { + "bbox": [ + 353, + 605, + 415, + 617 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 605, + 506, + 617 + ], + "score": 1.0, + "content": ", and then measure the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 254, + 629 + ], + "score": 1.0, + "content": "correlation of the original y with the", + "type": "text" + }, + { + "bbox": [ + 254, + 616, + 262, + 627 + ], + "score": 0.83, + "content": "\\hat { \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 615, + 315, + 629 + ], + "score": 1.0, + "content": "of generated", + "type": "text" + }, + { + "bbox": [ + 315, + 616, + 323, + 626 + ], + "score": 0.42, + "content": "\\hat { \\bf x }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 615, + 506, + 629 + ], + "score": 1.0, + "content": ". In Table 2, we give the correlation estimates", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "for our method and the Sup-VAE baselines. As we can see our method has a considerably higher", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "correlation score between the input and the obtained property than Sup-VAE. Conditional generation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 365, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 365, + 660 + ], + "score": 1.0, + "content": "seems considerably harder for the ZINC dataset for all methods.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 571, + 506, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 504, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 472, + 680 + ], + "score": 1.0, + "content": "To visualise the style transfer behavior of our model we randomly sample two molecules", + "type": "text" + }, + { + "bbox": [ + 473, + 668, + 504, + 677 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { A } , \\mathbf { x } _ { B }", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 241, + 690 + ], + "score": 1.0, + "content": "from the test set. We then sample", + "type": "text" + }, + { + "bbox": [ + 241, + 678, + 254, + 688 + ], + "score": 0.87, + "content": "\\mathbf { z } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 677, + 361, + 690 + ], + "score": 1.0, + "content": "from the learned posterior", + "type": "text" + }, + { + "bbox": [ + 361, + 677, + 400, + 689 + ], + "score": 0.92, + "content": "q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } _ { A } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 677, + 505, + 690 + ], + "score": 1.0, + "content": ". We subsequently decode", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 107, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 142, + 699 + ], + "score": 0.88, + "content": "\\left[ \\mathbf { z } _ { A } , \\mathbf { y } _ { B } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 687, + 146, + 700 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 146, + 689, + 160, + 699 + ], + "score": 0.61, + "content": "\\mathbf { y } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 687, + 235, + 700 + ], + "score": 1.0, + "content": "is the property of", + "type": "text" + }, + { + "bbox": [ + 236, + 690, + 249, + 699 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 687, + 353, + 700 + ], + "score": 1.0, + "content": ", and get a new molecule", + "type": "text" + }, + { + "bbox": [ + 353, + 689, + 373, + 699 + ], + "score": 0.89, + "content": "\\hat { \\mathbf { x } } _ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 687, + 505, + 700 + ], + "score": 1.0, + "content": ". Ideally, the obtained molecule", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 127, + 710 + ], + "score": 0.89, + "content": "\\hat { \\mathbf { x } } _ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 698, + 342, + 712 + ], + "score": 1.0, + "content": "should have a property value (logP) close to the target", + "type": "text" + }, + { + "bbox": [ + 343, + 700, + 357, + 710 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 698, + 426, + 712 + ], + "score": 1.0, + "content": "and be similar to", + "type": "text" + }, + { + "bbox": [ + 426, + 700, + 440, + 710 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". In Figure 4 we", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "give one such example. To put the results into context in Figure 8 in appendix, we give the results of a", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "virtual screening method, where we select from the full dataset five molecules which are structurally", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 146, + 397 + ], + "score": 1.0, + "content": "similar to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 146, + 387, + 160, + 396 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { A }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 160, + 384, + 277, + 397 + ], + "score": 1.0, + "content": "and have logP values close to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 278, + 387, + 291, + 396 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { B }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 292, + 384, + 505, + 397 + ], + "score": 1.0, + "content": ". As we can see the molecule that our model generates", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 396, + 161, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 161, + 408 + ], + "score": 1.0, + "content": "is a new one.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 663, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 128, + 66, + 475, + 315 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 66, + 475, + 315 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 66, + 475, + 315 + ], + "spans": [ + { + "bbox": [ + 128, + 66, + 475, + 315 + ], + "score": 0.975, + "type": "image", + "image_path": "ffa61574cd2f40b5b194d3b7515745269b23e87d9fcf063708e5b80200110fb9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 128, + 66, + 475, + 149.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 128, + 149.0, + 475, + 232.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 128, + 232.0, + 475, + 315.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 324, + 504, + 347 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 221, + 337 + ], + "score": 1.0, + "content": "Figure 5: Style transfer. The", + "type": "text" + }, + { + "bbox": [ + 221, + 326, + 228, + 334 + ], + "score": 0.48, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 322, + 386, + 337 + ], + "score": 1.0, + "content": "of the nine real molecules placed in the", + "type": "text" + }, + { + "bbox": [ + 387, + 325, + 393, + 334 + ], + "score": 0.3, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 322, + 505, + 337 + ], + "score": 1.0, + "content": "-axis is combined with 11 y", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 335, + 437, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 437, + 347 + ], + "score": 1.0, + "content": "property values, sampled in [-4.9, 4.9], the resulting pair is decoded to a molecule.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 504, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 146, + 397 + ], + "score": 1.0, + "content": "similar to", + "type": "text" + }, + { + "bbox": [ + 146, + 387, + 160, + 396 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 384, + 277, + 397 + ], + "score": 1.0, + "content": "and have logP values close to", + "type": "text" + }, + { + "bbox": [ + 278, + 387, + 291, + 396 + ], + "score": 0.86, + "content": "\\mathbf { y } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 384, + 505, + 397 + ], + "score": 1.0, + "content": ". As we can see the molecule that our model generates", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 396, + 161, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 161, + 408 + ], + "score": 1.0, + "content": "is a new one.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "To quantify the style transfer performance we proceed in exactly the same manner as we did to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 459, + 437 + ], + "score": 1.0, + "content": "quantify the conditional generation performance. However, now instead of sampling", + "type": "text" + }, + { + "bbox": [ + 459, + 425, + 466, + 434 + ], + "score": 0.34, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "from the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 434, + 504, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 217, + 448 + ], + "score": 1.0, + "content": "approximate learned prior,", + "type": "text" + }, + { + "bbox": [ + 218, + 435, + 241, + 446 + ], + "score": 0.9, + "content": "{ \\hat { q } } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 434, + 334, + 448 + ], + "score": 1.0, + "content": ", we first sample some", + "type": "text" + }, + { + "bbox": [ + 335, + 436, + 343, + 445 + ], + "score": 0.43, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 434, + 497, + 448 + ], + "score": 1.0, + "content": "from the test set and then we sample", + "type": "text" + }, + { + "bbox": [ + 497, + 437, + 504, + 444 + ], + "score": 0.39, + "content": "\\mathbf { z }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 214, + 459 + ], + "score": 1.0, + "content": "from the learned posterior", + "type": "text" + }, + { + "bbox": [ + 215, + 446, + 242, + 458 + ], + "score": 0.92, + "content": "q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 445, + 506, + 459 + ], + "score": 1.0, + "content": ". The results are in the second column of Table 2. As we can see", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "the correlation values are now lower than the ones we obtained in the simple conditional generation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 466, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 507, + 482 + ], + "score": 1.0, + "content": "case. This can be explained by the fact that now we are forcing a specific combination of structure,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "z comes from a real molecule, and property, which might simply be physically infeasible since the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "molecule space is discrete and not all combinations are possible. In addition, as it was the case for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 500, + 476, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 476, + 513 + ], + "score": 1.0, + "content": "the conditional generation, style transfer is considerably more difficult for the ZINC dataset.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "We further explore the style transfer and visualize how our model covers the combined space of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 497, + 541 + ], + "score": 1.0, + "content": "molecule structure and properties. We sample nine molecules from the QM9 test set, and get their", + "type": "text" + }, + { + "bbox": [ + 497, + 530, + 504, + 538 + ], + "score": 0.52, + "content": "\\mathbf { z }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 344, + 552 + ], + "score": 1.0, + "content": "encodings. For each such encoding we decode the vectors", + "type": "text" + }, + { + "bbox": [ + 345, + 539, + 430, + 551 + ], + "score": 0.9, + "content": "[ \\mathbf { z } , y ] , y \\in [ - 4 . 9 , 4 . 9 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 539, + 470, + 552 + ], + "score": 1.0, + "content": ", with the", + "type": "text" + }, + { + "bbox": [ + 470, + 541, + 477, + 551 + ], + "score": 0.67, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "(logP)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "interval sampled at 11 points. We give in Figure 5 the resulting valid molecules, each column there", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "corresponds to one of the nine original molecules, the ones surrounded by dotted rectangle, and their", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "decodings with different logP values.For each original molecule we give the generated molecules", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "ordered along the y axis according to the y property that they actually exhibit. The x-axis does not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 347, + 606 + ], + "score": 1.0, + "content": "provide an ordering of the original molecules according to", + "type": "text" + }, + { + "bbox": [ + 347, + 596, + 353, + 604 + ], + "score": 0.36, + "content": "\\mathbf { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 594, + 506, + 606 + ], + "score": 1.0, + "content": ", in fact we have ordered the original", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 324, + 618 + ], + "score": 1.0, + "content": "molecules by their y property. As we can see not all", + "type": "text" + }, + { + "bbox": [ + 324, + 605, + 347, + 617 + ], + "score": 0.91, + "content": "( \\mathbf { z } , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "combinations produce a result. These", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "holes can be explained either by the physical infeasibility of the combination and/or a limitation of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 627, + 183, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 183, + 638 + ], + "score": 1.0, + "content": "the learned model.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "We can use conditional generation to control in a fine manner the value of the desired property, to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "what can be seen as direct property optimization. We visualise the level of control we have on an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "experiment with a single molecule (with logP is -1.137), which we randomly sample from the test", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 180, + 689 + ], + "score": 1.0, + "content": "set. We obtain its", + "type": "text" + }, + { + "bbox": [ + 181, + 679, + 187, + 687 + ], + "score": 0.32, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "encoding and perform generations with increased logP taking values in 1000", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 162, + 701 + ], + "score": 1.0, + "content": "point grid in", + "type": "text" + }, + { + "bbox": [ + 162, + 687, + 215, + 699 + ], + "score": 0.37, + "content": "\\left[ - 1 . 1 3 7 , 4 . 9 \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 687, + 290, + 701 + ], + "score": 1.0, + "content": ". We then decode", + "type": "text" + }, + { + "bbox": [ + 291, + 688, + 316, + 700 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { z } , \\mathbf { y } _ { i } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "and compute the logP value of the generated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "molecules. Among the 1000 generated molecules only 19 are unique. We get an increase of logP of a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "very discrete nature, Figure 6. 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The", + "type": "text" + }, + { + "bbox": [ + 221, + 326, + 228, + 334 + ], + "score": 0.48, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 322, + 386, + 337 + ], + "score": 1.0, + "content": "of the nine real molecules placed in the", + "type": "text" + }, + { + "bbox": [ + 387, + 325, + 393, + 334 + ], + "score": 0.3, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 322, + 505, + 337 + ], + "score": 1.0, + "content": "-axis is combined with 11 y", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 335, + 437, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 335, + 437, + 347 + ], + "score": 1.0, + "content": "property values, sampled in [-4.9, 4.9], the resulting pair is decoded to a molecule.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 504, + 407 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 105, + 384, + 505, + 408 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "To quantify the style transfer performance we proceed in exactly the same manner as we did to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 459, + 437 + ], + "score": 1.0, + "content": "quantify the conditional generation performance. However, now instead of sampling", + "type": "text" + }, + { + "bbox": [ + 459, + 425, + 466, + 434 + ], + "score": 0.34, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "from the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 434, + 504, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 217, + 448 + ], + "score": 1.0, + "content": "approximate learned prior,", + "type": "text" + }, + { + "bbox": [ + 218, + 435, + 241, + 446 + ], + "score": 0.9, + "content": "{ \\hat { q } } _ { \\sigma } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 434, + 334, + 448 + ], + "score": 1.0, + "content": ", we first sample some", + "type": "text" + }, + { + "bbox": [ + 335, + 436, + 343, + 445 + ], + "score": 0.43, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 434, + 497, + 448 + ], + "score": 1.0, + "content": "from the test set and then we sample", + "type": "text" + }, + { + "bbox": [ + 497, + 437, + 504, + 444 + ], + "score": 0.39, + "content": "\\mathbf { z }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 214, + 459 + ], + "score": 1.0, + "content": "from the learned posterior", + "type": "text" + }, + { + "bbox": [ + 215, + 446, + 242, + 458 + ], + "score": 0.92, + "content": "q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 445, + 506, + 459 + ], + "score": 1.0, + "content": ". The results are in the second column of Table 2. As we can see", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "the correlation values are now lower than the ones we obtained in the simple conditional generation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 466, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 507, + 482 + ], + "score": 1.0, + "content": "case. This can be explained by the fact that now we are forcing a specific combination of structure,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "z comes from a real molecule, and property, which might simply be physically infeasible since the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "molecule space is discrete and not all combinations are possible. In addition, as it was the case for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 500, + 476, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 476, + 513 + ], + "score": 1.0, + "content": "the conditional generation, style transfer is considerably more difficult for the ZINC dataset.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 413, + 507, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "We further explore the style transfer and visualize how our model covers the combined space of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 497, + 541 + ], + "score": 1.0, + "content": "molecule structure and properties. We sample nine molecules from the QM9 test set, and get their", + "type": "text" + }, + { + "bbox": [ + 497, + 530, + 504, + 538 + ], + "score": 0.52, + "content": "\\mathbf { z }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 344, + 552 + ], + "score": 1.0, + "content": "encodings. For each such encoding we decode the vectors", + "type": "text" + }, + { + "bbox": [ + 345, + 539, + 430, + 551 + ], + "score": 0.9, + "content": "[ \\mathbf { z } , y ] , y \\in [ - 4 . 9 , 4 . 9 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 539, + 470, + 552 + ], + "score": 1.0, + "content": ", with the", + "type": "text" + }, + { + "bbox": [ + 470, + 541, + 477, + 551 + ], + "score": 0.67, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "(logP)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "interval sampled at 11 points. We give in Figure 5 the resulting valid molecules, each column there", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "corresponds to one of the nine original molecules, the ones surrounded by dotted rectangle, and their", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "decodings with different logP values.For each original molecule we give the generated molecules", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "ordered along the y axis according to the y property that they actually exhibit. The x-axis does not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 347, + 606 + ], + "score": 1.0, + "content": "provide an ordering of the original molecules according to", + "type": "text" + }, + { + "bbox": [ + 347, + 596, + 353, + 604 + ], + "score": 0.36, + "content": "\\mathbf { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 594, + 506, + 606 + ], + "score": 1.0, + "content": ", in fact we have ordered the original", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 324, + 618 + ], + "score": 1.0, + "content": "molecules by their y property. As we can see not all", + "type": "text" + }, + { + "bbox": [ + 324, + 605, + 347, + 617 + ], + "score": 0.91, + "content": "( \\mathbf { z } , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "combinations produce a result. These", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "holes can be explained either by the physical infeasibility of the combination and/or a limitation of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 627, + 183, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 183, + 638 + ], + "score": 1.0, + "content": "the learned model.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 516, + 506, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "We can use conditional generation to control in a fine manner the value of the desired property, to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "what can be seen as direct property optimization. We visualise the level of control we have on an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "experiment with a single molecule (with logP is -1.137), which we randomly sample from the test", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 180, + 689 + ], + "score": 1.0, + "content": "set. We obtain its", + "type": "text" + }, + { + "bbox": [ + 181, + 679, + 187, + 687 + ], + "score": 0.32, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "encoding and perform generations with increased logP taking values in 1000", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 162, + 701 + ], + "score": 1.0, + "content": "point grid in", + "type": "text" + }, + { + "bbox": [ + 162, + 687, + 215, + 699 + ], + "score": 0.37, + "content": "\\left[ - 1 . 1 3 7 , 4 . 9 \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 687, + 290, + 701 + ], + "score": 1.0, + "content": ". We then decode", + "type": "text" + }, + { + "bbox": [ + 291, + 688, + 316, + 700 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { z } , \\mathbf { y } _ { i } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "and compute the logP value of the generated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "molecules. Among the 1000 generated molecules only 19 are unique. We get an increase of logP of a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "very discrete nature, Figure 6. As already discussed not all combinations of structure and properties", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 470, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 470, + 733 + ], + "score": 1.0, + "content": "are possible. The generated molecules themselves are shown in the supplemental material.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 80, + 287, + 169 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 80, + 287, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 287, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 287, + 169 + ], + "score": 0.976, + "html": "
Modelz~qo(z)z~q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.71850.5005
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.74140.4715
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.38770.0880
Sup-VAE-3-GRU0.35140.1808
CGD-VAE-3-GRU0.39660.1559
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Given a start", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 302, + 202, + 502, + 215 + ], + "spans": [ + { + "bbox": [ + 302, + 202, + 430, + 215 + ], + "score": 1.0, + "content": "molecule (red), we combine its", + "type": "text" + }, + { + "bbox": [ + 430, + 205, + 437, + 213 + ], + "score": 0.42, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 202, + 459, + 215 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 459, + 203, + 502, + 214 + ], + "score": 0.55, + "content": "1 0 0 0 \\log \\mathrm { P }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 301, + 212, + 404, + 227 + ], + "spans": [ + { + "bbox": [ + 301, + 212, + 404, + 227 + ], + "score": 1.0, + "content": "values and decode (blue)", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + } + ], + "index": 17.25 + }, + { + "type": "image", + "bbox": [ + 327, + 230, + 463, + 327 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 327, + 230, + 463, + 327 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 327, + 230, + 463, + 327 + ], + "spans": [ + { + "bbox": [ + 327, + 230, + 463, + 327 + ], + "score": 0.733, + "type": "image", + "image_path": "6daa2bdd2b6b3d32bf6a2bb30e18d85a323167036361749ae8c33e10b946aa5e.jpg" + } + ] + } + ], + "index": 24.0, + "virtual_lines": [ + { + "bbox": [ + 327, + 230, + 463, + 278.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 327, + 278.5, + 463, + 327.0 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "index": 24.0 + }, + { + "type": "image", + "bbox": [ + 145, + 230, + 289, + 327 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 230, + 289, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 230, + 289, + 327 + ], + "spans": [ + { + "bbox": [ + 145, + 230, + 289, + 327 + ], + "score": 0.948, + "type": "image", + "image_path": "171721555e584b5336b980f58a15f7e4a7ff8c35dce2ba5cd515c171c0be165b.jpg" + } + ] + } + ], + "index": 23.0, + "virtual_lines": [ + { + "bbox": [ + 145, + 230, + 289, + 278.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 145, + 278.5, + 289, + 327.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 338, + 505, + 404 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "Figure 7: A comparison of simulated logP values and Tanimoto similarity to a target on the ZINC", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 347, + 507, + 363 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 507, + 363 + ], + "score": 1.0, + "content": "dataset. While the stacked LSTM model has high accuracy in terms of matching the desired property,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "it would require drawing many samples before finding any close matches to any particular desired", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "prototype. The CGD-VAE-3-GRU model represents a middle ground between a standard VAE model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "which does not condition on the property, and the stacked LSTM model which does not learn a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 393, + 204, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 204, + 405 + ], + "score": 1.0, + "content": "reusable representation.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + } + ], + "index": 25.75 + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 268, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 270, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 270, + 440 + ], + "score": 1.0, + "content": "4.1 CONDITIONAL LSTM BASELINE", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 506, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 507, + 462 + ], + "score": 1.0, + "content": "Finally, we consider a variant of the stacked LSTM model of Segler et al. (2017), with no latent space,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "where the model is modified to take a target logP value as an additional input at each generation step.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 469, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 507, + 484 + ], + "score": 1.0, + "content": "This model forms a very strong baseline for many distribution matching tasks (Liu et al., 2018; ?),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "though as best we are aware this has never been used directly for conditional generation given a target", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "property. We use a modification of the implementation provided by (?) with three layers and default", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 503, + 497, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 355, + 520 + ], + "score": 1.0, + "content": "settings, and fit the model by maximum likelihood training on", + "type": "text" + }, + { + "bbox": [ + 356, + 503, + 493, + 518 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( x _ { t } | x _ { 1 : t - 1 } , \\mathbf { y } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 504, + 497, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "Training on the ZINC dataset, we find the generated molecules from this model have a very high", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "correlation 0.975 with the target logP value, greatly outperforming any of the latent variable models", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "we consider. This suggests that such a model would be very useful for generating candidates globally,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 555, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 567 + ], + "score": 1.0, + "content": "but as the model has no latent variable it is not amenable to style transfer. We observe this in Figure 7,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "score": 1.0, + "content": "which samples 100 candidate molecules from both the stacked LSTM model and for CGD-VAE-3-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "GRU, conditioning on the property of one randomly-chosen test set example, while computing the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "Tanimoto similarity (computed using Morgan fingerprints of radius 2) to a second randomly-chosen", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "test set example, across 200 pairs. 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Our model allows also to condition generation on a prototype molecule with a desired", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "high-level structure. This work thus directly inverts the traditional relationship between molecules and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "their properties. 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Modelz~qo(z)z~q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.71850.5005
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.74140.4715
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.38770.0880
Sup-VAE-3-GRU0.35140.1808
CGD-VAE-3-GRU0.39660.1559
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Given a start", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 302, + 202, + 502, + 215 + ], + "spans": [ + { + "bbox": [ + 302, + 202, + 430, + 215 + ], + "score": 1.0, + "content": "molecule (red), we combine its", + "type": "text" + }, + { + "bbox": [ + 430, + 205, + 437, + 213 + ], + "score": 0.42, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 202, + 459, + 215 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 459, + 203, + 502, + 214 + ], + "score": 0.55, + "content": "1 0 0 0 \\log \\mathrm { P }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 301, + 212, + 404, + 227 + ], + "spans": [ + { + "bbox": [ + 301, + 212, + 404, + 227 + ], + "score": 1.0, + "content": "values and decode (blue)", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + } + ], + "index": 17.25 + }, + { + "type": "image", + "bbox": [ + 327, + 230, + 463, + 327 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 327, + 230, + 463, + 327 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 327, + 230, + 463, + 327 + ], + "spans": [ + { + "bbox": [ + 327, + 230, + 463, + 327 + ], + "score": 0.733, + "type": "image", + "image_path": "6daa2bdd2b6b3d32bf6a2bb30e18d85a323167036361749ae8c33e10b946aa5e.jpg" + } + ] + } + ], + "index": 24.0, + "virtual_lines": [ + { + "bbox": [ + 327, + 230, + 463, + 278.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 327, + 278.5, + 463, + 327.0 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "index": 24.0 + }, + { + "type": "image", + "bbox": [ + 145, + 230, + 289, + 327 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 230, + 289, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 230, + 289, + 327 + ], + "spans": [ + { + "bbox": [ + 145, + 230, + 289, + 327 + ], + "score": 0.948, + "type": "image", + "image_path": "171721555e584b5336b980f58a15f7e4a7ff8c35dce2ba5cd515c171c0be165b.jpg" + } + ] + } + ], + "index": 23.0, + "virtual_lines": [ + { + "bbox": [ + 145, + 230, + 289, + 278.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 145, + 278.5, + 289, + 327.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 338, + 505, + 404 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "Figure 7: A comparison of simulated logP values and Tanimoto similarity to a target on the ZINC", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 347, + 507, + 363 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 507, + 363 + ], + "score": 1.0, + "content": "dataset. While the stacked LSTM model has high accuracy in terms of matching the desired property,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "it would require drawing many samples before finding any close matches to any particular desired", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "prototype. The CGD-VAE-3-GRU model represents a middle ground between a standard VAE model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "which does not condition on the property, and the stacked LSTM model which does not learn a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 393, + 204, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 204, + 405 + ], + "score": 1.0, + "content": "reusable representation.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + } + ], + "index": 25.75 + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 268, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 270, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 270, + 440 + ], + "score": 1.0, + "content": "4.1 CONDITIONAL LSTM BASELINE", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 506, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 507, + 462 + ], + "score": 1.0, + "content": "Finally, we consider a variant of the stacked LSTM model of Segler et al. (2017), with no latent space,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "where the model is modified to take a target logP value as an additional input at each generation step.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 469, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 507, + 484 + ], + "score": 1.0, + "content": "This model forms a very strong baseline for many distribution matching tasks (Liu et al., 2018; ?),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "though as best we are aware this has never been used directly for conditional generation given a target", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "property. We use a modification of the implementation provided by (?) with three layers and default", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 503, + 497, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 355, + 520 + ], + "score": 1.0, + "content": "settings, and fit the model by maximum likelihood training on", + "type": "text" + }, + { + "bbox": [ + 356, + 503, + 493, + 518 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } ) = \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( x _ { t } | x _ { 1 : t - 1 } , \\mathbf { y } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 504, + 497, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 447, + 507, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "Training on the ZINC dataset, we find the generated molecules from this model have a very high", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "correlation 0.975 with the target logP value, greatly outperforming any of the latent variable models", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "we consider. This suggests that such a model would be very useful for generating candidates globally,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 555, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 567 + ], + "score": 1.0, + "content": "but as the model has no latent variable it is not amenable to style transfer. We observe this in Figure 7,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 578 + ], + "score": 1.0, + "content": "which samples 100 candidate molecules from both the stacked LSTM model and for CGD-VAE-3-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "GRU, conditioning on the property of one randomly-chosen test set example, while computing the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "Tanimoto similarity (computed using Morgan fingerprints of radius 2) to a second randomly-chosen", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "test set example, across 200 pairs. The VAE has higher Tanimoto similarities as it can condition on", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "the latent variable of the target molecule, representing a trade-off against the better adherence to the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 622, + 308, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 308, + 632 + ], + "score": 1.0, + "content": "target property value of the unconditioned LSTM.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 522, + 506, + 632 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 650, + 195, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 197, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 197, + 666 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "We presented a single step approach for the conditional generation of molecules with desired", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "properties. Our model allows also to condition generation on a prototype molecule with a desired", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "high-level structure. This work thus directly inverts the traditional relationship between molecules and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "their properties. We found that training the deep generative models conditional on target properties,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "following a supervised VAE approach, does not appreciably harm the quality of the unconditional", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "generative model as measured by validity, novelty, and uniqueness of samples. 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\\hat { \\mathbf { x } } ) = H ( \\mathbf { y } ) - H ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) } \\\\ & { \\qquad = H ( \\mathbf { y } ) + \\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } \\vert \\mathbf { y } ) } \\mathbb { E } _ { \\mathbf { y } ^ { \\prime } \\sim \\tilde { p } ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) } [ \\log \\tilde { p } ( \\mathbf { y } ^ { \\prime } \\vert \\hat { \\mathbf { x } } ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "f3def3078a70acd5de4fbedc7647e95f7ee8cf236f0ac3b8770154a6f4a8731b.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 168, + 415, + 184.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 195, + 184.0, + 415, + 200.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 206, + 505, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 255, + 219 + ], + "score": 1.0, + "content": "We do not know the true distribution", + "type": "text" + }, + { + "bbox": [ + 255, + 207, + 284, + 218 + ], + "score": 0.92, + "content": "\\tilde { p } ( \\mathbf { y } | \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 206, + 464, + 219 + ], + "score": 1.0, + "content": ", however, the RDKit provides an estimation", + "type": "text" + }, + { + "bbox": [ + 464, + 206, + 493, + 218 + ], + "score": 0.93, + "content": "p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 206, + 506, + 219 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 217, + 348, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 348, + 229 + ], + "score": 1.0, + "content": "the distribution assuming a Gaussian distribution over error:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 234, + 363, + 250 + ], + "lines": [ + { + "bbox": [ + 248, + 234, + 363, + 250 + ], + "spans": [ + { + "bbox": [ + 248, + 234, + 363, + 250 + ], + "score": 0.91, + "content": "p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) = \\mathcal { N } ( \\mathbf { y } \\vert f ( \\hat { \\mathbf { x } } ) , \\lambda _ { 1 } ^ { - 1 } \\mathbf { I } )", + "type": "interline_equation", + "image_path": "289e50d0f7798a72284b200d021c3baab2f2a34670611452c4a211a23b8db034.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 248, + 234, + 363, + 250 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 371, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 256, + 371, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 133, + 270 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 257, + 140, + 268 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 256, + 371, + 270 + ], + "score": 1.0, + "content": "is the molecule property estimator, i.e., RDKit. We have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 274, + 467, + 335 + ], + "lines": [ + { + "bbox": [ + 143, + 274, + 467, + 335 + ], + "spans": [ + { + "bbox": [ + 143, + 274, + 467, + 335 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle { \\bf I } ( { \\bf y } ; \\hat { \\bf x } ) = H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } [ \\log \\frac { \\tilde { p } ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) } { p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) } p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } \\ ~ } \\\\ { { \\displaystyle ~ = H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } [ D _ { k l } ( \\tilde { p } ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) \\vert \\vert p ( { \\bf y } ^ { \\prime } \\vert \\hat { \\bf x } ) ) + \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } \\log p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } \\ ~ } \\\\ { { \\displaystyle ~ \\geq H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } [ \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } \\log p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } } \\end{array}", + "type": "interline_equation", + "image_path": "9313d409114746a7bb4d13520d49b98c88c18a40042204504361c3ccec2df0c1.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 143, + 274, + 467, + 294.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 143, + 294.3333333333333, + 467, + 314.66666666666663 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 143, + 314.66666666666663, + 467, + 334.99999999999994 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 371, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 371, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 371, + 353 + ], + "score": 1.0, + "content": "By following the Lemma 5.1 given in Chen et al. (2016), we have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 358, + 406, + 372 + ], + "lines": [ + { + "bbox": [ + 204, + 358, + 406, + 372 + ], + "spans": [ + { + "bbox": [ + 204, + 358, + 406, + 372 + ], + "score": 0.88, + "content": "\\mathbf { I } ( \\mathbf { y } ; \\hat { \\mathbf { x } } ) \\geq H ( \\mathbf { y } ) + \\mathbb { E } _ { \\mathbf { y } \\sim p ( \\mathbf { y } ) , \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } ) } [ \\log p ( \\mathbf { y } | \\hat { \\mathbf { x } } ) ]", + "type": "interline_equation", + "image_path": "7f4de000aa662d02a9dc5ba4efae62b0270f7b9250580018179096fe3e0471b3.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 204, + 358, + 406, + 372 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 119, + 394 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 119, + 380, + 143, + 392 + ], + "score": 0.92, + "content": "H ( \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 378, + 237, + 394 + ], + "score": 1.0, + "content": "is constant, minimizing", + "type": "text" + }, + { + "bbox": [ + 238, + 380, + 339, + 393 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { i } ) } \\| f ( \\hat { \\mathbf { x } } ) - \\mathbf { y } _ { i } \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 378, + 449, + 394 + ], + "score": 1.0, + "content": "is equivalent to maximizing", + "type": "text" + }, + { + "bbox": [ + 450, + 380, + 479, + 392 + ], + "score": 0.91, + "content": "\\mathbf { I } ( \\mathbf { y } ; \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 378, + 506, + 394 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 390, + 288, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 186, + 406 + ], + "score": 1.0, + "content": "the assumption that", + "type": "text" + }, + { + "bbox": [ + 187, + 392, + 215, + 405 + ], + "score": 0.92, + "content": "p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 390, + 259, + 406 + ], + "score": 1.0, + "content": "is close to", + "type": "text" + }, + { + "bbox": [ + 259, + 393, + 288, + 405 + ], + "score": 0.91, + "content": "\\tilde { p } ( \\mathbf { y } | \\hat { \\mathbf { x } } )", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 106, + 419, + 381, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 382, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 382, + 432 + ], + "score": 1.0, + "content": "6.2 ARCHITECTURE AND TRAINING PROCEDURE DESCRIPTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "We use the same encoder and decoder network structure as Dai et al. (2018) with the only difference", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 317, + 465 + ], + "score": 1.0, + "content": "that our decoder takes as input the concatenation of", + "type": "text" + }, + { + "bbox": [ + 317, + 453, + 334, + 463 + ], + "score": 0.5, + "content": "\\mathbf { y } , \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 451, + 505, + 465 + ], + "score": 1.0, + "content": ". As GRU layers become computationally", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "expensive when the sequences length increase, we also examined the model using less layer GRU.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "To be precise, the decoder in Dai et al. (2018) takes the from of a dense hidden layer with ReLU", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "activation followed by three layers GRU Chung et al. (2014). We tried two different settings of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 341, + 509 + ], + "score": 1.0, + "content": "decoder: in the first setting, we feed the concatenation of", + "type": "text" + }, + { + "bbox": [ + 342, + 497, + 359, + 507 + ], + "score": 0.62, + "content": "\\mathbf { y } , \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "to dense layer then apply one layer", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 310, + 519 + ], + "score": 1.0, + "content": "GRU, in the second setting, to enhance the effect of", + "type": "text" + }, + { + "bbox": [ + 311, + 508, + 318, + 518 + ], + "score": 0.6, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "in the decoder, we feed y not only to the dense", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "layer but also to each layer of GRU. Furthermore, we set the dimension of the latent representation to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 250, + 541 + ], + "score": 1.0, + "content": "56. For the oracle function estimator", + "type": "text" + }, + { + "bbox": [ + 251, + 529, + 263, + 540 + ], + "score": 0.89, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 527, + 506, + 541 + ], + "score": 1.0, + "content": ", we use the same network architecture as the encoder (there is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "no parameter sharing) and we add one more fully connected layer followed by a Tanh transformation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 281, + 569 + ], + "score": 1.0, + "content": "To speed up convergence, we initialize the", + "type": "text" + }, + { + "bbox": [ + 281, + 556, + 294, + 568 + ], + "score": 0.89, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 556, + 441, + 569 + ], + "score": 1.0, + "content": "from a pre-training, where we train", + "type": "text" + }, + { + "bbox": [ + 442, + 557, + 454, + 568 + ], + "score": 0.89, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "on the well-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "trained (maximum 500 epochs with early stopping) supervised VAE’s decoder output to predict the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "molecules property value. We also initialize the parameters of the encoder/decoder networks with the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "partially trained supervised VAE model (after 40 epochs for QM9, 100 epochs for ZINC). We do not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 135, + 613 + ], + "score": 1.0, + "content": "update", + "type": "text" + }, + { + "bbox": [ + 135, + 602, + 143, + 610 + ], + "score": 0.77, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 600, + 160, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 600, + 177, + 612 + ], + "score": 0.9, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 600, + 443, + 613 + ], + "score": 1.0, + "content": "simultaneously, instead we do an alternate optimization. We update", + "type": "text" + }, + { + "bbox": [ + 443, + 602, + 452, + 610 + ], + "score": 0.76, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "continuously", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 228, + 624 + ], + "score": 1.0, + "content": "for five epochs while holding", + "type": "text" + }, + { + "bbox": [ + 229, + 611, + 245, + 622 + ], + "score": 0.9, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 610, + 369, + 624 + ], + "score": 1.0, + "content": "and do the same for updating", + "type": "text" + }, + { + "bbox": [ + 369, + 611, + 385, + 622 + ], + "score": 0.89, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 610, + 506, + 624 + ], + "score": 1.0, + "content": ". We set the hyper-parameter", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 130, + 634 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 131, + 622, + 142, + 633 + ], + "score": 0.88, + "content": "\\lambda _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 622, + 184, + 634 + ], + "score": 1.0, + "content": "to 50 and", + "type": "text" + }, + { + "bbox": [ + 185, + 622, + 196, + 633 + ], + "score": 0.88, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "to 1. The mini-batch size is set to 300 for QM9 and 100 for ZINC. We use", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "ADAM optimizer with learning rate 0.0001 and pytorch lr-schedular on the validation loss. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 643, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 490, + 658 + ], + "score": 1.0, + "content": "general training algorithm is described in below algorithm block1. In our experiment, to train", + "type": "text" + }, + { + "bbox": [ + 491, + 644, + 502, + 655 + ], + "score": 0.88, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 643, + 507, + 658 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 367, + 667 + ], + "score": 1.0, + "content": "we skipped the second term in step 7, which means we only train", + "type": "text" + }, + { + "bbox": [ + 367, + 655, + 379, + 667 + ], + "score": 0.88, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "on the training data but not the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "newly generated molecules obtained by permuting the property. The reason for this is that, during", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "the training, we found that it is easy for the model to learn to reconstruct but hard to conditionally", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "generate the molecules with given properties while we have no guidance of what the molecules", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 316, + 711 + ], + "score": 1.0, + "content": "should look like. 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\\hat { \\mathbf { x } } ) = H ( \\mathbf { y } ) - H ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) } \\\\ & { \\qquad = H ( \\mathbf { y } ) + \\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } \\vert \\mathbf { y } ) } \\mathbb { E } _ { \\mathbf { y } ^ { \\prime } \\sim \\tilde { p } ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) } [ \\log \\tilde { p } ( \\mathbf { y } ^ { \\prime } \\vert \\hat { \\mathbf { x } } ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "f3def3078a70acd5de4fbedc7647e95f7ee8cf236f0ac3b8770154a6f4a8731b.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 168, + 415, + 184.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 195, + 184.0, + 415, + 200.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 206, + 505, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 255, + 219 + ], + "score": 1.0, + "content": "We do not know the true distribution", + "type": "text" + }, + { + "bbox": [ + 255, + 207, + 284, + 218 + ], + "score": 0.92, + "content": "\\tilde { p } ( \\mathbf { y } | \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 206, + 464, + 219 + ], + "score": 1.0, + "content": ", however, the RDKit provides an estimation", + "type": "text" + }, + { + "bbox": [ + 464, + 206, + 493, + 218 + ], + "score": 0.93, + "content": "p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 206, + 506, + 219 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 217, + 348, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 348, + 229 + ], + "score": 1.0, + "content": "the distribution assuming a Gaussian distribution over error:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 106, + 206, + 506, + 229 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 234, + 363, + 250 + ], + "lines": [ + { + "bbox": [ + 248, + 234, + 363, + 250 + ], + "spans": [ + { + "bbox": [ + 248, + 234, + 363, + 250 + ], + "score": 0.91, + "content": "p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } ) = \\mathcal { N } ( \\mathbf { y } \\vert f ( \\hat { \\mathbf { x } } ) , \\lambda _ { 1 } ^ { - 1 } \\mathbf { I } )", + "type": "interline_equation", + "image_path": "289e50d0f7798a72284b200d021c3baab2f2a34670611452c4a211a23b8db034.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 248, + 234, + 363, + 250 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 371, + 268 + ], + "lines": [ + { + "bbox": [ + 106, + 256, + 371, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 133, + 270 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 257, + 140, + 268 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 256, + 371, + 270 + ], + "score": 1.0, + "content": "is the molecule property estimator, i.e., RDKit. We have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 256, + 371, + 270 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 274, + 467, + 335 + ], + "lines": [ + { + "bbox": [ + 143, + 274, + 467, + 335 + ], + "spans": [ + { + "bbox": [ + 143, + 274, + 467, + 335 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle { \\bf I } ( { \\bf y } ; \\hat { \\bf x } ) = H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } [ \\log \\frac { \\tilde { p } ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) } { p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) } p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } \\ ~ } \\\\ { { \\displaystyle ~ = H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } [ D _ { k l } ( \\tilde { p } ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) \\vert \\vert p ( { \\bf y } ^ { \\prime } \\vert \\hat { \\bf x } ) ) + \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } \\log p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } \\ ~ } \\\\ { { \\displaystyle ~ \\geq H ( { \\bf y } ) + \\mathbb { E } _ { \\hat { \\bf x } \\sim p _ { \\theta } ( { \\bf x } \\mid { \\bf y } ) } [ \\mathbb { E } _ { { \\bf y } ^ { \\prime } \\sim \\tilde { p } ( { \\bf y } \\mid \\hat { \\bf x } ) } \\log p ( { \\bf y } ^ { \\prime } \\mid \\hat { \\bf x } ) ] } } \\end{array}", + "type": "interline_equation", + "image_path": "9313d409114746a7bb4d13520d49b98c88c18a40042204504361c3ccec2df0c1.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 143, + 274, + 467, + 294.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 143, + 294.3333333333333, + 467, + 314.66666666666663 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 143, + 314.66666666666663, + 467, + 334.99999999999994 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 371, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 371, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 371, + 353 + ], + "score": 1.0, + "content": "By following the Lemma 5.1 given in Chen et al. (2016), we have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 338, + 371, + 353 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 358, + 406, + 372 + ], + "lines": [ + { + "bbox": [ + 204, + 358, + 406, + 372 + ], + "spans": [ + { + "bbox": [ + 204, + 358, + 406, + 372 + ], + "score": 0.88, + "content": "\\mathbf { I } ( \\mathbf { y } ; \\hat { \\mathbf { x } } ) \\geq H ( \\mathbf { y } ) + \\mathbb { E } _ { \\mathbf { y } \\sim p ( \\mathbf { y } ) , \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } ) } [ \\log p ( \\mathbf { y } | \\hat { \\mathbf { x } } ) ]", + "type": "interline_equation", + "image_path": "7f4de000aa662d02a9dc5ba4efae62b0270f7b9250580018179096fe3e0471b3.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 204, + 358, + 406, + 372 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 379, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 119, + 394 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 119, + 380, + 143, + 392 + ], + "score": 0.92, + "content": "H ( \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 378, + 237, + 394 + ], + "score": 1.0, + "content": "is constant, minimizing", + "type": "text" + }, + { + "bbox": [ + 238, + 380, + 339, + 393 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\hat { \\mathbf { x } } \\sim p _ { \\theta } ( \\mathbf { x } | \\mathbf { y } _ { i } ) } \\| f ( \\hat { \\mathbf { x } } ) - \\mathbf { y } _ { i } \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 378, + 449, + 394 + ], + "score": 1.0, + "content": "is equivalent to maximizing", + "type": "text" + }, + { + "bbox": [ + 450, + 380, + 479, + 392 + ], + "score": 0.91, + "content": "\\mathbf { I } ( \\mathbf { y } ; \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 378, + 506, + 394 + ], + "score": 1.0, + "content": "under", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 390, + 288, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 186, + 406 + ], + "score": 1.0, + "content": "the assumption that", + "type": "text" + }, + { + "bbox": [ + 187, + 392, + 215, + 405 + ], + "score": 0.92, + "content": "p ( \\mathbf { y } \\vert \\hat { \\mathbf { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 390, + 259, + 406 + ], + "score": 1.0, + "content": "is close to", + "type": "text" + }, + { + "bbox": [ + 259, + 393, + 288, + 405 + ], + "score": 0.91, + "content": "\\tilde { p } ( \\mathbf { y } | \\hat { \\mathbf { x } } )", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 378, + 506, + 406 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 419, + 381, + 431 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 382, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 382, + 432 + ], + "score": 1.0, + "content": "6.2 ARCHITECTURE AND TRAINING PROCEDURE DESCRIPTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 440, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "We use the same encoder and decoder network structure as Dai et al. (2018) with the only difference", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 317, + 465 + ], + "score": 1.0, + "content": "that our decoder takes as input the concatenation of", + "type": "text" + }, + { + "bbox": [ + 317, + 453, + 334, + 463 + ], + "score": 0.5, + "content": "\\mathbf { y } , \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 451, + 505, + 465 + ], + "score": 1.0, + "content": ". As GRU layers become computationally", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "expensive when the sequences length increase, we also examined the model using less layer GRU.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "To be precise, the decoder in Dai et al. (2018) takes the from of a dense hidden layer with ReLU", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "activation followed by three layers GRU Chung et al. (2014). We tried two different settings of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 341, + 509 + ], + "score": 1.0, + "content": "decoder: in the first setting, we feed the concatenation of", + "type": "text" + }, + { + "bbox": [ + 342, + 497, + 359, + 507 + ], + "score": 0.62, + "content": "\\mathbf { y } , \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "to dense layer then apply one layer", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 310, + 519 + ], + "score": 1.0, + "content": "GRU, in the second setting, to enhance the effect of", + "type": "text" + }, + { + "bbox": [ + 311, + 508, + 318, + 518 + ], + "score": 0.6, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "in the decoder, we feed y not only to the dense", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "layer but also to each layer of GRU. Furthermore, we set the dimension of the latent representation to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 527, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 250, + 541 + ], + "score": 1.0, + "content": "56. For the oracle function estimator", + "type": "text" + }, + { + "bbox": [ + 251, + 529, + 263, + 540 + ], + "score": 0.89, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 527, + 506, + 541 + ], + "score": 1.0, + "content": ", we use the same network architecture as the encoder (there is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "no parameter sharing) and we add one more fully connected layer followed by a Tanh transformation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 440, + 506, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 281, + 569 + ], + "score": 1.0, + "content": "To speed up convergence, we initialize the", + "type": "text" + }, + { + "bbox": [ + 281, + 556, + 294, + 568 + ], + "score": 0.89, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 556, + 441, + 569 + ], + "score": 1.0, + "content": "from a pre-training, where we train", + "type": "text" + }, + { + "bbox": [ + 442, + 557, + 454, + 568 + ], + "score": 0.89, + "content": "f _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "on the well-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "trained (maximum 500 epochs with early stopping) supervised VAE’s decoder output to predict the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "molecules property value. We also initialize the parameters of the encoder/decoder networks with the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "partially trained supervised VAE model (after 40 epochs for QM9, 100 epochs for ZINC). We do not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 135, + 613 + ], + "score": 1.0, + "content": "update", + "type": "text" + }, + { + "bbox": [ + 135, + 602, + 143, + 610 + ], + "score": 0.77, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 600, + 160, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 600, + 177, + 612 + ], + "score": 0.9, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 600, + 443, + 613 + ], + "score": 1.0, + "content": "simultaneously, instead we do an alternate optimization. We update", + "type": "text" + }, + { + "bbox": [ + 443, + 602, + 452, + 610 + ], + "score": 0.76, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "continuously", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 228, + 624 + ], + "score": 1.0, + "content": "for five epochs while holding", + "type": "text" + }, + { + "bbox": [ + 229, + 611, + 245, + 622 + ], + "score": 0.9, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 610, + 369, + 624 + ], + "score": 1.0, + "content": "and do the same for updating", + "type": "text" + }, + { + "bbox": [ + 369, + 611, + 385, + 622 + ], + "score": 0.89, + "content": "\\phi , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 610, + 506, + 624 + ], + "score": 1.0, + "content": ". We set the hyper-parameter", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 130, + 634 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 131, + 622, + 142, + 633 + ], + "score": 0.88, + "content": "\\lambda _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 622, + 184, + 634 + ], + "score": 1.0, + "content": "to 50 and", + "type": "text" + }, + { + "bbox": [ + 185, + 622, + 196, + 633 + ], + "score": 0.88, + "content": "\\lambda _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "to 1. The mini-batch size is set to 300 for QM9 and 100 for ZINC. We use", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "ADAM optimizer with learning rate 0.0001 and pytorch lr-schedular on the validation loss. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 643, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 490, + 658 + ], + "score": 1.0, + "content": "general training algorithm is described in below algorithm block1. In our experiment, to train", + "type": "text" + }, + { + "bbox": [ + 491, + 644, + 502, + 655 + ], + "score": 0.88, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 643, + 507, + 658 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 367, + 667 + ], + "score": 1.0, + "content": "we skipped the second term in step 7, which means we only train", + "type": "text" + }, + { + "bbox": [ + 367, + 655, + 379, + 667 + ], + "score": 0.88, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "on the training data but not the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "newly generated molecules obtained by permuting the property. The reason for this is that, during", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "the training, we found that it is easy for the model to learn to reconstruct but hard to conditionally", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "generate the molecules with given properties while we have no guidance of what the molecules", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 316, + 711 + ], + "score": 1.0, + "content": "should look like. Further more, some combination of", + "type": "text" + }, + { + "bbox": [ + 317, + 701, + 323, + 709 + ], + "score": 0.64, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 699, + 340, + 711 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 340, + 700, + 348, + 710 + ], + "score": 0.45, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "are physically not feasible. In this case,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 479, + 722 + ], + "score": 1.0, + "content": "when the conditional generation is not good enough yet during the training, we end up fitting", + "type": "text" + }, + { + "bbox": [ + 480, + 710, + 491, + 721 + ], + "score": 0.88, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 446, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 446, + 733 + ], + "score": 1.0, + "content": "the miss represented molecules representations and it makes the optimization harder.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 556, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 238, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 238, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 238, + 97 + ], + "score": 1.0, + "content": "Algorithm 1 Training algorithm", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 109, + 97, + 512, + 234 + ], + "lines": [ + { + "bbox": [ + 110, + 97, + 257, + 111 + ], + "spans": [ + { + "bbox": [ + 110, + 97, + 161, + 111 + ], + "score": 1.0, + "content": "1: Initialize", + "type": "text" + }, + { + "bbox": [ + 161, + 98, + 204, + 110 + ], + "score": 0.89, + "content": "p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } , \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 97, + 208, + 111 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 208, + 98, + 240, + 111 + ], + "score": 0.85, + "content": "q _ { \\phi } ( { \\bf z } | { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 97, + 244, + 111 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 245, + 99, + 257, + 110 + ], + "score": 0.67, + "content": "f _ { w }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 109, + 108, + 313, + 122 + ], + "spans": [ + { + "bbox": [ + 109, + 108, + 313, + 122 + ], + "score": 1.0, + "content": "2: for i=1,2, . . . , N (maximum epoch number) do", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 110, + 117, + 483, + 135 + ], + "spans": [ + { + "bbox": [ + 110, + 120, + 124, + 132 + ], + "score": 1.0, + "content": "3:", + "type": "text" + }, + { + "bbox": [ + 135, + 117, + 155, + 135 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 155, + 120, + 211, + 132 + ], + "score": 0.89, + "content": "{ \\bf j } = 1 , 2 , \\ldots , { \\bf L }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 117, + 294, + 135 + ], + "score": 1.0, + "content": ", sample a minibatch", + "type": "text" + }, + { + "bbox": [ + 295, + 119, + 411, + 133 + ], + "score": 0.91, + "content": "D = ( { \\bf X } , { \\bf Y } ) = \\{ { \\bf x } _ { i } , { \\bf y } _ { i } ) \\} _ { i = 1 } ^ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 117, + 483, + 135 + ], + "score": 1.0, + "content": "of M samples do", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 109, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 109, + 136, + 122, + 149 + ], + "score": 1.0, + "content": "4:", + "type": "text" + }, + { + "bbox": [ + 151, + 136, + 298, + 150 + ], + "score": 1.0, + "content": "randomly permute the property set", + "type": "text" + }, + { + "bbox": [ + 299, + 137, + 309, + 147 + ], + "score": 0.68, + "content": "\\mathbf { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 136, + 349, + 150 + ], + "score": 1.0, + "content": "to obtain", + "type": "text" + }, + { + "bbox": [ + 350, + 137, + 365, + 147 + ], + "score": 0.87, + "content": "\\mathbf { Y } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "and define a label permuted mini", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 122, + 145, + 271, + 162 + ], + "spans": [ + { + "bbox": [ + 122, + 145, + 147, + 162 + ], + "score": 1.0, + "content": "batch", + "type": "text" + }, + { + "bbox": [ + 147, + 147, + 271, + 160 + ], + "score": 0.89, + "content": "D ^ { * } = ( \\mathbf { X } , \\mathbf { \\bar { Y } } ^ { * } ) = \\{ \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ^ { * } \\} _ { i = 1 } ^ { \\bar { M } }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 109, + 162, + 512, + 232 + ], + "spans": [ + { + "bbox": [ + 109, + 165, + 122, + 180 + ], + "score": 1.0, + "content": "5:", + "type": "text" + }, + { + "bbox": [ + 151, + 162, + 512, + 232 + ], + "score": 0.53, + "content": "\\begin{array} { l } { { \\displaystyle \\theta ^ { j } = \\theta ^ { j - 1 } - \\gamma \\big ( - \\nabla _ { \\theta } \\mathcal { L } _ { E L B O } \\big ( \\theta , \\phi , \\omega \\big ) + \\frac { \\lambda _ { 1 } } { 2 } \\sum _ { \\substack { ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\in D ^ { * } } } \\mathbb { E } _ { q ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\nabla _ { \\theta } \\| f _ { \\omega } \\big ( g _ { \\theta } \\big ( \\mathbf { z } , \\mathbf { y } _ { i } \\big ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } \\big ) } } \\\\ { { \\displaystyle \\phi ^ { j } = \\phi ^ { j - 1 } - \\gamma \\big ( - \\nabla _ { \\phi } \\mathcal { L } _ { E L B O } \\big ( \\theta , \\phi , \\omega \\big ) \\big ) } } \\\\ { { \\displaystyle w ^ { j } = w ^ { j - 1 } - \\gamma \\big ( \\frac { \\lambda _ { 2 } } { 2 } \\sum _ { \\substack { ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\in D } } \\mathbb { E } _ { q ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\nabla _ { \\omega } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } + \\frac { \\lambda _ { 1 } } { 2 } \\sum _ { \\substack { ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\in D ^ { * } } } \\mathbb { E } _ { q ( \\mathbf { z } | \\mathbf { x } _ { i } ) } \\nabla _ { \\omega } \\| f _ { \\omega } \\big ( g _ { \\theta } ( \\mathbf { z } , \\mathbf { y } _ { i } ) \\big ) - \\mathbf { y } _ { i } \\| ^ { 2 } \\big ) } } \\end{array}", + "type": "inline_equation", + "image_path": "1f95c0933c9dfc06e4f321b402a28284b15d5ca73925931c96a05d80b0dccf31.jpg" + } + ], + "index": 6 + }, + { + "bbox": [ + 110, + 195, + 122, + 209 + ], + "spans": [ + { + "bbox": [ + 110, + 195, + 122, + 209 + ], + "score": 1.0, + "content": "6:", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 110, + 212, + 122, + 226 + ], + "spans": [ + { + "bbox": [ + 110, + 212, + 122, + 226 + ], + "score": 1.0, + "content": "7:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 270, + 395, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 396, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 396, + 282 + ], + "score": 1.0, + "content": "6.3 USE A FIXED PRE-TRAINED PROPERTY PREDICTION FUNCTION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 506, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 504, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 414, + 308 + ], + "score": 1.0, + "content": "We also investigate the case where we train the property prediction function", + "type": "text" + }, + { + "bbox": [ + 414, + 297, + 426, + 308 + ], + "score": 0.88, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 296, + 504, + 308 + ], + "score": 1.0, + "content": "on the well trained", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "supervised VAE output and keep it fixed during the training of the main model. We give the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 270, + 332 + ], + "score": 1.0, + "content": "performance in tables 3, 4. Using a fixed", + "type": "text" + }, + { + "bbox": [ + 270, + 319, + 282, + 330 + ], + "score": 0.88, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 318, + 506, + 332 + ], + "score": 1.0, + "content": ", in terms of reconstruction and generation performance,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "delivers mixed results 3. However, in terms of conditional generation 4, it does perform better than", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 339, + 470, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 376, + 354 + ], + "score": 1.0, + "content": "the baselines but worse compared to the case where we also update", + "type": "text" + }, + { + "bbox": [ + 376, + 341, + 388, + 352 + ], + "score": 0.89, + "content": "f _ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 339, + 470, + 354 + ], + "score": 1.0, + "content": "during the learning.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "table", + "bbox": [ + 106, + 377, + 505, + 464 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 377, + 505, + 464 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 377, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 464 + ], + "score": 0.982, + "html": "
QM9ZINC
ModelReconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE3.6110.30190.0044.600.70-100
GVAE96.0060.20=80.9053.707.20100
SD-VAE97.8498.4099.2891.9776.2043.50-、
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8495.61100
CGD-VAE-1-GRU98.9695.0392.7789.7467.4617.3882.44100
Sup-VAE-3-GRU97.8197.995.0989.4782.4036.1686.26100
CGD-VAE-3-GRU96.993.898.2989.5581.8037.7898.70100
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Modelz~qq(z)z ~ q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.63310.3835
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.66650.4507
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.29810.0866
Sup-VAE-3-GRU0.36380.1818
CGD-VAE-3-GRU0.37650.1310
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QM9ZINC
ModelReconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE3.6110.30190.0044.600.70-100
GVAE96.0060.20=80.9053.707.20100
SD-VAE97.8498.4099.2891.9776.2043.50-、
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8495.61100
CGD-VAE-1-GRU98.9695.0392.7789.7467.4617.3882.44100
Sup-VAE-3-GRU97.8197.995.0989.4782.4036.1686.26100
CGD-VAE-3-GRU96.993.898.2989.5581.8037.7898.70100
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Modelz~qq(z)z ~ q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.63310.3835
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.66650.4507
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.29810.0866
Sup-VAE-3-GRU0.36380.1818
CGD-VAE-3-GRU0.37650.1310
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As we can see the molecule that our model generates is a new one.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "image", + "bbox": [ + 109, + 156, + 467, + 222 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 156, + 467, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 156, + 467, + 222 + ], + "spans": [ + { + "bbox": [ + 109, + 156, + 467, + 222 + ], + "score": 0.962, + "type": "image", + "image_path": "a7e485939ef9875fdba8293f1eb15bcfd2c0bfca822f4dd0ac966cbb93a44c64.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 109, + 156, + 467, + 178.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 109, + 178.0, + 467, + 200.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 109, + 200.0, + 467, + 222.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 238, + 504, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "Figure 8: Molecules selected with virtual screening over the full dataset in a manner that they are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "structurally similar to the A molecule of figure 4 while they have a logP value which is close to the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 260, + 288, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 288, + 272 + ], + "score": 1.0, + "content": "logP value of the B molecule also of figure 4", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 290, + 275, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 276, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 246, + 304 + ], + "score": 1.0, + "content": "6.5 METRICS AS FUNCTION OF", + "type": "text" + }, + { + "bbox": [ + 246, + 290, + 276, + 302 + ], + "score": 0.66, + "content": "\\mathbf { y } - \\mathbf { y } ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 506, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 507, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 507, + 324 + ], + "score": 1.0, + "content": "The validity and novelty were mainly used to assess purely the generative model performance.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "However, it is also interesting to see if the model capable of generating valid and novel molecules", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 475, + 345 + ], + "score": 1.0, + "content": "if we start from an existing molecules and drift away from the original label, i.e., if we get", + "type": "text" + }, + { + "bbox": [ + 475, + 335, + 482, + 343 + ], + "score": 0.42, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 134, + 356 + ], + "score": 0.94, + "content": "q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 343, + 303, + 357 + ], + "score": 1.0, + "content": ", and then compare different values from", + "type": "text" + }, + { + "bbox": [ + 303, + 343, + 344, + 356 + ], + "score": 0.93, + "content": "p ( \\mathbf { \\bar { x } } | \\mathbf { z } , \\mathbf { y } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 343, + 357, + 357 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 357, + 344, + 367, + 355 + ], + "score": 0.85, + "content": "\\mathbf { y } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "moves far from y. 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Here we", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 690, + 450, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 452, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 452, + 701 + ], + "score": 1.0, + "content": "6.6 SAMPLING FROM APPROXIMATED MARGINAL POSTERIOR FOR GENERATION", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 222, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 223, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 223, + 94 + ], + "score": 1.0, + "content": "6.4 VIRTUAL SCREENING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "score": 1.0, + "content": "The figure 8 displays the results of a virtual screening method, where we select from the full dataset", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 292, + 127 + ], + "score": 1.0, + "content": "five molecules which are structurally similar to", + "type": "text" + }, + { + "bbox": [ + 292, + 115, + 306, + 126 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "in figure 4 in section 4 and have logP values close", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 403, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 116, + 137 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 127, + 131, + 137 + ], + "score": 0.83, + "content": "\\mathbf { y } _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 125, + 403, + 137 + ], + "score": 1.0, + "content": ". 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ModelQM9ZINC
Reconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE Gómez-Bombarelli et al. (2016)3.6110.30=90.044.60.70·100
GVAE Kusner et al. (2017)96.0060.20=80.9053.707.20=100
SD-VAE Dai et al.(2018)97.8498.4099.2891.9776.2043.50--
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8494.61100
CGD-VAE-1-GRU99.2795.6193.6587.8788.6429.0099.24100
Sup-VAE-3-GRU97.8197.9095.0989.4782.4036.1686.26100
CGD-VAE-3-GRU99.3197,8098.7796.2181.8037.7898.75100
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CGD-VAE-1-GRU0.38770.0880
Sup-VAE-3-GRU0.35140.1808
CGD-VAE-3-GRU0.39660.1559
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QM9ZINC
ModelReconstruction %Valid%Unique %Novel %Reconstruction %Valid%Unique %Novel %
CVAE3.6110.30190.0044.600.70-100
GVAE96.0060.20=80.9053.707.20100
SD-VAE97.8498.4099.2891.9776.2043.50-、
Sup-VAE-1-GRU97.5393.6691.3092.0574.1232.8495.61100
CGD-VAE-1-GRU98.9695.0392.7789.7467.4617.3882.44100
Sup-VAE-3-GRU97.8197.995.0989.4782.4036.1686.26100
CGD-VAE-3-GRU96.993.898.2989.5581.8037.7898.70100
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Modelz~qq(z)z ~ q(z|x)
QM9Sup-VAE-1-GRU0.54200.2526
CGD-VAE-1-GRU0.63310.3835
Sup-VAE-3-GRU0.69580.4204
CGD-VAE-3-GRU0.66650.4507
ZINCSup-VAE-1-GRU0.23010.0481
CGD-VAE-1-GRU0.29810.0866
Sup-VAE-3-GRU0.36380.1818
CGD-VAE-3-GRU0.37650.1310
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1: Initialize pe(x|z,y),q(z|x), fw
2:for i=1,2,...,N (maximum epoch number) do 3: for j= 1,2,...,L,sample a minibatch D = (X,Y) = {xi,yi)}≧1 of M samples do
4: randomly permute the property set Y to obtain Y* and define a label permuted mini
0j=0j-1-γ(-∀θLELBo(0,,w)+ 5: M Eq(z|xi)Vθllfω(ge(z,yi))-yill2) (xi,yi)ED*
6: =j-1-γ(-∀LELBo(0,,ω))
7: wj=wj-1-γ(²，∑Eq(z|xi)Vωllfu(ge(z,yi))-yil²+，∑Eq(z|x)Vulfu(gθ(z,yi))-yil²) (xi,yi)ED (xi,yi)ED*
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Multi-task learning has emerged as a promising approach for sharing structure across multiple tasks to enable more efficient learning. However, the multi-task setting presents a number of optimization challenges, making it difficult to realize large efficiency gains compared to learning tasks independently. The reasons why multi-task learning is so challenging compared to single task learning are not fully understood. Motivated by the insight that gradient interference causes optimization challenges, we develop a simple and general approach for avoiding interference between gradients from different tasks, by altering the gradients through a technique we refer to as “gradient surgery”. We propose a form of gradient surgery that projects the gradient of a task onto the normal plane of the gradient of any other task that has a conflicting gradient. On a series of challenging multi-task supervised and multi-task reinforcement learning problems, we find that this approach leads to substantial gains in efficiency and performance. Further, it can be effectively combined with previously-proposed multi-task architectures for enhanced performance in a model-agnostic way. ", + "bbox": [ + 233, + 242, + 766, + 492 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 515, + 336, + 531 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While deep learning and deep reinforcement learning (RL) have shown considerable promise in enabling systems to perform complex tasks, the data requirements of current methods make it difficult to learn a breadth of capabilities particularly when all tasks are learned individually from scratch. A natural approach to such multi-task learning problems is to train a single network on all tasks jointly, with the aim of discovering shared structure across the tasks in a way that achieves greater efficiency and performance than solving the tasks individually. However, learning multiple tasks all at once results in a difficult optimization problem, sometimes leading to worse overall performance and data efficiency compared to learning tasks individually (Parisotto et al., 2015; Rusu et al., 2016a). These optimization challenges are so prevalent that multiple multi-task RL algorithms have considered using independent training as a subroutine of the algorithm before distilling the independent models into a multi-tasking model (Levine et al., 2016; Parisotto et al., 2015; Rusu et al., 2016a; Ghosh et al., 2017; Teh et al., 2017), producing a multi-task model but losing out on the efficiency gains over independent training. If we could tackle the optimization challenges of multi-task learning effectively, we may be able to actually realize the hypothesized benefits of multi-task learning without the cost in final performance. ", + "bbox": [ + 174, + 541, + 825, + 750 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While there has been a significant amount of research in multi-task learning (Caruana, 1997; Ruder, 2017), the optimization challenges are not well understood. Prior work has described varying learning speeds of different tasks (Chen et al., 2017) and plateaus in the optimization landscape (Schaul et al., 2019) as potential causes, while a range of other works have focused on the model architecture (Misra et al., 2016b; Liu et al., 2018). In this work, we instead hypothesize that the central optimization issue in multi-task learning arises from gradients from different tasks conflicting with one another. In particular, we define two gradients to be conflicting if they point away from one another (i.e., have a negative cosine similarity). As a concrete example, consider the 2D optimization landscapes of two task objectives shown in Figure 1. The optimization landscape of each task consists of a deep valley, as has been characterized of neural network optimization landscapes in the past (Goodfellow et al., 2014). When considering the combined optimization landscape for multiple tasks, SGD produces gradients that struggle to efficiently find the optimum. This occurs due to a gradient thrashing phenomenon, where the gradient of one task destabilizes optimization in the valley. We can observe this in Figure 1(d) when the optimization reaches the deep valley of task 1, but is prevented from traversing the valley to an optimum. In Section 6.2, we find experimentally that this thrashing phenomenon also occurs in a neural network multi-task learning problem. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/61de57b0fe5017235f4ce4df8cb4be8d30719ff3e91a41a59f711ae8cb6ecd69.jpg", + "image_caption": [ + "Figure 1: Visualization of PCGrad’s effect on a 2D multi-task optimization problem. (a) A multi-task objective landscape. (b) & (c) Contour plots of the individual task objectives that comprise the multi-task objective. (d) Trajectory of gradient updates on the multi-task objective using the Adam optimizer. The gradient vectors of the two tasks at the end of the trajectory are indicated by blue and red arrows, where the relative lengths are on a log scale.(e) Trajectory of gradient updates on the multi-task objective using Adam with PCGrad. For (d) and (e), the optimization trajectory goes from black to yellow. " + ], + "image_footnote": [], + "bbox": [ + 174, + 103, + 820, + 223 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 321, + 825, + 378 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The core contribution of this work is a method for mitigating gradient interference by altering the gradients directly, i.e. by performing “gradient surgery”. If two gradients are conflicting, we alter the gradients by projecting each onto the normal plane of the other, preventing the interfering components of the gradient from being applied to the network. We refer to this particular form of gradient surgery as projecting conflicting gradients (PCGrad). PCGrad is model-agnostic, requiring only a single modification to the application of gradients. Hence, it is easy to apply to a range of problem settings, including multi-task supervised learning and multi-task reinforcement learning, and can also be readily combined with other multi-task learning approaches, such as those that modify the architecture. We evaluate PCGrad on multi-task CIFAR classification, multi-objective scene understanding, a challenging multi-task RL domain, and goal-conditioned RL. Across the board, we find PCGrad leads to significant improvements in terms of data efficiency, optimization speed, and final performance compared to prior approaches. Further, on multi-task supervised learning tasks, PCGrad can be successfully combined with prior state-of-the-art methods for multi-task learning for even greater performance. ", + "bbox": [ + 174, + 386, + 825, + 579 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 590, + 339, + 607 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The goal of multi-task learning is to find parameters $\\theta$ of a model $f _ { \\theta }$ that achieve high average performance across all the training tasks drawn from a distribution of tasks $p ( \\tau )$ . More formally, we aim to solve the problem: $\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { \\mathcal { T } _ { i } \\sim p ( \\mathcal { T } ) } \\left[ \\mathcal { L } _ { i } ( f _ { \\theta } ) \\right]$ , where $\\mathcal { L } _ { i }$ is a loss function for the $i$ -th task $\\mathcal { T } _ { i }$ that we want to minimize. To obtain a model that solves a specific task from the task distribution $p ( \\mathcal { T } )$ , we define a task-conditioned model $f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { y } | \\boldsymbol { x } , z _ { i } )$ , with input $x$ , output $y$ , and encoding $z _ { i }$ for task $\\mathcal { T } _ { i }$ , which could be provided as a one-hot vector or in any other form. ", + "bbox": [ + 173, + 614, + 826, + 705 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 MULTI-TASK LEARNING VIA GRADIENT SURGERY ", + "text_level": 1, + "bbox": [ + 174, + 715, + 627, + 732 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While the multi-task problem can in principle be solved by simply applying a standard single-task algorithm with a suitable task identifier provided to the model or a simple multi-head or multi-output model, a number of prior works (Parisotto et al., 2015; Rusu et al., 2016a; Sener & Koltun, 2018) have found this learning problem to be difficult, especially in the reinforcement learning setting. We hypothesize that one of the main challenges of multi-task learning can be characterized by conflicting and thrashing gradients, and find that this can significantly impede learning progress, especially when combined with iterative data collection. We identify possible causes for this problem and propose a simple and general approach to mitigate it. ", + "bbox": [ + 174, + 739, + 825, + 852 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3.1 THRASHING GRADIENTS IN MULTI-TASK OPTIMIZATION LANDSCAPES ", + "text_level": 1, + "bbox": [ + 176, + 859, + 707, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We hypothesize that a key optimization issue in multi-task learning arises when gradients from multiple tasks are in conflict with one another, i.e. when gradients point away from one another. More specifically, we hypothesize that such conflict may lead to gradient thrashing. Concretely, gradient thrashing refers to the phenomenon where a large gradient for one task changes the parameter vectors in a way that substantially decreases performance on another task. Since worse performance typically leads to larger gradients, this results in alternating gradient directions, where, at the next iteration, the second task will have large gradients that dominate and reduce performance on the former task. This issue can be particularly pronounced for neural network optimization, since neural network loss landscapes are known to resemble long narrow valleys (Goodfellow et al., 2014), where the gradient perpendicular to the direction of the valley will be small. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/28c8389133d913290b204c119be99e66cf5a7eed386a660afd2ba9a69abc4cfd.jpg", + "image_caption": [ + "Figure 2: Visual depiction of conflicting gradients and PCGrad. In (a), we see that tasks A and B have conflicting gradient directions, which can lead to destructive interference and unstable learning. In (b), we illustrate the PCGrad algorithm in cases where gradients are conflicting. PCGrad projects the gradient of task A onto the normal vector of task B’s gradient. In (c), we show that tasks with non-conflicting gradients are not altered under PCGrad, thereby keeping tasks with constructive interference. " + ], + "image_footnote": [], + "bbox": [ + 287, + 103, + 707, + 205 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 281, + 825, + 378 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We aim to study this hypothesis through two toy examples. First, consider the two-dimensional optimization landscape illustrated in Fig. 1a, where the landscape for each task objective corresponds to a deep and curved valley (Fig. 1b and 1c). The optima of this multi-task objective correspond to where the two valleys meet. More details on the optimization landscape are in Appendix B. We observe that the gradient thrashing hypothesis is consistent with what we observe when running Adam (Kingma & Ba, 2014) on this landscape in Fig. 1d, where we observe that Adam does not traverse one valley towards the other, preventing it from reaching an optimum. ", + "bbox": [ + 174, + 386, + 825, + 484 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We also aim to detect if a similar phenomenon occurs in multi-task learning with a neural network with thousands of parameters on a toy regression problem. To measure the extent of gradient thrashing, we plot the cosine similarity between the gradients of two tasks throughout the beginning of learning in Fig. 4 (left). We indeed observe a significant level of gradient thrashing at every iteration, where the cosine similarity varies between $- 0 . 7 5$ and 0.75 at a very high frequency. ", + "bbox": [ + 174, + 491, + 825, + 560 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Motivated by these observations, we develop an algorithm that aims to alleviate the optimization challenges caused by gradient thrashing by preventing such gradient conflict between tasks. ", + "bbox": [ + 173, + 566, + 823, + 595 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 PCGRAD: PROJECTING CONFLICTING GRADIENTS ", + "text_level": 1, + "bbox": [ + 174, + 604, + 566, + 619 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We aim to prevent gradient thrashing by directly altering the gradients themselves, i.e. through “gradient surgery.” To be maximally effective and maximally applicable, we must perform surgery in a way that still allows for positive interactions between the task gradients and does not introduce any assumptions on the form of the model. ", + "bbox": [ + 173, + 627, + 825, + 683 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We start by first detecting whether two gradients are in conflict, by measuring whether they point away from one another. More concretely, we characterize two tasks as conflicting for the current parameter setting if they yield a negative cosine similarity between their respective gradients. The goal of PCGrad is to modify the gradients for each task so as to minimize negative conflict with other task gradients, which will in turn mitigate gradient thrashing. ", + "bbox": [ + 174, + 690, + 825, + 760 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To deconflict gradients during optimization, PCGrad adopts a simple procedure: if the gradients between two tasks are in conflict, i.e. their cosine similarity is negative, we project the gradient from one task onto the normal plane of the gradient of the other task. This amounts to removing the conflicting component of the gradient for the task, thereby reducing the amount of destructive gradient interference between tasks. A pictorial description of this idea is shown in Fig. 2. Suppose the gradient for task $\\mathcal { T } _ { i }$ is $\\mathbf { g } _ { i }$ , and the gradient for task $\\tau _ { j }$ is $\\mathbf { g } _ { j }$ . PCGrad proceeds as follows: (1) First, it determines whether $\\mathbf { g } _ { i }$ conflicts with ${ \\bf { g } } _ { j }$ by computing the cosine similarity between vectors $\\mathbf { g } _ { i }$ and $\\mathbf { g } _ { j }$ , where negative values indicate conflicting gradients. (2) If the cosine similarity is negative, we replace $\\mathbf { g } _ { i }$ by its projection onto the normal plane of $\\mathbf { g } _ { j }$ : $\\begin{array} { r } { \\mathbf { g } _ { i } = \\mathbf { g } _ { i } - \\frac { \\mathbf { g } _ { i } \\cdot \\mathbf { g } _ { j } } { \\| \\mathbf { g } _ { j } \\| ^ { 2 } } \\mathbf { g } _ { j } } \\end{array}$ gi·gjkgjk2 gj . If the gradients are not in conflict, i.e. cosine similarity is non-negative, the original gradient $\\mathbf { g } _ { i }$ remains unaltered. (3) PCGrad repeats this process across all of the other tasks sampled in random order from the current ", + "bbox": [ + 174, + 766, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 1 PCGrad Update Rule ", + "text_level": 1, + "bbox": [ + 174, + 118, + 390, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Require: Current model parameters $\\theta$ \n1: Sample mini-batch of tasks $B = \\{ \\mathcal { T } _ { k } \\} \\sim p ( \\mathcal { T } )$ \n2: for $\\bar { \\mathcal { T } } _ { i } \\sim \\mathcal { B }$ in sequence do \n3: Compute gradient ${ \\bf g } _ { i }$ of $\\mathcal { T } _ { i }$ as $\\mathbf { g } _ { i } = \\nabla _ { \\theta } \\mathcal { L } _ { i } ( f _ { \\theta } )$ \n4: for ${ \\tau _ { j } } \\stackrel { \\mathrm { u n i f o r m l y } } { \\sim } { B }$ in random order do \n5: Compute gradient $\\mathbf { g } _ { j }$ of task $\\mathcal { T } _ { j }$ as $\\mathbf { g } _ { j } = \\nabla _ { \\theta } \\mathcal { L } _ { j } ( f _ { \\theta } )$ \n6: Compute cosine similarity between $\\mathbf { g } _ { i }$ as $\\mathbf { g } _ { j }$ as $\\begin{array} { r } { \\cos ( \\phi _ { i j } ) = \\frac { \\mathbf { g } _ { i } \\cdot \\mathbf { g } _ { j } } { \\| \\mathbf { g } _ { i } \\| \\| \\mathbf { g } _ { j } \\| } } \\end{array}$ . \n7: 8: if $\\begin{array} { r l } & { \\cos ( \\phi _ { i j } ) < 0 \\mathrm { t h e n } } \\\\ & { \\mathrm { S e t } \\mathbf { g } _ { i } = \\mathbf { g } _ { i } - \\frac { \\mathbf { g } _ { i } \\cdot \\mathbf { g } _ { j } } { \\| \\mathbf { g } _ { j } \\| ^ { 2 } } \\mathbf { g } _ { j } } \\end{array}$ // Subtract the projection of ${ \\bf g } _ { i }$ onto $\\mathbf { g } _ { j }$ \n9: end if \n10: end for \n11: Store $\\mathbf { g } _ { i } ^ { \\mathrm { p r o j } } = \\mathbf { g } _ { i }$ \n12: end for \n13: return update $\\begin{array} { r } { \\Delta \\theta = \\sum _ { i } \\mathbf { g } _ { i } ^ { \\mathrm { p r o j } } } \\end{array}$ ", + "bbox": [ + 179, + 137, + 696, + 337 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "batch $\\mathcal { T } _ { j } \\ \\forall \\ j \\ \\neq i$ , resulting in the gradient ${ \\bf g } _ { i } ^ { \\mathrm { p r o j } }$ that is applied for task $\\mathcal { T } _ { i }$ . We perform the same procedure for all tasks in the batch to obtain their respective gradients. The full update procedure is described in Algorithm 1 and a discussion on using a random task order is included in Appendix D. ", + "bbox": [ + 174, + 371, + 825, + 414 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This procedure, while simple to implement, ensures that the gradients that we apply for each task per batch interfere minimally with the other tasks in the batch, mitigating the thrashing gradient problem, producing a variant on standard first-order gradient descent in the multi-objective setting. In practice, the PCGrad gradient surgery method can be combined with any gradient-based optimizer, including commonly used methods such as SGD with momentum and Adam (Kingma & Ba, 2014), by simply passing the computed update to the respective optimizer instead of the original gradient. Our experimental results verify the hypothesis that this procedure reduces the problem of thrashing gradients, and find that, as a result, learning progress is substantially improved. ", + "bbox": [ + 174, + 420, + 825, + 532 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Finally, we analyze the convergence of this procedure in Theorem 1 in the two-task setting, to ensure that the procedure is sensible under the standard assumptions in optimization. ", + "bbox": [ + 173, + 539, + 821, + 568 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1. Consider two task loss functions $\\mathcal { L } _ { 1 } : \\mathbb { R } ^ { n } \\mathbb { R }$ and $\\mathcal { L } _ { 2 } : \\mathbb { R } ^ { n } \\mathbb { R }$ which are convex and differentiable. For all $\\theta \\in \\mathbb { R } ^ { n }$ , let $\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) = \\mathcal { L } _ { 1 } ( \\boldsymbol { \\theta } ) + \\mathcal { L } _ { 2 } ( \\boldsymbol { \\theta } ) , } \\end{array}$ , i.e. $\\mathcal { L }$ is a multi-task objective. Let $\\phi$ be the angle between $\\nabla { \\mathcal { L } } _ { 1 } ( \\theta )$ and $\\nabla { \\mathcal { L } } _ { 2 } ( \\theta )$ . Suppose $\\mathcal { L }$ is differentiable and that its gradient is Lipschitz continuous with constant $L > 0$ , i.e. we have $| | \\nabla \\mathcal { L } ( \\bar { \\theta _ { 1 } } ) - \\nabla \\mathcal { L } ( \\theta _ { 2 } ) | | _ { 2 } \\leq L | | \\theta _ { 1 } - \\theta _ { 2 } | | _ { 2 } f o r$ any $\\theta _ { 1 } , \\theta _ { 2 }$ . Then, the PCGrad update rule with step size $\\begin{array} { r } { t \\le \\frac { 1 } { L } } \\end{array}$ will converge to either $( l )$ a location in the optimization landscape where $\\cos ( \\phi ) = - 1$ or (2) the optimal value $\\mathcal { L } ( \\theta ^ { \\ast } )$ . ", + "bbox": [ + 174, + 571, + 825, + 655 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proof. See Appendix A. ", + "bbox": [ + 174, + 666, + 334, + 681 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1 states that application of the PCGrad update in the two-task setting with a convex and Lipschitz multi-task loss function $\\mathcal { L }$ leads to convergence to either the minimizer of $\\mathcal { L }$ or a potentially sub-optimal objective value. A sub-optimal solution occurs when the cosine similarity between the gradients of the two tasks is $- 1$ , i.e. the gradients directly conflict, leading to zero gradient after applying PCGrad. However, in practice, since we are using SGD, which is a noisy estimate of the true batch gradients, the cosine similarity between the gradients of two tasks in a minibatch is unlikely to be $- 1$ , thus avoiding this scenario. ", + "bbox": [ + 174, + 685, + 825, + 784 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 THE PRACTICAL OPERATIONS OF PCGRAD", + "text_level": 1, + "bbox": [ + 174, + 795, + 565, + 811 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We apply PCGrad to both supervised learning and reinforcement learning problem settings with multiple tasks or goals. In this section, we discuss the practical instantiations of PCGrad in those settings. Further implementation details are included in Section 6. ", + "bbox": [ + 174, + 820, + 825, + 862 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 MULTI-TASK SUPERVISED LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 871, + 477, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In multi-task supervised learning, each task $\\mathcal { T } _ { i } \\sim p ( \\mathcal { T } )$ has a corresponding training dataset $\\mathcal { D } _ { i }$ consisting of $N _ { i }$ labeled training examples, i.e. $\\mathcal { D } _ { i } = \\{ ( x , y ) _ { n } \\} _ { n = 1 } ^ { N _ { i } }$ . The objective for each task in this supervised setting is then defined as $\\mathcal { L } _ { i } ( f _ { \\theta } ) = \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } _ { i } } \\left[ - \\log f _ { \\theta } ( y \\vert x , z _ { i } ) \\right]$ , where $z _ { i }$ is a one-hot encoding of task $\\mathcal { T } _ { i }$ . ", + "bbox": [ + 173, + 892, + 825, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "At each training step, we randomly sample a batch of data points $\\boldsymbol { B }$ from the whole dataset $\\textstyle \\bigcup _ { i } { \\mathcal { D } } _ { i }$ and then group the sampled data with the same task encoding into small batches denoted as $B _ { i }$ for each $\\mathcal { T } _ { i }$ represented in $\\boldsymbol { B }$ . We denote the set of tasks appearing in $\\boldsymbol { B }$ as $B _ { T }$ . After sampling, we precompute the gradient of each task in $B _ { T }$ as ", + "bbox": [ + 173, + 138, + 825, + 194 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6f78a9cfdc999e372577f0823206dc2bb72800f1f9a1c2255fec1a0e71855d3c.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla _ { \\theta } \\mathcal { L } _ { i } ( f _ { \\theta } ) = \\mathbb { E } _ { ( { x } , { y } ) \\sim \\mathcal { B } _ { i } } \\left[ - \\nabla _ { \\theta } \\log f _ { \\theta } ( { y } | { x } , { z } _ { i } ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 202, + 651, + 219 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Given the set of precomputed gradients $\\nabla _ { \\theta } \\mathcal { L } _ { i } ( f _ { \\theta } )$ , we also precompute the cosine similarity between all pairs of the gradients in the set. Using the pre-computed gradients and their similarities, we can obtain the PCGrad update by following Algorithm 1, without re-computing task gradients nor backpropagating into the network. ", + "bbox": [ + 174, + 233, + 825, + 290 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since the PCGrad procedure is only modifying the gradients of shared parameters in the optimization step, it is model-agnostic and can be readily applied to any architecture designed for supervised multi-task learning. In Section 6, we combine PCGrad with two state-of-the-art architectures for multi-task learning, which leads to noticeable improvement over their original performance. ", + "bbox": [ + 173, + 296, + 825, + 353 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 MULTI-TASK AND GOAL-CONDITIONED REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 361, + 699, + 376 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For multi-task reinforcement learning, PCGrad can be readily applied to policy gradient methods by directly updating the computed policy gradient of each task, following Algorithm 1, analogous to the supervised learning setting. For actor-critic algorithms, it is also straightforward to apply PCGrad: we simply replace the task gradients for both the actor and the critic by their gradients computed via PCGrad. Hence, PCGrad can be readily incorporated into a variety of model-free RL algorithms. When applying PCGrad to goal-conditioned RL, we represent $p ( \\mathcal T )$ as a distribution of goals and let $z _ { i }$ be the encoding of a goal. Similar to the multi-task supervised learning setting discussed above, PCGrad may be combined with various architectures designed for multi-task and goal-conditioned RL (Fernando et al., 2017; Devin et al., 2016), where PCGrad operates on the gradients of shared parameters, leaving task-specific parameters untouched. ", + "bbox": [ + 173, + 382, + 826, + 522 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In our experiments, we apply PCGrad to the soft actor-critic (SAC) algorithm (Haarnoja et al., 2018), a recently proposed off-policy actor-critic algorithm that has shown significant gains in sample efficiency and asymptotic performance across many different domains. In SAC, we employ a $\\mathrm { Q } \\mathrm { - }$ learning style gradient to compute the gradient of the Q-function network, $Q _ { \\phi } ( s , a , z _ { i } )$ , often known as the critic, and a reparameterization-style gradient to compute the gradient of the policy network $\\pi _ { \\boldsymbol { \\theta } } ( a | s , z _ { i } )$ , often known as the actor. For sampling, we instantiate a set of replay buffers $\\{ \\mathcal { D } _ { i } \\} _ { \\mathcal { T } _ { i } \\sim p ( \\mathcal { T } ) }$ . Training and data collection are alternated throughout training. During a data collection step, we run the policy $\\pi _ { \\theta }$ on all the tasks $\\mathcal { T } _ { i } \\sim p ( \\mathcal { T } )$ to collect an equal number of paths for each task and store the paths of each task $\\mathcal { T } _ { i }$ into the corresponding replay buffer $\\mathcal { D } _ { i }$ . At each training step, we sample an equal amount of data from each replay buffer $\\mathcal { D } _ { i }$ to form a stratified batch. For each task $\\mathcal { T } _ { i } \\sim p ( \\mathcal { T } )$ , the parameters of the critic $\\theta$ are optimized to minimize the soft Bellman residual: ", + "bbox": [ + 173, + 529, + 826, + 683 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4d45e30dbf2fe62bcaabae70a310ca426ab2e69c77390a1d79b751aca907386e.jpg", + "text": "$$\n\\begin{array} { r l } & { J _ { Q } ^ { ( i ) } ( \\phi ) = \\mathbb { E } _ { ( s _ { t } , a _ { t } , z _ { i } ) \\sim \\mathcal { D } _ { i } } \\left[ Q _ { \\phi } ( s _ { t } , a _ { t } , z _ { i } ) - ( r ( s _ { t } , a _ { t } , z _ { i } ) + \\gamma V _ { \\bar { \\phi } } ( s _ { t + 1 } , z _ { i } ) ) \\right] , } \\\\ & { V _ { \\bar { \\phi } } ( s _ { t + 1 } , z _ { i } ) = \\mathbb { E } _ { a _ { t + 1 } \\sim \\pi _ { \\theta } } \\left[ Q _ { \\bar { \\phi } } ( s _ { t + 1 } , a _ { t + 1 } , z _ { i } ) - \\alpha \\log \\pi _ { \\theta } ( a _ { t + 1 } | s _ { t + 1 } , z _ { i } ) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 691, + 725, + 747 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\gamma$ is the discount factor, $\\bar { \\phi }$ are the delayed parameters, and $\\alpha$ is a learnable temperature that automatically adjusts the weight of the entropy term. For each task $\\mathcal { T } _ { i } \\sim p ( \\mathcal { T } )$ , the parameters of the policy $\\pi _ { \\theta }$ are trained to minimize the following objective ", + "bbox": [ + 174, + 757, + 823, + 800 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/475646734cccabb50b6351b9efbb0de4497c121807e7e90335d8938b8c54a392.jpg", + "text": "$$\nJ _ { \\pi } ^ { ( i ) } ( \\theta ) = \\mathbb { E } _ { s _ { t } \\sim \\mathcal { D } _ { i } } \\left[ \\mathbb { E } _ { a _ { t } \\sim \\pi _ { \\theta } ( a _ { t } \\mid s _ { t } , z _ { i } ) ) } \\left[ \\alpha \\log \\pi _ { \\theta } ( a _ { t } \\mid s _ { t } , z _ { i } ) - Q _ { \\phi } ( { s _ { t } , a _ { t } , z _ { i } } ) \\right] \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 810, + 723, + 830 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We compute $\\nabla _ { \\phi } J _ { Q } ^ { ( i ) } ( \\phi )$ and $\\nabla _ { \\theta } J _ { \\pi } ^ { ( i ) } ( \\theta )$ for all $\\begin{array} { r } { \\mathcal { T } _ { i } \\sim p ( \\mathcal { T } ) } \\end{array}$ and apply PCGrad to both following Algorithm 1. ", + "bbox": [ + 176, + 842, + 821, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the context of SAC specifically, we further study how the temperature $\\alpha$ should be adjusted. If we use a single learnable temperature for adjusting entropy of the multi-task policy $\\pi _ { \\boldsymbol { \\theta } } ( a | s , z _ { i } )$ , SAC may stop exploring once all easier tasks are solved, leading to poor performance on tasks that are harder or require more exploration. To address this issue, we propose to learn the temperature on a per-task basis, i.e. using a parametrized model to represent $\\alpha _ { \\psi } ( z _ { i } )$ (which we abbreviate as PA for per-task alpha). This allows the method to control the entropy of $\\dot { \\pi } _ { \\boldsymbol { \\theta } } ( a | s , z _ { i } )$ per-task. We optimize the parameters of $\\alpha _ { \\psi } ( z _ { i } )$ using the same constrained optimization framework as in Haarnoja et al. (2018). ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 190, + 344, + 208 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithms for multi-task learning typically consider how to train a single model that can solve a variety of different tasks (Caruana, 1997; Bakker & Heskes, 2003; Ruder, 2017). The multi-task formulation has been applied to many different settings, including supervised learning (Zhang et al., 2014; Long & Wang, 2015; Yang & Hospedales, 2016; Sener & Koltun, 2018; Zamir et al., 2018) and reinforcement-learning (Espeholt et al., 2018; Wilson et al., 2007), as well as many different domains, such as vision (Bilen & Vedaldi, 2016; Misra et al., 2016a; Kokkinos, 2017; Liu et al., 2018; Zamir et al., 2018), language (Collobert & Weston, 2008; Dong et al., 2015; McCann et al., 2018; Radford et al., 2019) and robotics (Riedmiller et al., 2018; Wulfmeier et al., 2019; Hausman et al., 2018). While multi-task learning has the promise of accelerating acquisition of large task repertoires, in practice it presents a challenging optimization problem, which has been tackled in several ways in prior work. ", + "bbox": [ + 174, + 222, + 826, + 375 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A number of architectural solutions have been proposed to the multi-task learning problem based on multiple modules or paths (Fernando et al., 2017; Devin et al., 2016; Misra et al., 2016b; Rusu et al., 2016b; Rosenbaum et al., 2018; Vandenhende et al., 2019; Rosenbaum et al., 2018), or using attention-based architectures (Liu et al., 2018; Maninis et al., 2019). Our work is agnostic to the model architecture and can be combined with prior architectural approaches in a complementary fashion. A different set of multi-task learning approaches aim to decompose the problem into multiple local problems, often corresponding to each task, that are significantly easier to learn, akin to divide and conquer algorithms (Levine et al., 2016; Rusu et al., 2016a; Parisotto et al., 2015; Teh et al., 2017; Ghosh et al., 2017; Czarnecki et al., 2019). Eventually, the local models are combined into a single, multi-task policy using different distillation techniques (outlined in (Hinton et al., 2015; Czarnecki et al., 2019)). In contrast to these methods, we propose a simple and cogent scheme for multi-task learning that allows us to learn the tasks simultaneously using a single, shared model without the need for network distillation. ", + "bbox": [ + 174, + 382, + 825, + 563 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Similarly to our work, a number of prior approaches have observed the difficulty of optimization in the multi-task learning setting (Hessel et al., 2019; Chen et al., 2018; Kendall et al., 2018b; Schaul et al., 2019). Our work, in contrast to many of these optimization schemes, suggests that the challenge in multi-task learning may be attributed to the problem of gradient thrashing, which we address directly by introducing a simple and practical algorithm that de-conflicts gradients from different tasks. Prior work (Sener & Koltun, 2018) alternatively proposes a gradient-based multi-objective optimization problem for multi-task learning to address the problem of optimizing possibly conflicting objectives. As noted in Alg 2 in (Sener & Koltun, 2018), it learns a constant scaling factor for per-task gradient to avoid conflicting, while our method corrects both the scaling factor and the direction of per-task gradient, which can more effectively deconflict gradients. Prior work has also used the cosine similarity between gradients to define when an auxiliary task might be useful for single-task learning (Du et al., 2018). We similarly use cosine similarity between gradients to determine if the gradients between a pair of tasks are in conflict. Unlike Du et al. (2018), we use this measure of gradient conflict as a part of gradient surgery in the context of multi-task learning applications. ", + "bbox": [ + 174, + 569, + 825, + 763 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A number of works in continual learning have studied how to make gradient updates that do not adversely affect other tasks by projecting the gradients into a space that do not conflict with previous tasks (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2018). Those methods focus on the continual learning setting, and either need to solve for the gradient projections using quadratic programming (Lopez-Paz & Ranzato, 2017), or only projecting the gradient onto the normal plane of the average of the gradients of past tasks (Chaudhry et al., 2018). In contrast, our work focuses on multi-task learning, does not require solving any QP, and iteratively projects the gradients of each task onto the normal plane of the gradients of each of the other tasks instead of averaging. Finally, our method is distinct from and solves a different problem than the projected gradient method (Calamai & More, 1987), which is an approach for constrained optimization that projects gradients onto the ´ constraint manifold. ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/fd487e1226e8625528f23a719200a0346be4dbf560626ca0509476293a46fb8b.jpg", + "image_caption": [ + "Figure 3: We show visualization of 50 tasks used in MT50 from Meta-World (Yu et al., 2019), which we use for our multi-task RL experiments. MT10 is a subset of the total 50 tasks, which includes reach, push, pick & place, open drawer, close drawer, open door, press button top, open window, close window, and insert peg inside. " + ], + "image_footnote": [], + "bbox": [ + 173, + 103, + 820, + 393 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 449, + 326, + 465 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The goal of our experiments is to study the following questions: (1) Are conflicting gradients a major factor in making optimization for multi-task learning challenging? (2) Does PCGrad make the optimization problems easier for various multi-task learning problems including supervised, reinforcement, and goal-conditioned reinforcement learning settings across different task families? (3) Can PCGrad be combined with other multi-task learning approaches to further improve performance? ", + "bbox": [ + 174, + 476, + 825, + 545 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 559, + 375, + 573 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To evaluate our method experimentally, we consider both a multi-task supervised learning and a multi-task reinforcement learning problem setup. For supervised learning, we first consider the MultiMNIST dataset (Sener & Koltun, 2018), which contains two tasks: classifying the digit on the top left and on the bottom right in an overlaid image. Beyond digit classification, we also use the CIFAR-100 dataset (Krizhevsky et al., 2009) where each of the 20 label superclasses are treated as distinct tasks, following Rosenbaum et al. (2018). We also conduct experiments on the NYUv2 dataset (Silberman et al., 2012), which consists of RGB-D indoor scene images. Following Liu et al. (2018), we evaluate our method on 3 tasks: 13-class semantic segmentation, depth estimation, and surface normal prediction. In the case of multi-task reinforcement learning, we evaluate our algorithm on the recently proposed Meta-World benchmark (Yu et al., 2019). This benchmark includes a variety of simulated robotic manipulation tasks contained in a shared, table-top environment with a simulated Sawyer arm (visualized as the ”Push” environment in Fig. 3). In particular, we use the multi-task benchmarks MT10 and MT50, which consists of the 10 tasks and 50 tasks respectively depicted in Fig. 3 that require diverse strategies to solve them, which makes them difficult to optimize jointly with a single policy. Note that MT10 is a subset of MT50. To evaluate goal-conditioned RL scenarios, we consider goal-conditioned robotic pushing with a Sawyer robot. This domain is representative of challenges in learning goal-conditioned policies over a wide distribution of goals. For details on the experimental set-up and model architectures see Appendix E. ", + "bbox": [ + 174, + 582, + 825, + 830 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.2 ANALYSIS OF CONFLICTING GRADIENTS ", + "text_level": 1, + "bbox": [ + 174, + 844, + 496, + 858 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To answer question (1), we consider a simple regression problem, where each task is regressing the input to the output of a sine function. The amplitude and the phase of each task are varied. We construct 10 tasks with the amplitude uniformly sampled from the range [0, 5] and the phase uniformly sampled from the range $[ 0 , \\pi ]$ . The input is also uniformly sampled from the range $[ 0 , 5 ]$ and is concatenated with the one-hot task encoding. For training, we use a 3-layer fully-connected neural network with 100 hidden units. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/6be492001dd85363da9fcd707c39223ebe8895c02351127cc23ecc0ee1a381e0.jpg", + "image_caption": [ + "Figure 4: An analysis of the gradients during the first 1000 updates of training, on a toy 10-task sinusoid regression problem. Left: The cosine similarity between the gradients of 2 of the 10 tasks (selected arbitrarily, and fixed throughout this plot). We observe a substantial amount of thrashing with standard Adam training, while Adam with PCGrad reduces the thrashing and leads to more closely aligned updates. Right: Adam with PCGrad improves performance compared to standard Adam. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 821, + 217 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 310, + 821, + 339 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare the performance of the network trained with Adam and the network trained with Adam with PCGrad-modified gradients while plotting the cosine similarity between a pair of tasks during training as shown in Figure 4. The plot on the left in Figure 4 demonstrates that the cosine similarity of Adam gradients between a pair of tasks has high variance, which leads to the gradient thrashing problem, while the cosine similarity of the gradient projected by PCGrad yields positive values diminishing the conflicting-gradients problem. As shown in the plot on the right in Figure 4, Adam with PCGrad leads to faster learning over Adam, which implies that gradient thrashing is indeed a problem in multi-task optimization and reducing it can result in considerable performance boost. ", + "bbox": [ + 174, + 347, + 825, + 458 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.3 MULTI-TASK AND MULTI-OBJECTIVE SUPERVISED LEARNING WITH PCGRAD ", + "text_level": 1, + "bbox": [ + 174, + 469, + 756, + 483 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To answer question (3), we perform experiments on three standard multi-task supervised learning datasets: MultiMNIST, multi-task CIFAR-100 and NYUv2. We include the results on MultiMNIST in Appendix C. ", + "bbox": [ + 176, + 492, + 825, + 534 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For CIFAR-100, we follow (Rosenbaum et al., 2018) to treat 20 coarse labels in the dataset as distinct tasks and create a dataset with 20 tasks and 2500 training instances as well as 500 test instances per task. We combine PCGrad with a powerful multi-task learning architecture, routing networks (Rosenbaum et al., 2018; 2019), by simply projecting gradients of the shared parameters in routing networks. As shown in Table 1, applying PCGrad to a single network achieves $71 \\%$ classification accuracy, which outperforms most of the prior methods such as independent training and cross-stitch (Misra et al., 2016b). Though routing networks achieve better performance than PCGrad on its own, PCGrad is complementary to routing networks and combining PCGrad with routing networks leads to a $2 . 8 \\%$ absolute improvement in test accuracy averaged over 3 runs. ", + "bbox": [ + 173, + 540, + 825, + 666 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also combine PCGrad with another state-of-art multi-task learning algorithm, MTAN (Liu et al., 2018), and evaluate the performance on a more challenging indoor scene dataset, NYUv2, which contains 3 tasks as described in Section 6.1. We compare MTAN with PCGrad to a list of methods mentioned in Section 6.1, where each method is trained with three different weighting schemes as in (Liu et al., 2018), equal weighting, weight uncertainty (Kendall et al., 2018a), and DWA (Liu et al., 2018). We only run MTAN with PCGrad with weight uncertainty as we find weight uncertainty as the most effective scheme for training MTAN. The results comparing Cross-Stitch, MTAN and MTAN $^ +$ PCGrad are presented in Table 2 while the full comparison can be found in Table 4 in the Appendix E.3. MTAN with PCGrad is able to achieve the best scores in 8 out of the 9 categories where there are 3 categories per task. ", + "bbox": [ + 174, + 672, + 825, + 813 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our multi-task supervised learning results demonstrate that PCGrad can be seamlessly combined with state-of-art multi-task learning architectures and further improve their results on established supervised multi-task learning benchmarks. ", + "bbox": [ + 174, + 819, + 825, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.4 MULTI-TASK REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 872, + 511, + 887 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To answer question (2), we test all methods on 10 and 50 manipulation tasks respectively shown in Figure 3. At each data collection step, we collect 600 samples for each task, and at each training step, ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/8422e47507603dbedc09dc524e98c40a0530b1bd7de6d445ebe2838dc5f02d41.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
% accuracy
task specific-1-fc (Rosenbaum et al., 2018)42
task specific-all-fc (Rosenbaum et al., 2018)49
cross stitch-all-fc (Misra et al.,2016b) routing-all-fc + WPL (Rosenbaum et al.,2019)53
independent74.7
PCGrad (ours)67.7
71
routing-all-fc + WPL + PCGrad (ours)[77.5
", + "bbox": [ + 305, + 102, + 692, + 224 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/1e0079a5e814421afb35412ca245a7935b5adeb8553aabeeb807bf1b10aae17b.jpg", + "table_caption": [ + "Table 1: CIFAR-100 multi-task results. We apply PCGrad to the routing networks and achieve a significant improvement in classfication accuracy. " + ], + "table_footnote": [], + "table_body": "
#P.ArchitectureWeightingSegmentationDepthSurface Normal
(Higher Better) mIoUPix Acc(Lower Better) Abs ErrRel ErrAngle Distance (Lower Better) Mean1Median11.25Within t° (Higher Better) 22.530
~3Cros-Stitch+Equal Weights14.7150.230.64810.287133.5628.5820.0840.5451.97
Uncert.Weights*15.6952.600.62770.270232.6927.2621.6342.8454.45
DWA†,T=216.1153.190.59220.261132.3426.9121.8143.1454.92
Equal Weights17.7255.320.59060.257731.4425.3723.1745.6557.48
1.77MTANtUncert.Weights*17.6755.610.59270.259231.2525.5722.9945.8357.67
DWA+,T=217.1554.970.59560.256931.6025.4622.4844.8657.24
1.77MTAN++ PCGrad (ours）Uncert.Weights*20.1756.650.59040.246730.0124.8322.2846.1258.77
", + "bbox": [ + 179, + 262, + 815, + 407 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 2: We present the results on three tasks on the NYUv2 dataset: 13-class semantic segmentation, depth estimation, and surface normal prediction results. #P shows the total number of network parameters. We highlight the best performing combination of multi-task architecture and weighting in bold. The top validation scores for each task are annotated with boxes. The symbols indicate prior methods: ∗: (Kendall et al., 2018a), †: (Liu et al., 2018), ‡: (Misra et al., 2016b). Performance of other methods as reported in (Liu et al., 2018). ", + "bbox": [ + 173, + 419, + 825, + 483 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/377976d48312e8cb7dfcb0de421a3d07cf0558c92f93eb0fe433c37dc3e4830e.jpg", + "image_caption": [ + "Figure 5: Learning curve on MT10, MT50 and goal-conditioned pushing. PCGrad outperforms the other methods in the three settings in terms of both success rates / average distance to the goal and data efficiency. " + ], + "image_footnote": [], + "bbox": [ + 183, + 500, + 813, + 613 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "we sample 128 datapoints per task from corresponding replay buffers. The results are shown in the two plots on the left in Figure 5. We measure success according to the metrics used in the Meta-World benchmark where the reported the success rates are averaged across tasks. For all methods, we apply PA as discussed in Section 4 to learn a separate alpha term per task as the task encoding in MT10 and MT50 is just a one-hot encoding. PCGrad combined with SAC learns all tasks with the best data efficiency and successfully solves all of the 10 tasks in MT10 and about $70 \\%$ of the 50 tasks in MT50. Training a single SAC policy and a multi-head policy turns out to be unable to acquire half of the skills in both MT10 and MT50, suggesting that eliminating gradient interference across tasks can significantly boost performance of multi-task RL. Training independent SAC agents is able to eventually solve all tasks in MT10 and $70 \\%$ of the tasks in MT50, but requires about 2 millions and 15 millions more samples than PCGrad with SAC in MT10 and MT50 respectively, implying that applying PCGrad can result in leveraging shared structure among tasks that expedites multi-task learning. ", + "bbox": [ + 173, + 666, + 825, + 847 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As noted by $\\mathrm { Y u }$ et al. (2019), these tasks involve fairly distinct behavior motions, which makes learning all of them with a single policy challenging as demonstrated by poor baseline performance. The ability to learn these tasks together opens the door for a number of interesting extensions to meta-learning, goal conditioned RL and generalization to novel task families. We present the results of PCGrad on goal-conditioned RL in the following subsection. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We also provide an ablation study on the importance of correcting the gradient direction and scaling the gradient magnitudes in PCGrad. We construct two variants of PCGrad: (1) only applying the gradient direction corrected with PCGrad while keeping the gradient magnitude unchanged and (2) only applying the gradient magnitude computed by PCGrad while keeping the gradient direction unchanged. As shown in the plot on the left in Figure 6, both variants perform worse than PCGrad and the variant where we only vary the gradient magnitudes is much worse than PCGrad. We also compare PCGrad to a prior method GradNorm (Chen et al., 2018), which scales the magnitude of gradients of all the tasks. As shown in the plot on the right in Figure 6, PCGrad significantly outperforms GradNorm. We also notice that the variant of PCGrad where only the gradient magnitudes change gets comparable results to GradNorm, which suggests that its important to modify both the gradient directions and magnitudes to eliminate interference and achieve good multi-task learning results. ", + "bbox": [ + 173, + 103, + 825, + 256 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/412941b5dca2cd83f58b0c272e922c1522ccbaa8b55a58fb2ffb379383c3a3c8.jpg", + "image_caption": [ + "Figure 6: Ablation study on only using the magnitude and the direction of the gradients modified by PCGrad (left) and comparison between PCGrad and GradNorm (Chen et al., 2018) (right). PCGrad outperforms both ablations and GradNorm with a large margin, indicating the importance of modifying both the gradient directions and magnitudes in multi-task learning. " + ], + "image_footnote": [], + "bbox": [ + 205, + 267, + 790, + 393 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6.5 GOAL-CONDITIONED REINFORCEMENT LEARNING", + "text_level": 1, + "bbox": [ + 174, + 463, + 568, + 477 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "For our goal-conditioned RL evaluation, we use the robot-pushing environment described in Sec. 6.1 where the goals are represented as the concatenations of the initial positions of the puck to be pushed and the its goal location, both of which are uniformly sampled (details in Appendix E.2). We also apply PA as discussed in Section 4 to predict the temperature for entropy term given the goal. We summarize the results in the plot on the right in Figure 5. PCGrad with SAC and PA achieves the best performance in terms of average distance to the goal position, while PCGrad with SAC improves over the baseline and a vanilla SAC agent is struggling to successfully accomplish the task. This suggests that PCGrad is able to ease the RL optimization problem also when the task distribution is continuous. ", + "bbox": [ + 174, + 482, + 825, + 607 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 617, + 320, + 632 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this work, we identified one of the major challenges in multi-task optimization: conflicting gradients across tasks. We proposed a simple algorithm (PCGrad) to mitigate the challenge of conflicting gradients via “gradient surgery”. PCGrad provides a simple way to project gradients to be orthogonal in a multi-task setting, which substantially improves optimization performance, since the task gradients are prevented from negating each other. We provide some simple didactic examples and analysis of how this procedure works in simple settings, and subsequently show significant improvement in optimization for a variety of multi-task supervised learning and reinforcement learning problems. We show that, once some of the optimization challenges of multi-task learning are alleviated by PCGrad, we can obtain the hypothesized benefits in efficiency and asymptotic performance that are believed to be possible in multi-task settings. ", + "bbox": [ + 173, + 638, + 825, + 777 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "While we studied multi-task supervised learning and multi-task reinforcement learning in this work, we suspect the problem of conflicting gradients to be prevalent in a range of other settings and applications, such as meta-learning, continual learning, multi-goal imitation learning (Codevilla et al., 2018), and multi-task problems in natural language processing applications (McCann et al., 2018). Due to its simplicity and model-agnostic nature, we expect that applying PCGrad in these domains to be a promising avenue for future investigation. Further, the general idea of gradient surgery may be an important ingredient for alleviating a broader class of optimization challenges in deep learning, such as the challenges in the stability challenges in two-player games (Roth et al., 2017) and multi-agent optimizations (Nedic & Ozdaglar, 2009). 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", + "bbox": [ + 173, + 103, + 825, + 133 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 176, + 150, + 398, + 166 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. We will use the shorthand $| | \\cdot | |$ to denote the $L _ { 2 }$ -norm and $\\nabla { \\mathcal { L } } = \\nabla _ { \\theta } { \\mathcal { L } }$ , where $\\theta$ is the parameter vector. Let $\\mathbf { g _ { 1 } } = \\nabla \\mathcal { L } _ { 1 }$ , $\\mathbf { g _ { 2 } } = \\nabla \\mathcal { L } _ { 2 }$ , and $\\phi$ be the angle between $\\bf { g _ { 1 } }$ and $\\mathbf { g _ { 2 } }$ . ", + "bbox": [ + 173, + 174, + 823, + 205 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "At each PCGrad update, we have two cases: $c o s ( \\phi ) \\geq 0$ or $\\cos ( \\phi < 0 )$ . ", + "bbox": [ + 173, + 210, + 643, + 227 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "If $\\cos ( \\phi ) \\geq 0$ , then we apply the standard gradient descent update using $\\begin{array} { r } { t \\leq \\frac { 1 } { L } } \\end{array}$ , which leads to a strict decrease in the objective function value ${ \\mathcal { L } } ( \\phi )$ unless $\\nabla \\mathcal { L } ( \\phi ) = 0$ , which occurs only when $\\theta = \\theta ^ { * }$ (Boyd & Vandenberghe, 2004). ", + "bbox": [ + 174, + 232, + 825, + 275 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In the case that $\\cos ( \\phi ) < 0$ , we proceed as follows: ", + "bbox": [ + 173, + 281, + 511, + 296 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Our assumption that $\\nabla \\mathcal { L }$ is Lipschitz continuous with constant $L$ implies that $\\nabla ^ { 2 } { \\mathcal { L } } ( \\theta ) - L I$ is a negative semidefinite matrix. Using this fact, we can perform a quadratic expansion of $\\mathcal { L }$ around $\\mathcal { L } ( \\boldsymbol { \\theta } )$ and obtain the following inequality: ", + "bbox": [ + 176, + 303, + 823, + 344 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/da45ea21b36ff60aa246c9e39a5dcba8ac2afd13e991ae913a40925423a0fd31.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } ( { \\boldsymbol { \\theta } } ^ { + } ) \\leq \\mathcal { L } ( { \\boldsymbol { \\theta } } ) + \\nabla \\mathcal { L } ( { \\boldsymbol { \\theta } } ) ^ { T } ( { \\boldsymbol { \\theta } } ^ { + } - { \\boldsymbol { \\theta } } ) + \\frac { 1 } { 2 } \\nabla ^ { 2 } \\mathcal { L } ( { \\boldsymbol { \\theta } } ) \\vert \\vert { \\boldsymbol { \\theta } } ^ { + } - { \\boldsymbol { \\theta } } \\vert \\vert ^ { 2 } } \\\\ { \\leq \\mathcal { L } ( { \\boldsymbol { \\theta } } ) + \\nabla \\mathcal { L } ( { \\boldsymbol { \\theta } } ) ^ { T } ( { \\boldsymbol { \\theta } } ^ { + } - { \\boldsymbol { \\theta } } ) + \\frac { 1 } { 2 } L \\vert \\vert { \\boldsymbol { \\theta } } ^ { + } - { \\boldsymbol { \\theta } } \\vert \\vert ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 351, + 696, + 412 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Now, we can plug in the PCGrad update by letting θ+ = θ − t(∇L(θ) − g1·g2||g1||2 g1 − | g1·g2|g2||2 g2). We then get: ", + "bbox": [ + 176, + 417, + 821, + 450 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/5601f984611c4c47acb213c31dc465fdf08d993d3e758cc0d7b6ffbeea3bce35.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ( \\boldsymbol { \\theta } ^ { + } ) \\leq \\mathcal { L } ( \\boldsymbol { \\theta } ) + t ( \\nabla \\mathcal { L } ( \\boldsymbol { \\theta } ) ) ^ { T } ( - \\nabla \\mathcal { L } ( \\boldsymbol { \\theta } ) + \\frac { \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } } { | | \\mathbf { g _ { 1 } } | | ^ { 2 } } \\mathbf { g _ { 1 } } + \\frac { \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } } { | | \\mathbf { g _ { 2 } } | | ^ { 2 } } \\mathbf { g _ { 2 } } ) } \\\\ & { \\qquad + \\frac { 1 } { 2 } L t ^ { 2 } | | \\nabla \\mathcal { L } ( \\boldsymbol { \\theta } ) - \\frac { \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } } { | | \\mathbf { g _ { 1 } } | | ^ { 2 } } \\mathbf { g _ { 1 } } - \\frac { \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } } { | | \\mathbf { g _ { 2 } } | | ^ { 2 } } \\mathbf { g _ { 2 } } | | ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 455, + 684, + 522 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/48620aed8f353196343a6503a5c1b848cc130cc1b4ac899e929fc66aee2ccd1a.jpg", + "text": "$$\n\\begin{array} { l } { = \\displaystyle \\mathcal { L } ( \\boldsymbol { \\theta } ) + t ( - | | \\mathbf { g _ { 1 } } | | ^ { 2 } - | | \\mathbf { g _ { 2 } } | | ^ { 2 } + 2 \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } + \\frac { ( \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } ) ^ { 2 } } { | | \\mathbf { g _ { 1 } } | | ^ { 2 } } + \\frac { ( \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } ) ^ { 2 } } { | | \\mathbf { g _ { 2 } } | | ^ { 2 } } ) } \\\\ { + \\displaystyle \\frac { 1 } { 2 } L t ^ { 2 } | | \\mathbf { g _ { 1 } } + \\mathbf { g _ { 2 } } - \\frac { \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } } { | | \\mathbf { g _ { 1 } } | | ^ { 2 } } \\mathbf { g _ { 1 } } - \\frac { \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } } { | | \\mathbf { g _ { 2 } } | | ^ { 2 } } \\mathbf { g _ { 2 } } | | ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 574, + 754, + 646 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(Expanding further and re-arranging terms) ", + "bbox": [ + 287, + 662, + 570, + 679 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c2b9ec75d30e2be3a5d20e1e00352d4776e6375a34b4e2d5f551e7958df00957.jpg", + "text": "$$\n{ \\begin{array} { l } { = { \\mathcal { L } } ( \\theta ) - ( t - { \\frac { 1 } { 2 } } L t ^ { 2 } ) ( | | \\mathbf { g _ { 1 } } | | ^ { 2 } + | | \\mathbf { g _ { 2 } } | | ^ { 2 } - { \\frac { \\left( \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } \\right) } { | | \\mathbf { g _ { 1 } } | | ^ { 2 } } } - { \\frac { \\left( \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } \\right) } { | | \\mathbf { g _ { 2 } } | | ^ { 2 } } } ) } \\\\ { = { \\mathcal { L } } t ^ { 2 } ( \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } - { \\frac { \\left( \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } \\right) ^ { 2 } } { | | \\mathbf { g _ { 1 } } | | ^ { 2 } | | \\mathbf { g _ { 2 } } | | ^ { 2 } } } \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } ) } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 287, + 696, + 722, + 770 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(Using the identity $\\cos ( \\phi ) = { \\frac { \\mathbf { g _ { 1 } } \\cdot \\mathbf { g _ { 2 } } } { | | \\mathbf { g _ { 1 } } | | | | \\mathbf { g _ { 2 } } | | } } )$ ", + "bbox": [ + 284, + 786, + 560, + 819 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7577bcd38ed0c92f9ac8dbb4d304e974ee398b07457b57fbd2d09ffa460115d7.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle = \\mathcal { L } ( \\theta ) - ( t - \\frac { 1 } { 2 } L t ^ { 2 } ) [ ( 1 - \\cos ^ { 2 } ( \\phi ) ) | | { \\bf g _ { 1 } } | ] ^ { 2 } + ( 1 - \\cos ^ { 2 } ( \\phi ) ) | | { \\bf g _ { 2 } } | | ^ { 2 } } \\} } \\\\ { { \\displaystyle - \\left. L t ^ { 2 } ( 1 - \\cos ^ { 2 } ( \\phi ) ) | | { \\bf g _ { 1 } } | | | { \\bf g _ { 2 } } | | \\cos ( \\phi ) \\right. } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 837, + 738, + 888 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(Note that $\\cos ( \\phi ) < 0$ so the final term is non-negative) ", + "bbox": [ + 284, + 905, + 651, + 920 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Using $\\begin{array} { r } { t \\le \\frac { 1 } { L } } \\end{array}$ , we know that $\\begin{array} { r } { - ( 1 - \\frac 1 2 L t ) = \\frac 1 2 L t - 1 \\le \\frac 1 2 L ( 1 / L ) - 1 = \\frac { - 1 } { 2 } } \\end{array}$ and $L t ^ { 2 } \\leq t$ ", + "bbox": [ + 171, + 101, + 759, + 121 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Plugging this into the last expression above, we can conclude the following: ", + "bbox": [ + 176, + 125, + 673, + 140 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/766c441f1b57efec78c1a581652c1af8a92e749623230ec937def1ef7b4c005e.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } ( \\boldsymbol { \\theta } ^ { + } ) \\leq \\mathcal { L } ( \\boldsymbol { \\theta } ) - \\frac { 1 } { 2 } t [ ( 1 - \\cos ^ { 2 } ( \\boldsymbol { \\phi } ) ) | | \\mathbf { g } _ { 1 } | ] ^ { 2 } + ( 1 - \\cos ^ { 2 } ( \\boldsymbol { \\phi } ) ) | | \\mathbf { g } _ { 2 } | ] ^ { 2 } ] } \\\\ { \\displaystyle - t ( 1 - \\cos ^ { 2 } ( \\boldsymbol { \\phi } ) ) | | \\mathbf { g } _ { 1 } | | | | \\mathbf { g } _ { 2 } | | \\cos ( \\boldsymbol { \\phi } ) } \\\\ { \\displaystyle = \\mathcal { L } ( \\boldsymbol { \\theta } ) - \\frac { 1 } { 2 } t ( 1 - \\cos ^ { 2 } ( \\boldsymbol { \\phi } ) ) | | | \\mathbf { g } _ { 1 } | | ^ { 2 } + 2 | | \\mathbf { g } _ { 1 } | | | | \\mathbf { g } _ { 2 } | | \\cos ( \\boldsymbol { \\phi } ) + | | \\mathbf { g } _ { 2 } | | ^ { 2 } ] } \\\\ { \\displaystyle = \\mathcal { L } ( \\boldsymbol { \\theta } ) - \\frac { 1 } { 2 } t ( 1 - \\cos ^ { 2 } ( \\boldsymbol { \\phi } ) ) [ | | \\mathbf { g } _ { 1 } | | ^ { 2 } + 2 \\mathbf { g } _ { 1 } \\cdot \\mathbf { g } _ { 2 } + | | \\mathbf { g } _ { 2 } | | ^ { 2 } ] } \\\\ { \\displaystyle = \\mathcal { L } ( \\boldsymbol { \\theta } ) - \\frac { 1 } { 2 } t ( 1 - \\cos ^ { 2 } ( \\boldsymbol { \\phi } ) ) | | \\mathbf { g } _ { 1 } + \\mathbf { g } _ { 2 } | | ^ { 2 } } \\\\ { \\displaystyle = \\mathcal { L } ( \\boldsymbol { \\theta } ) - \\frac { 1 } { 2 } t ( 1 - \\cos ^ { 2 } ( \\boldsymbol { \\phi } ) ) | | \\nabla \\mathcal { L } ( \\boldsymbol { \\theta } ) | | ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 143, + 725, + 316 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "If $\\cos ( \\phi ) > - 1$ , then $\\begin{array} { r l } { { \\frac { 1 } { 2 } t ( 1 - \\cos ^ { 2 } ( \\phi ) ) \\| \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } } } \\end{array}$ will always be positive unless $\\nabla { \\mathcal { L } } ( \\theta ) = 0$ . This inequality implies that the objective function value strictly decreases with each iteration where $\\cos ( \\phi ) > - 1$ . ", + "bbox": [ + 174, + 327, + 825, + 371 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Hence repeatedly applying PCGrad process can either reach the optimal value ${ \\mathcal { L } } ( \\theta ) = { \\mathcal { L } } ( \\theta ^ { * } )$ or $\\cos ( \\phi ) = - 1$ , in which case $\\begin{array} { r } { \\frac { 1 } { 2 } t ( 1 - \\bar { \\cos ^ { 2 } ( \\phi ) } ) \\| \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } = 0 , } \\end{array}$ . Note that this result only holds when we choose $t$ to be small enough, i.e. $\\begin{array} { r } { t \\le \\frac { 1 } { L } } \\end{array}$ . ", + "bbox": [ + 173, + 376, + 825, + 422 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B 2D OPTIMIZATION LANDSCAPE DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 460, + 552, + 477 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To produce the 2D optimization visualizations in Figure 1, we used a parameter vector $\\theta = \\left[ \\theta _ { 1 } , \\theta _ { 2 } \\right] \\in$ $\\mathbb { R } ^ { 2 }$ and the following task loss functions: ", + "bbox": [ + 173, + 484, + 825, + 513 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/70ec97f2c015e7e95a9ab21b9c63fbf1322d0d6ac1a861ef0542b71b348cf2b9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { 1 } ( \\theta ) = 2 0 \\log ( \\operatorname* { m a x } ( | . 5 \\theta _ { 1 } + \\operatorname { t a n h } ( \\theta _ { 2 } ) | , 0 . 0 0 0 0 0 5 ) ) } \\\\ & { \\mathcal { L } _ { 2 } ( \\theta ) = 2 5 \\log ( \\operatorname* { m a x } ( | . 5 \\theta _ { 1 } - \\operatorname { t a n h } ( \\theta _ { 2 } ) + 2 | , 0 . 0 0 0 0 0 5 ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 517, + 687, + 554 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The multi-task objective is $\\mathcal { L } ( \\theta ) = \\mathcal { L } _ { 1 } ( \\theta ) + \\mathcal { L } _ { 2 } ( \\theta )$ . We initialized $\\theta = [ 0 . 5 , - 3 ]$ and performed 500,000 gradient updates to minimize $\\mathcal { L }$ using the Adam optimizer with learning rate 0.001. We compared using Adam for each update to using Adam in conjunction with the PCGrad method presented in Section 3.2. ", + "bbox": [ + 173, + 556, + 825, + 613 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C EXPERIMENTAL RESULTS ON MULTIMNIST", + "text_level": 1, + "bbox": [ + 173, + 622, + 578, + 640 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Following the same set-up in Sener & Koltun (2018), for each image, we sample a different one uniformly at random. Then we put one of the image on the top left and the other one on the bottom right. The two tasks in the multi-task learning problem are to classify the digits on the top left (task-L) and bottom right (task-R) respectively. We construct such $6 0 \\mathrm { K }$ examples. We combine PCGrad with the same backbone architecture used in (Sener & Koltun, 2018) and compare its performance to Sener & Koltun (2018) by running the open-sourced code provided in (Sener & Koltun, 2018). As shown in Table 3, our method results $0 . 1 3 \\%$ and $0 . 5 5 \\%$ improvement over Sener & Koltun (2018) in left and right digit accuracy respectively. ", + "bbox": [ + 173, + 646, + 826, + 758 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/40944757bf48fbabb24a8057dc05cc39314b780d0d6993b24894f8a4cc4df537.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
left digitright digit
Sener & Koltun (2018)96.4594.95
PCGrad (ours)96.5895.50
", + "bbox": [ + 344, + 770, + 651, + 832 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table 3: MultiMNIST results. PCGrad achieves improvements over Sener & Koltun (2018) in both left and right digit classfication accuracy. ", + "bbox": [ + 173, + 842, + 823, + 868 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D ABLATION STUDY ON THE TASK ORDER ", + "text_level": 1, + "bbox": [ + 174, + 872, + 540, + 888 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "As stated on line 4 in Algorithm 1, we sample the tasks from the batch and randomly shuffle the order of the tasks before performing the update steps in PCGrad. With random shuffling, we make ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "PCGrad symmetric w.r.t. the task order in expectation. In Figure 7, we observe that PCGrad with a random task order achieves better performance between PCGrad with a fixed task order in the setting of MT50 where the number of tasks is large and the conflicting gradient phenomenon is much more likely to happen. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/807b5fd27120d7c92f596dd5dff23d51259e68ce83646668c231b41e2475cb34.jpg", + "image_caption": [ + "Figure 7: Ablation study on using a fixed task order during PCGrad. PCGrad with a random task order does significantly better PCGrad with a fixed task order in MT50 benchmark. " + ], + "image_footnote": [], + "bbox": [ + 305, + 174, + 687, + 329 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E EXPERIMENT DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 383, + 398, + 398 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E.1 DETAILED EXPERIMENT SET-UP ", + "bbox": [ + 178, + 409, + 441, + 422 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For our CIFAR-100 multi-task experiment, we adopt the architecture used in Rosenbaum et al. (2019), which is a convolutional neural network that consists of 3 convolutional layers with $1 6 0 3 \\times 3$ filters each layer and 2 fully connected layers with 320 hidden units. As for experiments on the NYUv2 dataset, we follow Liu et al. (2018) to use SegNet (Badrinarayanan et al., 2017) as the backbone architecture. ", + "bbox": [ + 174, + 431, + 825, + 501 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Our reinforcement learning experiments all use the SAC (Haarnoja et al., 2018) algorithm as the base algorithm, where the actor and the critic are represented as 6-layer fully-connected feedforward neural networks for all methods. The numbers of hidden units of each layer of the neural networks are 160, 300 and 200 for MT10, MT50 and goal-conditioned RL respectively. ", + "bbox": [ + 174, + 507, + 825, + 563 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use five algorithms as baselines in the CIFAR-100 multi-task experiment: task specific-1-fc (Rosenbaum et al., 2018): a convolutional neural network shared across tasks except that each task has a separate last fully-connected layer, task specific-1-fc (Rosenbaum et al., 2018) : all the convolutional layers shared across tasks with separate fully-connected layers for each task, cross stitch-all-fc (Misra et al., 2016b): one convolutional neural network per task along with cross-stitch units to share features across tasks, routing-all-fc ${ \\bf \\Pi } _ { \\bf { \\Pi } } + { \\bf W } { \\bf P } { \\bf L }$ (Rosenbaum et al., 2019): a network that employs a trainable router trained with multi-agent RL algorithm (WPL) to select trainable functions for each task, independent: training separate neural networks for each task. ", + "bbox": [ + 173, + 570, + 825, + 681 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For comparisons on the NYUv2 dataset, we consider 5 baselines: Single Task, One Task: the vanilla SegNet used for single-task training, Single Task, STAN (Liu et al., 2018): the single-task version of MTAN as mentioned below, Multi-Task, Split, Wide / Deep (Liu et al., 2018): the standard SegNet shared for all three tasks except that each task has a separate last layer for final task-specific prediction with two variants Wide and Deep specified in Liu et al. (2018), Multi-Task Dense: a shared network followed by separate task-specific networks, Multi-Task Cross-Stitch (Misra et al., 2016b): similar to the baseline used in CIFAR-100 experiment but with SegNet as the backbone, MTAN (Liu et al., 2018): a shared network with a soft-attention module for each task. ", + "bbox": [ + 173, + 689, + 825, + 800 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "On the multi-task and goal-conditioned RL domain, we apply PCGrad to the vanilla SAC algorithm with task encoding as part of the input to the actor and the critic as described in Section 4 and compare our method to the vanilla SAC without PCGrad and training actors and critics for each task individually (Independent). ", + "bbox": [ + 174, + 806, + 825, + 863 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E.2 GOAL-CONDITIONED EXPERIMENT DETAILS", + "text_level": 1, + "bbox": [ + 174, + 873, + 526, + 887 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We use the pushing environment from the Meta-World benchmark (Yu et al., 2019) as shown in Figure 3. In this environment, the table spans from $[ - 0 . 4 , 0 . 2 ]$ to [0.4, 1.0] in the 2D space. To construct the goals, we sample the intial positions of the puck from the range $[ - 0 . 2 , 0 . 6 ]$ to [0.2, 0.7] on the table and the goal positions from the range $[ - 0 . 2 , 0 . 8 5 ]$ to $[ 0 . 2 , 0 . 9 5 ]$ on the table. The goal is represented as a concatenation of the initial puck position and the goal position. Since in the goal-conditioned setting, the task distribution is continuous, we sample a minibatch of 9 goals and 128 samples per goal at each training iteration and also sample 600 samples per goal in the minibatch at each data collection step. ", + "bbox": [ + 176, + 895, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E.3 FULL NYUV2 RESULTS", + "text_level": 1, + "bbox": [ + 174, + 196, + 383, + 212 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We provide the full comparison on the NYUv2 dataset in Table 4. ", + "bbox": [ + 173, + 218, + 602, + 233 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/e6d43772b8d5b6a739ebe5b491dd43f7ddfbc5652b9b27093753f371a57fbbf4.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Type#P.Architecture WeightingSegmentationDepthSurface Normal
mIoU Pix Acc Abs Err Rel Err(Higher Better） (Lower Better)Angle Distance (Lower Better) Mean MedianWithin t (Higher Better) 11.2522.5
Single Task3 One Task15.10 51.540.7508 0.3266531.7625.5122.12 45.3330 57.13
4.56 STANtn.a. n.a.15.7352.890.69350.289132.0926.3221.4944.3856.51
Multi Task1.75 Split, WideEqual Weights15.8951.190.64940.280433.6928.9118.5439.9152.02
Uncert.Weights*15.8651.120.60400.257032.3326.6221.6843.5955.36
DWA+,T=216.9253.720.61250.254632.3427.1020.6942.7354.74
2Split, DeepEqual Weights13.0341.470.78360.332638.2836.559.5027.1139.63
Uncert.Weights*14.5343.690.77050.334035.1432.1314.6934.5246.94
DWA+,T= 213.6344.410.75810.322736.4134.1212.8231.1243.48
4.95 DenseEqual Weights16.0652.730.64880.287133.5828.0120.0741.5053.35
Uncert.Weights*16.4854.400.62820.276131.6825.6821.7344.5856.65
DWAt,T = 216.1554.350.60590.259332.4427.4020.5342.7654.27
≈3 Cross-StitchtEqual Weights14.71 15.6950.230.64810.287133.5628.5820.08 40.5451.97
Uncert. Weights*52.600.62770.270232.6927.2621.63 21.8142.8454.45 54.92
DWA+,T= 216.1153.190.59220.261132.3426.9143.14
1.77 MTANtEqual Weights 17.7255.320.59060.257731.4425.3723.1745.6557.48
Uncert.Weights*17.6755.610.59270.259231.2525.5722.9945.8357.67
DWAt,T=217.1554.970.59560.256931.6025.4622.48 44.8657.24
", + "bbox": [ + 181, + 244, + 815, + 523 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Table 4: We present the full results on three tasks on the NYUv2 dataset: 13-class semantic segmentation, depth estimation, and surface normal prediction results. #P shows the total number of network parameters. We highlight the best performing combination of multi-task architecture and weighting in bold. The top validation scores for each task are annotated with boxes. The symbols indicate prior methods: ∗: (Kendall et al., 2018a), †: (Liu et al., 2018), $^ \\ddag$ : (Misra et al., 2016b). Performance of other methods taken from (Liu et al., 2018). 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#P.ArchitectureWeightingSegmentationDepthSurface Normal
(Higher Better) mIoUPix Acc(Lower Better) Abs ErrRel ErrAngle Distance (Lower Better) Mean1Median11.25Within t° (Higher Better) 22.530
~3Cros-Stitch+Equal Weights14.7150.230.64810.287133.5628.5820.0840.5451.97
Uncert.Weights*15.6952.600.62770.270232.6927.2621.6342.8454.45
DWA†,T=216.1153.190.59220.261132.3426.9121.8143.1454.92
Equal Weights17.7255.320.59060.257731.4425.3723.1745.6557.48
1.77MTANtUncert.Weights*17.6755.610.59270.259231.2525.5722.9945.8357.67
DWA+,T=217.1554.970.59560.256931.6025.4622.4844.8657.24
1.77MTAN++ PCGrad (ours）Uncert.Weights*20.1756.650.59040.246730.0124.8322.2846.1258.77
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% accuracy
task specific-1-fc (Rosenbaum et al., 2018)42
task specific-all-fc (Rosenbaum et al., 2018)49
cross stitch-all-fc (Misra et al.,2016b) routing-all-fc + WPL (Rosenbaum et al.,2019)53
independent74.7
PCGrad (ours)67.7
71
routing-all-fc + WPL + PCGrad (ours)[77.5
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Type#P.Architecture WeightingSegmentationDepthSurface Normal
mIoU Pix Acc Abs Err Rel Err(Higher Better） (Lower Better)Angle Distance (Lower Better) Mean MedianWithin t (Higher Better) 11.2522.5
Single Task3 One Task15.10 51.540.7508 0.3266531.7625.5122.12 45.3330 57.13
4.56 STANtn.a. n.a.15.7352.890.69350.289132.0926.3221.4944.3856.51
Multi Task1.75 Split, WideEqual Weights15.8951.190.64940.280433.6928.9118.5439.9152.02
Uncert.Weights*15.8651.120.60400.257032.3326.6221.6843.5955.36
DWA+,T=216.9253.720.61250.254632.3427.1020.6942.7354.74
2Split, DeepEqual Weights13.0341.470.78360.332638.2836.559.5027.1139.63
Uncert.Weights*14.5343.690.77050.334035.1432.1314.6934.5246.94
DWA+,T= 213.6344.410.75810.322736.4134.1212.8231.1243.48
4.95 DenseEqual Weights16.0652.730.64880.287133.5828.0120.0741.5053.35
Uncert.Weights*16.4854.400.62820.276131.6825.6821.7344.5856.65
DWAt,T = 216.1554.350.60590.259332.4427.4020.5342.7654.27
≈3 Cross-StitchtEqual Weights14.71 15.6950.230.64810.287133.5628.5820.08 40.5451.97
Uncert. Weights*52.600.62770.270232.6927.2621.63 21.8142.8454.45 54.92
DWA+,T= 216.1153.190.59220.261132.3426.9143.14
1.77 MTANtEqual Weights 17.7255.320.59060.257731.4425.3723.1745.6557.48
Uncert.Weights*17.6755.610.59270.259231.2525.5722.9945.8357.67
DWAt,T=217.1554.970.59560.256931.6025.4622.48 44.8657.24
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b/parse/train/HJgeEh09KQ/HJgeEh09KQ_span.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:321c9bafd6d5ecfd111ebbd5967d416964f1d506a50846ecf61a2d7c046c8906 +size 548953 diff --git a/parse/train/HJlSmC4FPS/HJlSmC4FPS.md b/parse/train/HJlSmC4FPS/HJlSmC4FPS.md new file mode 100644 index 0000000000000000000000000000000000000000..c55c025b650e171a9fe21a0db389d4a449d0823d --- /dev/null +++ b/parse/train/HJlSmC4FPS/HJlSmC4FPS.md @@ -0,0 +1,310 @@ +# ROBUST AND INTERPRETABLE BLIND IMAGE DENOISING VIA BIAS-FREE CONVOLUTIONAL NEURAL NETWORKS + +Sreyas Mohan∗ Center for Data Science New York University sm7582@nyu.edu + +Zahra Kadkhodaie∗ Center for Data Science New York University zk388@nyu.edu + +# Carlos Fernandez-Granda + +Eero P. Simoncelli +Center for Neural Science, and +Howard Hughes Medical Institute +New York University +eero.simoncelli@nyu.edu + +Center for Data Science, and Courant Inst. of Mathematical Sciences New York University cfgranda@cims.nyu.edu + +# ABSTRACT + +We study the generalization properties of deep convolutional neural networks for image denoising in the presence of varying noise levels. We provide extensive empirical evidence that current state-of-the-art architectures systematically overfit to the noise levels in the training set, performing very poorly at new noise levels. We show that strong generalization can be achieved through a simple architectural modification: removing all additive constants. The resulting "bias-free" networks attain state-of-the-art performance over a broad range of noise levels, even when trained over a narrow range. They are also locally linear, which enables direct analysis with linear-algebraic tools. We show that the denoising map can be visualized locally as a filter that adapts to both image structure and noise level. In addition, our analysis reveals that deep networks implicitly perform a projection onto an adaptively-selected low-dimensional subspace, with dimensionality inversely proportional to noise level, that captures features of natural images. + +# 1 INTRODUCTION AND CONTRIBUTIONS + +The problem of denoising consists of recovering a signal from measurements corrupted by noise, and is a canonical application of statistical estimation that has been studied since the 1950’s. Achieving high-quality denoising results requires (at least implicitly) quantifying and exploiting the differences between signals and noise. In the case of photographic images, the denoising problem is both an important application, as well as a useful test-bed for our understanding of natural images. In the past decade, convolutional neural networks (LeCun et al., 2015) have achieved state-of-the-art results in image denoising (Zhang et al., 2017; Chen & Pock, 2017). Despite their success, these solutions are mysterious: we lack both intuition and formal understanding of the mechanisms they implement. Network architecture and functional units are often borrowed from the image-recognition literature, and it is unclear which of these aspects contributes to, or limits, the denoising performance. The goal of this work is advance our understanding of deep-learning models for denoising. Our contributions are twofold: First, we study the generalization capabilities of deep-learning models across different noise levels. Second, we provide novel tools for analyzing the mechanisms implemented by neural networks to denoise natural images. + +An important advantage of deep-learning techniques over traditional methodology is that a single neural network can be trained to perform denoising at a wide range of noise levels. Currently, this is achieved by simulating the whole range of noise levels during training (Zhang et al., 2017). Here, we show that this is not necessary. Neural networks can be made to generalize automatically across noise levels through a simple modification in the architecture: removing all additive constants. We find this holds for a variety of network architectures proposed in previous literature. We provide extensive empirical evidence that the main state-of-the-art denoising architectures systematically overfit to the noise levels in the training set, and that this is due to the presence of a net bias. Suppressing this bias makes it possible to attain state-of-the-art performance while training over a very limited range of noise levels. + +![](images/fabece6dfe4190ead780e139a603bf1d53d62157f9a737a2b4029c927bb45599.jpg) +Figure 1: First-order analysis of the residual of a denoising convolutional neural network as a function of noise level. The plots show the norms of the residual and the net bias averaged over $1 0 0 \ : 2 0 \times 2 0$ natural-image patches for networks trained over different training ranges. The range of noises used for training is highlighted in blue. (a) When the network is trained over the full range of noise levels $( \sigma \in [ 0 , 1 \bar { 0 } 0 ] )$ the net bias is small, growing slightly as the noise increases. (b-c) When the network is trained over the a smaller range ( ${ \bf \sigma } _ { \sigma } \in [ 0 , 5 5 ]$ and $\sigma \in [ 0 , 3 0 ] )$ , the net bias grows explosively for noise levels beyond the training range. This coincides with a dramatic drop in performance, reflected in the difference between the magnitudes of the residual and the true noise. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results as shown in Figure 8. + +The data-driven mechanisms implemented by deep neural networks to perform denoising are almost completely unknown. It is unclear what priors are being learned by the models, and how they are affected by the choice of architecture and training strategies. Here, we provide novel linear-algebraic tools to visualize and interpret these strategies through a local analysis of the Jacobian of the denoising map. The analysis reveals locally adaptive properties of the learned models, akin to existing nonlinear filtering algorithms. In addition, we show that the deep networks implicitly perform a projection onto an adaptively-selected low-dimensional subspace capturing features of natural images. + +# 2 RELATED WORK + +The classical solution to the denoising problem is the Wiener filter (Wiener, 1950), which assumes a translation-invariant Gaussian signal model. The main limitation of Wiener filtering is that it over-smoothes, eliminating fine-scale details and textures. Modern filtering approaches address this issue by adapting the filters to the local structure of the noisy image (e.g. Tomasi & Manduchi (1998); Milanfar (2012)). Here we show that neural networks implement such strategies implicitly, learning them directly from the data. + +In the 1990’s powerful denoising techniques were developed based on multi-scale ("wavelet") transforms. These transforms map natural images to a domain where they have sparser representations. This makes it possible to perform denoising by applying nonlinear thresholding operations in order to discard components that are small relative to the noise level (Donoho & Johnstone, 1995; Simoncelli & Adelson, 1996; Chang et al., 2000). From a linear-algebraic perspective, these algorithms operate by projecting the noisy input onto a lower-dimensional subspace that contains plausible signal content. The projection eliminates the orthogonal complement of the subspace, which mostly contains noise. This general methodology laid the foundations for the state-of-the-art models in the 2000’s (e.g. (Dabov et al., 2006)), some of which added a data-driven perspective, learning sparsifying transforms (Elad & Aharon, 2006), and nonlinear shrinkage functions (Hel-Or & Shaked, 2008; Raphan & Simoncelli, 2008), directly from natural images. Here, we show that deep-learning models learn similar priors in the form of local linear subspaces capturing image features. + +![](images/766b675863ec4d135ea0868be988519d4bfb8aa88e9dd9072ec2d550ff1f8dcb.jpg) +Figure 2: Denoising of an example natural image by a CNN and its bias-free counterpart (BF-CNN), both trained over noise levels in the range $\sigma \in [ 0 , 1 0 ]$ (image intensities are in the range [0, 255]). The CNN performs poorly at high noise levels $\sigma = 9 0$ , far beyond the training range), whereas BF-CNN performs at state-of-the-art levels. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Section 5). + +In the past decade, purely data-driven models based on convolutional neural networks (LeCun et al., 2015) have come to dominate all previous methods in terms of performance. These models consist of cascades of convolutional filters, and rectifying nonlinearities, which are capable of representing a diverse and powerful set of functions. Training such architectures to minimize mean square error over large databases of noisy natural-image patches achieves current state-of-the-art results (Zhang et al., 2017; Huang et al., 2017; Ronneberger et al., 2015; Zhang et al., 2018a). + +# 3 NETWORK BIAS IMPAIRS GENERALIZATION + +We assume a measurement model in which images are corrupted by additive noise: $y = x + n$ , where $\boldsymbol { x } \in \mathbb { R } ^ { N }$ is the original image, containing $N$ pixels, $n$ is an image of i.i.d. samples of Gaussian noise with variance $\sigma ^ { 2 }$ , and $y$ is the noisy observation. The denoising problem consists of finding a function $f : \mathbb { R } ^ { N } \to \mathbb { R } ^ { N }$ , that provides a good estimate of the original image, $x$ . Commonly, one minimizes the mean squared error : $\begin{array} { r } { f = \arg \operatorname* { m i n } _ { g } E | | x - g ( y ) | | ^ { 2 } } \end{array}$ , where the expectation is taken over some distribution over images, $x$ , as well as over the distribution of noise realizations. In deep learning, the denoising function $g$ is parameterized by the weights of the network, so the optimization is over these parameters. If the noise standard deviation, $\sigma$ , is unknown, the expectation must also be taken over a distribution of $\sigma$ . This problem is often called blind denoising in the literature. In this work, we study the generalization performance of CNNs across noise levels $\sigma$ , i.e. when they are tested on noise levels not included in the training set. + +Feedforward neural networks with rectified linear units (ReLUs) are piecewise affine: for a given activation pattern of the ReLUs, the effect of the network on the input is a cascade of linear transformations (convolutional or fully connected layers, $W _ { k }$ ), additive constants $( b _ { k } )$ , and pointwise multiplications by a binary mask corresponding to the fixed activation pattern $( R )$ . Since each of these is affine, the entire cascade implements a single affine transformation. For a fixed noisy input image $\boldsymbol { y } \in \mathbb { R } ^ { N }$ with $N$ pixels, the function $f : \mathbb { R } ^ { N } \xrightarrow [ ] { } \mathbb { R } ^ { N }$ computed by a denoising neural network may be written + +$$ +f ( y ) = W _ { L } R ( W _ { L - 1 } . . . . R ( W _ { 1 } y + b _ { 1 } ) + . . . b _ { L - 1 } ) + b _ { L } = A _ { y } y + b _ { y } , +$$ + +where $A _ { y } \in \mathbb R ^ { N \times N }$ is the Jacobian of $f ( \cdot )$ evaluated at input $y$ , and $b _ { y } \in \mathbb { R } ^ { N }$ represents the net bias. The subscripts on $A _ { y }$ and $b _ { y }$ serve as a reminder that both depend on the ReLU activation patterns, which in turn depend on the input vector $y$ . + +Based on equation 1 we can perform a first-order decomposition of the error or residual of the neural network for a specific input: $\dot { y } - f ( y ) = ( I - A _ { y } ) y - b _ { y }$ . Figure 1 shows the magnitude of the residual and the constant, which is equal to the net bias $b _ { y }$ , for a range of noise levels. Over the training range, the net bias is small, implying that the linear term is responsible for most of the denoising (see Figures 9 and 10 for a visualization of both components). However, when the network is evaluated at noise levels outside of the training range, the norm of the bias increases dramatically, and the residual is significantly smaller than the noise, suggesting a form of overfitting. Indeed, network performance generalizes very poorly to noise levels outside the training range. This is illustrated for an example image in Figure 2, and demonstrated through extensive experiments in Section 5. + +![](images/aa7b1aeaed1b81a05c927bc5cac02088ee2ed9668f33db4475822b1668937454.jpg) +Figure 3: Comparison of the performance of a CNN and a BF-CNN with the same architecture for the experimental design described in Section 5. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR. Both networks are trained over a fixed ranges of noise levels indicated by a blue background. In all cases, the performance of BF-CNN generalizes robustly beyond the training range, while that of the CNN degrades significantly. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figures 11 and 12). + +# 4 PROPOSED METHODOLOGY: BIAS-FREE NETWORKS + +Section 3 shows that CNNs overfit to the noise levels present in the training set, and that this is associated with wild fluctuations of the net bias $b _ { y }$ . This suggests that the overfitting might be ameliorated by removing additive (bias) terms from every stage of the network, resulting in a biasfree CNN (BF-CNN). Note that bias terms are also removed from the batch-normalization used during training. This simple change in the architecture has an interesting consequence. If the CNN has ReLU activations the denoising map is locally homogeneous, and consequently invariant to scaling: rescaling the input by a constant value simply rescales the output by the same amount, just as it would for a linear system. + +Lemma 1. Let $f _ { \mathrm { B F } } : \mathbb { R } ^ { N } \to \mathbb { R } ^ { N }$ be a feedforward neural network with ReLU activation functions and no additive constant terms in any layer. For any input $y \in \mathbb R$ and any nonnegative constant $\alpha$ , + +$$ +f _ { \mathrm { B F } } ( \alpha y ) = \alpha f _ { \mathrm { B F } } ( y ) . +$$ + +Proof. We can write the action of a bias-free neural network with $L$ layers in terms of the weight matrix $W _ { i }$ , $1 \leq i \leq L$ , of each layer and a rectifying operator $\mathcal { R }$ , which sets to zero any negative entries in its input. Multiplying by a nonnegative constant does not change the sign of the entries of a vector, so for any $z$ with the right dimension and any $\alpha > 0$ $\mathcal { R } ( \alpha z ) = \bar { \alpha } \mathcal { R } ( z )$ , which implies + +$$ +f _ { \mathrm { B F } } ( \alpha y ) = W _ { L } \mathcal { R } ( W _ { L - 1 } \cdot \cdot \cdot \mathcal { R } ( W _ { 1 } \alpha y ) ) = \alpha W _ { L } \mathcal { R } ( W _ { L - 1 } \cdot \cdot \cdot \mathcal { R } ( W _ { 1 } y ) ) = \alpha f _ { \mathrm { B F } } ( y ) . +$$ + +Note that networks with nonzero net bias are not scaling invariant because scaling the input may change the activation pattern of the ReLUs. Scaling invariance is intuitively desireable for a denoising method operating on natural images; a rescaled image is still an image. Note that Lemma 1 holds for networks with skip connections where the feature maps are concatenated or added, because both of these operations are linear. + +In the following sections we demonstrate that removing all additive terms in CNN architectures has two important consequences: (1) the networks gain the ability to generalize to noise levels not encountered during training (as illustrated by Figure 2 the improvement is striking), and (2) the denoising mechanism can be analyzed locally via linear-algebraic tools that reveal intriguing ties to more traditional denoising methodology such as nonlinear filtering and sparsity-based techniques. + +![](images/b0a76b776318596f885de52b9a72ab7cecf15a0b17c307724c032d5509daf174.jpg) +Figure 4: Visualization of the linear weighting functions (rows of $A _ { y }$ in equation 4) of a BF-CNN for three example pixels of an input image, and three levels of noise. The images in the three rightmost columns show the weighting functions used to compute each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (note that some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. As the noise level $\sigma$ increases, the spatial extent of the weight functions increases in order to average out the noise, while respecting boundaries between different regions in the image, which results in dramatically different functions for each pixel. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figure 13). + +# 5 BIAS-FREE NETWORKS GENERALIZE ACROSS NOISE LEVELS + +In order to evaluate the effect of removing the net bias in denoising CNNs, we compare several state-ofthe-art architectures to their bias-free counterparts, which are exactly the same except for the absence of any additive constants within the networks (note that this includes the batch-normalization additive parameter). These architectures include popular features of existing neural-network techniques in image processing: recurrence, multiscale filters, and skip connections. More specifically, we examine the following models (see Section A for additional details): + +• DnCNN (Zhang et al., 2017): A feedforward CNN with 20 convolutional layers, each consisting of $3 \times 3$ filters, 64 channels, batch normalization (Ioffe & Szegedy, 2015), a ReLU nonlinearity, and a skip connection from the initial layer to the final layer. Recurrent CNN: A recurrent architecture inspired by Zhang et al. (2018a) where the basic module is a CNN with 5 layers, $3 \times 3$ filters and 64 channels in the intermediate layers. The order of the recurrence is 4. +• UNet (Ronneberger et al., 2015): A multiscale architecture with 9 convolutional layers and skip connections between the different scales. +• Simplified DenseNet: CNN with skip connections inspired by the DenseNet architecture (Huang et al., 2017; Zhang et al., 2018b). + +We train each network to denoise images corrupted by i.i.d. Gaussian noise over a range of standard deviations (the training range of the network). We then evaluate the network for noise levels that are both within and beyond the training range. Our experiments are carried out on $1 8 0 \times 1 8 0$ natural images from the Berkeley Segmentation Dataset (Martin et al., 2001) to be consistent with previous results (Schmidt & Roth, 2014; Chen & Pock, 2017; Zhang et al., 2017). Additional details about the dataset and training procedure are provided in Section B. + +![](images/3e83acb0b73db0ce69e3ecd87f63c557fe9868a70c632e4a15cf3d159d405582.jpg) +Figure 5: Analysis of the SVD of the Jacobian of a BF-CNN for ten natural images, corrupted by noise of standard deviation $\sigma = 5 0$ . (a) Singular value distributions. For all images, a large proportion of the values are near zero, indicating (approximately) a projection onto a subspace (the signal subspace). (b) Histogram of dot products (cosine of angle) between the left and right singular vectors that lie within the signal subspaces. (c) Effective dimensionality of the signal subspaces (computed as sum of squared singular values) as a function of noise level. For comparison, the total dimensionality of the space is 1600 ( $4 0 \times 4 0$ pixels). Average dimensionality (red curve) falls approximately as the inverse of $\sigma$ (dashed curve). The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figure 17). + +Figures 3, 11 and 12 show our results. For a wide range of different training ranges, and for all architectures, we observe the same phenomenon: the performance of CNNs is good over the training range, but degrades dramatically at new noise levels; in stark contrast, the corresponding BF-CNNs provide strong denoising performance over noise levels outside the training range. This holds for both PSNR and the more perceptually-meaningful Structural Similarity Index (Wang et al., 2004) (see Figure 12). Figure 2 shows an example image, demonstrating visually the striking difference in generalization performance between a CNN and its corresponding BF-CNN. Our results provide strong evidence that removing net bias in CNN architectures results in effective generalization to noise levels out of the training range. + +# 6 REVEALING THE DENOISING MECHANISMS LEARNED BY BF-CNNS + +In this section we perform a local analysis of BF-CNN networks, which reveals the underlying denoising mechanisms learned from the data. A bias-free network is strictly linear, and its net action can be expressed as + +$$ +f _ { \mathrm { B F } } ( y ) = W _ { L } R ( W _ { L - 1 } . . . R ( W _ { 1 } y ) ) = A _ { y } y , +$$ + +where $A _ { y }$ is the Jacobian of $f _ { \mathrm { B F } } ( \cdot )$ evaluated at $y$ . The Jacobian at a fixed input provides a local characterization of the denoising map. In order to study the map we perform a linear-algebraic analysis of the Jacobian. Our approach is similar in spirit to visualization approaches– proposed in the context of image classification– that differentiate neural-network functions with respect to their input (e.g. Simonyan et al. (2013); Montavon et al. (2017)). + +# 6.1 NONLINEAR ADAPTIVE FILTERING + +The linear representation of the denoising map given by equation 4 implies that the ith pixel of the output image is computed as an inner product between the $i$ th row of $A _ { y }$ , denoted $a _ { y } ( i )$ , and the input image: + +$$ +f _ { \mathrm { B F } } ( y ) ( i ) = \sum _ { j = 1 } ^ { N } A _ { y } ( i , j ) y ( j ) = a _ { y } ( i ) ^ { T } y . +$$ + +![](images/16b082547c2cb6eb78a62cd5e5653354fe3fe299de582ebb63c0f2eb9d9af4c3.jpg) +Figure 6: Visualization of left singular vectors of the Jacobian of a BF-CNN, evaluated on two different images (top and bottom rows), corrupted by noise with standard deviation $\sigma = 5 0$ . The left column shows original (clean) images. The next three columns show singular vectors corresponding to non-negligible singular values. The vectors capture features from the clean image. The last three columns on the right show singular vectors corresponding to singular values that are almost equal to zero. These vectors are noisy and unstructured. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figure 16). + +The vectors $a _ { y } ( i )$ can be interpreted as adaptive filters that produce an estimate of the denoised pixel via a weighted average of noisy pixels. Examination of these filters reveals their diversity, and their relationship to the underlying image content: they are adapted to the local features of the noisy image, averaging over homogeneous regions of the image without blurring across edges. This is shown for two separate examples and a range of noise levels in Figures 4, 13, 14 and 15 for the architectures described in Section 5. We observe that the equivalent filters of all architectures adapt to image structure. + +Classical Wiener filtering (Wiener, 1950) denoises images by computing a local average dependent on the noise level. As the noise level increases, the averaging is carried out over a larger region. As illustrated by Figures 4, 13, 14 and 15, the equivalent filters of BF-CNNs also display this behavior. The crucial difference is that the filters are adaptive. The BF-CNNs learn such filters implicitly from the data, in the spirit of modern nonlinear spatially-varying filtering techniques designed to preserve fine-scale details such as edges (e.g. Tomasi & Manduchi (1998), see also Milanfar (2012) for a comprehensive review, and Choi et al. (2018) for a recent learning-based approach). + +# 6.2 PROJECTION ONTO ADAPTIVE LOW-DIMENSIONAL SUBSPACES + +The local linear structure of a BF-CNN facilitates analysis of its functional capabilities via the singular value decomposition (SVD). For a given input $y$ , we compute the SVD of the Jacobian matrix: $\begin{array} { r } { A _ { y } = U S V ^ { T } } \end{array}$ , with $U$ and $V$ orthogonal matrices, and $S$ a diagonal matrix. We can decompose the effect of the network on its input in terms of the left singular vectors $\{ U _ { 1 } , U _ { 2 } \ldots , U _ { N } \}$ (columns of $U _ { . }$ ), the singular values $\lbrace s _ { 1 } , s _ { 2 } \ldots , s _ { N } \rbrace$ (diagonal elements of $S$ ), and the right singular vectors $\{ V _ { 1 } , V _ { 2 } , \ldots V _ { N } \}$ (columns of $V$ ): + +$$ +f _ { \mathrm { B F } } ( y ) = A _ { y } y = U S V ^ { T } y = \sum _ { i = 1 } ^ { N } s _ { i } ( V _ { i } ^ { T } y ) U _ { i } . +$$ + +The output is a linear combination of the left singular vectors, each weighted by the projection of the input onto the corresponding right singular vector, and scaled by the corresponding singular value. + +Analyzing the SVD of a BF-CNN on a set of ten natural images reveals that most singular values are very close to zero (Figure 5a). The network is thus discarding all but a very low-dimensional portion of the input image. We also observe that the left and right singular vectors corresponding to the singular values with non-negligible amplitudes are approximately the same (Figure 5b). This means that the Jacobian is (approximately) symmetric, and we can interpret the action of the network as projecting the noisy signal onto a low-dimensional subspace, as is done in wavelet thresholding schemes. This is confirmed by visualizing the singular vectors as images (Figure 6). The singular vectors corresponding to non-negligible singular values are seen to capture features of the input image; those corresponding to near-zero singular values are unstructured. The BF-CNN therefore implements an approximate projection onto an adaptive signal subspace that preserves image structure, while suppressing the noise. + +![](images/bfd30d8deaf464b4270100538b296bae32b80a0126072796e84c43d2074f0a75.jpg) +Figure 7: Signal subspace properties. Left: Signal subspace, computed from Jacobian of a BF-CNN evaluated at a particular noise level, contains the clean image. Specifically, the fraction of squared $\ell _ { 2 }$ norm preserved by projection onto the subspace is nearly one as $\sigma$ grows from 10 to 100 (relative to the image pixels, which lie in the range [0, 255]). Results are averaged over 50 example clean images. Right: Signal subspaces at different noise levels are nested. The subspace axes for a higher noise level lie largely within the subspace obtained for the lowest noise level $\sigma = 1 0$ ), as measured by the sum of squares of their projected norms. Results are shown for 10 example clean images. + +We can define an "effective dimensionality" of the signal subspace as $d : = \textstyle \sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 }$ , the amount of variance captured by applying the linear map to an $N$ -dimensional Gaussian noise vector with variance $\sigma ^ { 2 }$ , normalized by the noise variance. The remaining variance equals + +$$ +E _ { n } | | A _ { y } n | | ^ { 2 } = E _ { n } | | U _ { y } S _ { y } V _ { y } ^ { T } n | | ^ { 2 } = E _ { n } | | S _ { y } n | | ^ { 2 } = E _ { n } \sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } n _ { i } ^ { 2 } = \sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } E _ { n } ( n _ { i } ^ { 2 } ) \approx \sigma ^ { 2 } \sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } , +$$ + +$E _ { n }$ $n$ $\begin{array} { r } { d = E _ { n } \| \boldsymbol { A } _ { y } n \| ^ { 2 } / \sigma ^ { 2 } = \sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } . } \end{array}$ + +When we examine the preserved signal subspace, we find that the clean image lies almost completely within it. For inputs of the form $y : = x + n$ (where $x$ is the clean image and $n$ the noise), we find that the subspace spanned by the singular vectors up to dimension $d$ contains $x$ almost entirely, in the sense that projecting $x$ onto the subspace preserves most of its energy. This holds for the whole range of noise levels over which the network is trained (Figure 7). + +We also find that for any given clean image, the effective dimensionality of the signal subspace $( d )$ decreases systematically with noise level (Figure 5c). At lower noise levels the network detects a richer set of image features, and constructs a larger signal subspace to capture and preserve them. Empirically, we found that (on average) $d$ is approximately proportional to $\frac { 1 } { \sigma }$ (see dashed line in Figure 5c). These signal subspaces are nested: the subspaces corresponding to lower noise levels contain more than $9 5 \%$ of the subspace axes corresponding to higher noise levels (Figure 7). + +Finally, we note that this behavior of the signal subspace dimensionality, combined with the fact that it contains the clean image, explains the observed denoising performance across different noise levels (Figure 3). Specifically, if we assume $d \approx \alpha / \sigma$ , the mean squared error is proportional to $\sigma$ : + +$$ +\begin{array} { r l } & { \mathrm { M S E } = E _ { n } \vert \vert A _ { y } ( x + n ) - x \vert \vert ^ { 2 } } \\ & { ~ \approx E _ { n } \vert \vert A _ { y } n \vert \vert ^ { 2 } } \\ & { ~ \approx \sigma ^ { 2 } d } \\ & { ~ \approx \alpha \sigma } \end{array} +$$ + +Note that this result runs contrary to the intuitive expectation that MSE should be proportional to the noise variance, which would be the case if the denoiser operated by projecting onto a fixed subspace. The scaling of MSE with the square root of the noise variance implies that the PSNR of the denoised image should be a linear function of the input PSNR, with a slope of $1 / 2$ , consistent with the empirical results shown in Figure 3. Note that this behavior holds even when the networks are trained only on modest levels of noise (e.g., $\sigma \in [ 0 , 1 0 ] \rangle$ . + +# 7 DISCUSSION + +In this work, we show that removing constant terms from CNN architectures ensures strong generalization across noise levels, and also provides interpretability of the denoising method via linear-algebra techniques. We provide insights into the relationship between bias and generalization through a set of observations. Theoretically, we argue that if the denoising network operates by projecting the noisy observation onto a linear space of “clean” images, then that space should include all rescalings of those images, and thus, the origin. This property can be guaranteed by eliminating bias from the network. Empirically, in networks that allow bias, the net bias of the trained network is quite small within the training range. However, outside the training range the net bias grows dramatically resulting in poor performance, which suggests that the bias may be the cause of the failure to generalize. In addition, when we remove bias from the architecture, we preserve performance within the training range, but achieve near-perfect generalization, even to noise levels more than $1 0 \mathrm { x }$ those in the training range. These observations do not fully elucidate how our network achieves its remarkable generalization- only that bias prevents that generalization, and its removal allows it. + +It is of interest to examine whether bias removal can facilitate generalization in noise distributions beyond Gaussian, as well as other image-processing tasks, such as image restoration and image compression. We have trained bias-free networks on uniform noise and found that they generalize outside the training range. In fact, bias-free networks trained for Gaussian noise generalize well when tested on uniform noise (Figures 18 and 19). In addition, we have applied our methodology to image restoration (simultaneous deblurring and denoising). Preliminary results indicate that bias-free networks generalize across noise levels for a fixed blur level, whereas networks with bias do not (Figure 20). An interesting question for future research is whether it is possible to achieve generalization across blur levels. Our initial results indicate that removing bias is not sufficient to achieve this. + +Finally, our linear-algebraic analysis uncovers interesting aspects of the denoising map, but these interpretations are very local: small changes in the input image change the activation patterns of the network, resulting in a change in the corresponding linear mapping. Extending the analysis to reveal global characteristics of the neural-network functionality is a challenging direction for future research. + +# ACKNOWLEDGEMENTS + +This work was partially supported by the Howard Hughes Medical Institute (HHMI). + +# REFERENCES + +S Grace Chang, Bin Yu, and Martin Vetterli. Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Processing, 9(9):1532–1546, 2000. + +Yunjin Chen and Thomas Pock. Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration. IEEE Trans. Patt. Analysis and Machine Intelligence, 39(6): 1256–1272, 2017. + +Sungjoon Choi, John Isidoro, Pascal Getreuer, and Peyman Milanfar. Fast, trainable, multiscale denoising. In 2018 25th IEEE International Conference on Image Processing (ICIP), pp. 963–967. IEEE, 2018. + +Kostadin Dabov, Alessandro Foi, Vladimir Katkovnik, and Karen Egiazarian. Image denoising with block-matching and 3d filtering. 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URL http://arxiv.org/abs/1812.10477. + +# A DESCRIPTION OF DENOISING ARCHITECTURES + +In this section we describe the denoising architectures used for our computational experiments in more detail. + +# A.1 DNCNN + +We implement BF-DnCNN based on the architecture of the Denoising CNN (DnCNN) (Zhang et al., 2017). DnCNN consists of 20 convolutional layers, each consisting of $3 \times 3$ filters and 64 channels, batch normalization (Ioffe & Szegedy, 2015), and a ReLU nonlinearity. It has a skip connection from the initial layer to the final layer, which has no nonlinear units. To construct a bias-free DnCNN (BF-DnCNN) we remove all sources of additive bias, including the mean parameter of the batch-normalization in every layer (note however that the scaling parameter is preserved). + +# A.2 RECURRENT CNN + +Inspired by Zhang et al. (2018a), we consider a recurrent framework that produces a denoised image estimate of the form $\hat { x } _ { t } = f ( \hat { x } _ { t - 1 } , y _ { \mathrm { n o i s y } } )$ , at time $t$ where $f$ is a neural network. We use a 5-layer fully convolutional network with $3 \times 3$ filters in all layers and 64 channels in each intermediate layer to implement $f$ . We initialize the denoised estimate as the noisy image, i.e ${ \hat { x } } _ { 0 } : = y _ { \mathrm { n o i s y } }$ . For the version of the network with net bias, we add trainable additive constants to every filter in all but the last layer. During training, we run the recurrence for a maximum of $T$ times, sampling $T$ uniformly at random from $\{ 1 , 2 , 3 , 4 \}$ for each mini-batch. At test time we fix $T = 4$ . + +# A.3 UNET + +Our UNet model (Ronneberger et al., 2015) has the following layers: + +1. conv1 - Takes in input image and maps to 32 channels with $5 \times 5$ convolutional kernels. +2. conv2 - Input: 32 channels. Output: 32 channels. $3 \times 3$ convolutional kernels. +3. conv3 - Input: 32 channels. Output: 64 channels. $3 \times 3$ convolutional kernels with stride 2. +4. conv4- Input: 64 channels. Output: 64 channels. $3 \times 3$ convolutional kernels. +5. conv5- Input: 64 channels. Output: 64 channels. $3 \times 3$ convolutional kernels with dilation factor of 2. +6. conv6- Input: 64 channels. Output: 64 channels. $3 \times 3$ convolutional kernels with dilation factor of 4. +7. conv7- Transpose Convolution layer. Input: 64 channels. Output: 64 channels. $4 \times 4$ filters with stride 2. +8. conv8- Input: 96 channels. Output: 64 channels. $3 \times 3$ convolutional kernels. The input to this layer is the concatenation of the outputs of layer conv7 and conv2. +9. conv9- Input: 32 channels. Output: 1 channels. $5 \times 5$ convolutional kernels. + +The structure is the same as in Zhang et al. (2018a), but without recurrence. For the version with bias, we add trainable additive constants to all the layers other than conv9. This configuration of UNet assumes even width and height, so we remove one row or column from images in with odd height or width. + +# A.4 SIMPLIFIED DENSENET + +Our simplified version of the DenseNet architecture (Huang et al., 2017) has 4 blocks in total. Each block is a fully convolutional 5-layer CNN with $3 \times 3$ filters and 64 channels in the intermediate layers with ReLU nonlinearity. The first three blocks have an output layer with 64 channels while the last block has an output layer with only one channel. The output of the $i ^ { t h }$ block is concatenated with the input noisy image and then fed to the $( i + 1 ) ^ { t h }$ block, so the last three blocks have 65 input channels. In the version of the network with bias, we add trainable additive parameters to all the layers except for the last layer in the final block. + +# B DATASETS AND TRAINING PROCEDURE + +Our experiments are carried out on $1 8 0 \times 1 8 0$ natural images from the Berkeley Segmentation Dataset (Martin et al., 2001). We use a training set of 400 images. The training set is augmented via downsampling, random flips, and random rotations of patches in these images (Zhang et al., 2017). A test set containing 68 images is used for evaluation. We train the DnCNN and it’s bias free model on patches of size $5 0 \times 5 0$ , which yields a total of 541,600 clean training patches. For the remaining architectures, we use patches of size $1 2 8 \times 1 2 8$ for a total of 22,400 training patches. + +We train DnCNN and its bias-free counterpart using the Adam Optimizer (Kingma & Ba, 2014) over 70 epochs with an initial learning rate of $\mathrm { 1 0 ^ { - 3 } }$ and a decay factor of 0.5 at the $\mathbf { \bar { 5 } 0 } ^ { t h }$ and $6 0 ^ { t h }$ epochs, with no early stopping. We train the other models using the Adam optimizer with an initial learning rate of $1 0 ^ { - 3 }$ and train for 50 epochs with a learning rate schedule which decreases by a factor of 0.25 if the validation PSNR decreases from one epoch to the next. We use early stopping and select the model with the best validation PSNR. + +# C ADDITIONAL RESULTS + +In this section we report additional results of our computational experiments: + +• Figure 8 shows the first-order analysis of the residual of the different architectures described in Section A, except for DnCNN which is shown in Figure 1. +• Figures 9 and 10 visualize the linear and net bias terms in the first-order decomposition of an example image at different noise levels. +• Figure 11 shows the PSNR results for the experiments described in Section 5. +• Figure 12 shows the SSIM results for the experiments described in Section 5. +• Figures 13, 14 and 15 show the equivalent filters at several pixels of two example images for different architectures (see Section 6.1). +• Figure 16 shows the singular vectors of the Jacobian of different BF-CNNs (see Section 6.2). Figure 17 shows the singular values of the Jacobian of different BF-CNNs (see Section 6.2). +Figure 18 and 19 shows that networks trained on noise samples drawn from Gaussian distribution with 0 mean generalizes to noise drawn from uniform distribution with 0 mean during test time. Experiments follow the procedure described in Section 5 except that the networks are evaluated on a different noise distribution during the test time. +Figure 20 shows the application of BF-CNN and CNN to the task of image restoration, where the image is corrupted with both noise and blur at the same time. We show that BF-CNNs can generalize outside the training range for noise levels for a fixed blur level, but do not outperform CNN when generalizing to unseen blur levels. + +![](images/99874d0030f5b131a9708e7019ffd08fffde12f32af9cb61e5a9bf661839aee7.jpg) +Figure 8: First-order analysis of the residual of Recurrent-CNN (Section A.2), UNet (Section A.3) and DenseNet (Section A.4) as a function of noise level. The plots show the magnitudes of the residual and the net bias averaged over 68 images in Set68 test set of Berkeley Segmentation Dataset (Martin et al., 2001) for networks trained over different training ranges. The range of noises used for training is highlighted in gray. (left) When the network is trained over the full range of noise levels $\mathit { \sigma } _ { \mathcal { \sigma } } \in [ 0 , 1 0 0 ] $ ) the net bias is small, growing slightly as the noise increases. (middle and right) When the network is trained over the a smaller range $( \sigma \in [ 0 , 5 5 ]$ and $\sigma \in [ 0 , 3 0 ] )$ , the net bias grows explosively for noise levels outside the training range. This coincides with the dramatic drop in performance due to overfitting, reflected in the difference between the residual and the true noise. + +![](images/e5fc7c006db5b0e106b6a9a140bd37fa5e52057dd779277911dd6a96245eb993.jpg) +Figure 9: Visualization of the decomposition of output of DnCNN trained for noise range [0, 55] into linear part and net bias. The noise level $\sigma = 7 0$ (highlighted by $^ *$ ) is outside the training range. Over the training range, the net bias is small, and the linear part is responsible for most of the denoising effort. However, when the network is evaluated out of the training range, the contribution of the bias increases dramatically, which coincides with a significant drop in denoising performance. + +![](images/f820b3943690865fa7a59dcb037d199c6ab17fcd38920470b2c2338db528af7c.jpg) +Figure 10: Visualization of the decomposition of output of Recurrent-CNN (Section A.2, UNet (Section A.3) and DenseNet (Section A.4) trained for noise range [0, 55] into linear part and net bias. The noise level $\sigma = 9 0$ (highlighted by $^ *$ ) is outside the training range. Over the training range, the net bias is small, and the linear part is responsible for most of the denoising effort. However, when the network is evaluated out of the training range, the contribution of the bias increases dramatically, which coincides with a significant drop in denoising performance. + +![](images/89814a7a607446d1ce837255a96e5cd0f61ec833e6ce82fd547639b8347f107f.jpg) +Figure 11: Comparisons of architectures with (red curves) and without (blue curves) a net bias for the experimental design described in Section 5. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels. (a) Deep Convolutional Neural Network, DnCNN (Zhang et al., 2017). (b) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (c) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (d) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). + +![](images/22a873533c511400a20f0be326a3c082c3f6113c21e8128eaf8fa4792abf9477.jpg) +Figure 12: Comparisons of architectures with (red curves) and without (blue curves) a net bias for the experimental design described in Section 5. The performance is quantified by the SSIM of the denoised image as a function of the input SSIM of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels. (a) Deep Convolutional Neural Network, DnCNN (Zhang et al., 2017). (b) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (c) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (d) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). + +![](images/0bc851c120b9b9c6a975d32b70f50b3dfa9c9a1e6b3ccdaade89f7717b0f53e4.jpg) +Figure 13: Visualization of the linear weighting functions (rows of $A _ { y }$ ) of Bias-Free Recurrent-CNN (top 2 rows) (Section A.2), Bias-Free UNet (next 2 rows) (Section A.3) and Bias-Free DenseNet (bottom 2 rows) (Section A.4) for three example pixels of a noisy input image (left). The next image is the denoised output. The three images on the right show the linear weighting functions corresponding to each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (although some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. Each row corresponds to a noisy input with increasing $\sigma$ and the filters adapt by averaging over a larger region. + +![](images/94994115094756ebf4076ef017149f91680027ba3e68ecbe2ffebf6fca71a4a1.jpg) +Figure 14: Visualization of the linear weighting functions (rows of $A _ { y . }$ ) of a BF-DnCNN for three example pixels of a noisy input image (left). The next image is the denoised output. The three images on the right show the linear weighting functions corresponding to each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (although some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. Each row corresponds to a noisy input with increasing $\sigma$ and the filters adapt by averaging over a larger region. + +![](images/d3aa2b6f29e268f36288778aef4914b553be2fe8dd24c956bb9845a2e5292297.jpg) +Figure 15: Visualization of the linear weighting functions (rows of $A _ { y }$ ) of Bias-Free Recurrent-CNN (top 2 rows) (Section A.2), Bias-Free UNet (next 2 rows) (Section A.3) and Bias-Free DenseNet (bottom 2 rows) (Section A.4) for three example pixels of a noisy input image (left). The next image is the denoised output. The three images on the right show the linear weighting functions corresponding to each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (although some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. Each row corresponds to a noisy input with increasing $\sigma$ and the filters adapt by averaging over a larger region. + +![](images/44fb6f8d36b9f7ad7e6e80a4141748b5ec9bb00d0167b3f68a530ef80daffe9a.jpg) +Figure 16: Visualization of left singular vectors of the Jacobian of a BF Recurrent CNN (top 2 rows), BF UNet (next 2 rows) and BF DenseNet (bottom 2 rows) evaluated on three different images, corrupted by noise with standard deviation $\sigma = 2 5$ . The left column shows original (clean) images. The next three columns show singular vectors corresponding to non-negligible singular values. The vectors capture features from the clean image. The last three columns on the right show singular vectors corresponding to singular values that are almost equal to zero. These vectors are noisy and unstructured. + +![](images/460223787582bc2925179c1185aac04d273d98e03459b69d964ece8ece43bc00.jpg) +Figure 17: Analysis of the SVD of the Jacobian of BF-CNN for ten natural images, corrupted by noise of standard deviation $\sigma = 5 0$ . For all images, a large proportion of the singular values are near zero, indicating (approximately) a projection onto a subspace (the signal subspace). (a) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (b) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (c) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). + +![](images/e0c2db51a56a6c78406c9cd03e489455d59948648fffe9de854084945110df5a.jpg) +Figure 18: Comparison of the performance of a CNN and a BF-CNN with the same architecture for the experimental design described in Section 5. The networks are trained using i.i.d. Gaussian noise but evaluated on noise drawn i.i.d. from a uniform distribution with mean 0. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels, i.e. they are able to generalize across the two different noise distributions. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figures 19). + +![](images/4e3fbf39bc440f03916b2ef2f01bcd4439e441c5544651506c506e585b668fd8.jpg) +Figure 19: Comparisons of architectures with (red curves) and without (blue curves) a net bias for the experimental design described in Section 5. The networks are trained using i.i.d. Gaussian noise but evaluated on noise drawn i.i.d. from a uniform distribution with mean 0. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels, i.e. they are able to generalize across the two different noise distributions. (a) Deep Convolutional Neural Network, DnCNN (Zhang et al., 2017). (b) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (c) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (d) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). + +![](images/a79f7d18cb7a353ab54a56266e0ade542a4c3778358b1ec216d285a54d9ce429.jpg) +Figure 20: Comparison of the performance of DnCNN and a corresponding BF-CNN for image restoration. Training is carried out on data corrupted with Gaussian noise $\sigma _ { \mathrm { n o i s e } } \in [ 0 , 5 5 ]$ and Gaussian blur $\sigma _ { \mathrm { b l u r } } \in [ 0 , 4 ]$ . Performance is measured on test data for inside and outside the training ranges. Left: The difference in performance measured in $\Delta \mathrm { P S N R } = \mathrm { P S N R } _ { \mathrm { B F - C N N } } - \mathrm { P S N R } _ { \mathrm { D n C N N } }$ . The training region is illustrated by the rectangular boundary. Bias-free network generalizes across noise levels for each fixed blur levels, whereas DnCNN does not. However, BF-CNN does not generalize across blur levels. Right: A horizontal slice of the left plot for a fixed blur level of $\sigma _ { \mathrm { { b l u r } } } = 2 $ . BF-CNN generalizes robustly beyond the training range, while the performance of DnCNN degrades significantly. \ No newline at end of file diff --git a/parse/train/HJlSmC4FPS/HJlSmC4FPS_content_list.json b/parse/train/HJlSmC4FPS/HJlSmC4FPS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..61646894453fcf82eabbb3f88b398fc9177a21d0 --- /dev/null +++ b/parse/train/HJlSmC4FPS/HJlSmC4FPS_content_list.json @@ -0,0 +1,1560 @@ +[ + { + "type": "text", + "text": "ROBUST AND INTERPRETABLE BLIND IMAGE DENOISING VIA BIAS-FREE CONVOLUTIONAL NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 101, + 714, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sreyas Mohan∗ Center for Data Science New York University sm7582@nyu.edu ", + "bbox": [ + 184, + 195, + 343, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zahra Kadkhodaie∗ Center for Data Science New York University zk388@nyu.edu ", + "bbox": [ + 486, + 195, + 645, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Carlos Fernandez-Granda ", + "text_level": 1, + "bbox": [ + 491, + 273, + 676, + 286 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Eero P. Simoncelli \nCenter for Neural Science, and \nHoward Hughes Medical Institute \nNew York University \neero.simoncelli@nyu.edu ", + "bbox": [ + 183, + 272, + 410, + 342 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Center for Data Science, and Courant Inst. of Mathematical Sciences New York University cfgranda@cims.nyu.edu ", + "bbox": [ + 490, + 287, + 751, + 342 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 378, + 544, + 393 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We study the generalization properties of deep convolutional neural networks for image denoising in the presence of varying noise levels. We provide extensive empirical evidence that current state-of-the-art architectures systematically overfit to the noise levels in the training set, performing very poorly at new noise levels. We show that strong generalization can be achieved through a simple architectural modification: removing all additive constants. The resulting \"bias-free\" networks attain state-of-the-art performance over a broad range of noise levels, even when trained over a narrow range. They are also locally linear, which enables direct analysis with linear-algebraic tools. We show that the denoising map can be visualized locally as a filter that adapts to both image structure and noise level. In addition, our analysis reveals that deep networks implicitly perform a projection onto an adaptively-selected low-dimensional subspace, with dimensionality inversely proportional to noise level, that captures features of natural images. ", + "bbox": [ + 233, + 409, + 766, + 589 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION AND CONTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 614, + 524, + 630 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The problem of denoising consists of recovering a signal from measurements corrupted by noise, and is a canonical application of statistical estimation that has been studied since the 1950’s. Achieving high-quality denoising results requires (at least implicitly) quantifying and exploiting the differences between signals and noise. In the case of photographic images, the denoising problem is both an important application, as well as a useful test-bed for our understanding of natural images. In the past decade, convolutional neural networks (LeCun et al., 2015) have achieved state-of-the-art results in image denoising (Zhang et al., 2017; Chen & Pock, 2017). Despite their success, these solutions are mysterious: we lack both intuition and formal understanding of the mechanisms they implement. Network architecture and functional units are often borrowed from the image-recognition literature, and it is unclear which of these aspects contributes to, or limits, the denoising performance. The goal of this work is advance our understanding of deep-learning models for denoising. Our contributions are twofold: First, we study the generalization capabilities of deep-learning models across different noise levels. Second, we provide novel tools for analyzing the mechanisms implemented by neural networks to denoise natural images. ", + "bbox": [ + 174, + 645, + 825, + 839 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "An important advantage of deep-learning techniques over traditional methodology is that a single neural network can be trained to perform denoising at a wide range of noise levels. Currently, this is achieved by simulating the whole range of noise levels during training (Zhang et al., 2017). Here, we show that this is not necessary. Neural networks can be made to generalize automatically across noise levels through a simple modification in the architecture: removing all additive constants. We find this holds for a variety of network architectures proposed in previous literature. We provide extensive empirical evidence that the main state-of-the-art denoising architectures systematically overfit to the noise levels in the training set, and that this is due to the presence of a net bias. Suppressing this bias makes it possible to attain state-of-the-art performance while training over a very limited range of noise levels. ", + "bbox": [ + 174, + 847, + 823, + 902 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/fabece6dfe4190ead780e139a603bf1d53d62157f9a737a2b4029c927bb45599.jpg", + "image_caption": [ + "Figure 1: First-order analysis of the residual of a denoising convolutional neural network as a function of noise level. The plots show the norms of the residual and the net bias averaged over $1 0 0 \\ : 2 0 \\times 2 0$ natural-image patches for networks trained over different training ranges. The range of noises used for training is highlighted in blue. (a) When the network is trained over the full range of noise levels $( \\sigma \\in [ 0 , 1 \\bar { 0 } 0 ] )$ the net bias is small, growing slightly as the noise increases. (b-c) When the network is trained over the a smaller range ( ${ \\bf \\sigma } _ { \\sigma } \\in [ 0 , 5 5 ]$ and $\\sigma \\in [ 0 , 3 0 ] )$ , the net bias grows explosively for noise levels beyond the training range. This coincides with a dramatic drop in performance, reflected in the difference between the magnitudes of the residual and the true noise. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results as shown in Figure 8. " + ], + "image_footnote": [], + "bbox": [ + 178, + 103, + 818, + 243 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 422, + 825, + 507 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The data-driven mechanisms implemented by deep neural networks to perform denoising are almost completely unknown. It is unclear what priors are being learned by the models, and how they are affected by the choice of architecture and training strategies. Here, we provide novel linear-algebraic tools to visualize and interpret these strategies through a local analysis of the Jacobian of the denoising map. The analysis reveals locally adaptive properties of the learned models, akin to existing nonlinear filtering algorithms. In addition, we show that the deep networks implicitly perform a projection onto an adaptively-selected low-dimensional subspace capturing features of natural images. ", + "bbox": [ + 174, + 513, + 825, + 611 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 633, + 344, + 650 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The classical solution to the denoising problem is the Wiener filter (Wiener, 1950), which assumes a translation-invariant Gaussian signal model. The main limitation of Wiener filtering is that it over-smoothes, eliminating fine-scale details and textures. Modern filtering approaches address this issue by adapting the filters to the local structure of the noisy image (e.g. Tomasi & Manduchi (1998); Milanfar (2012)). Here we show that neural networks implement such strategies implicitly, learning them directly from the data. ", + "bbox": [ + 174, + 666, + 825, + 750 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the 1990’s powerful denoising techniques were developed based on multi-scale (\"wavelet\") transforms. These transforms map natural images to a domain where they have sparser representations. This makes it possible to perform denoising by applying nonlinear thresholding operations in order to discard components that are small relative to the noise level (Donoho & Johnstone, 1995; Simoncelli & Adelson, 1996; Chang et al., 2000). From a linear-algebraic perspective, these algorithms operate by projecting the noisy input onto a lower-dimensional subspace that contains plausible signal content. The projection eliminates the orthogonal complement of the subspace, which mostly contains noise. This general methodology laid the foundations for the state-of-the-art models in the 2000’s (e.g. (Dabov et al., 2006)), some of which added a data-driven perspective, learning sparsifying transforms (Elad & Aharon, 2006), and nonlinear shrinkage functions (Hel-Or & Shaked, 2008; Raphan & Simoncelli, 2008), directly from natural images. Here, we show that deep-learning models learn similar priors in the form of local linear subspaces capturing image features. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/766b675863ec4d135ea0868be988519d4bfb8aa88e9dd9072ec2d550ff1f8dcb.jpg", + "image_caption": [ + "Figure 2: Denoising of an example natural image by a CNN and its bias-free counterpart (BF-CNN), both trained over noise levels in the range $\\sigma \\in [ 0 , 1 0 ]$ (image intensities are in the range [0, 255]). The CNN performs poorly at high noise levels $\\sigma = 9 0$ , far beyond the training range), whereas BF-CNN performs at state-of-the-art levels. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Section 5). " + ], + "image_footnote": [], + "bbox": [ + 183, + 106, + 815, + 252 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the past decade, purely data-driven models based on convolutional neural networks (LeCun et al., 2015) have come to dominate all previous methods in terms of performance. These models consist of cascades of convolutional filters, and rectifying nonlinearities, which are capable of representing a diverse and powerful set of functions. Training such architectures to minimize mean square error over large databases of noisy natural-image patches achieves current state-of-the-art results (Zhang et al., 2017; Huang et al., 2017; Ronneberger et al., 2015; Zhang et al., 2018a). ", + "bbox": [ + 174, + 357, + 825, + 441 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 NETWORK BIAS IMPAIRS GENERALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 460, + 571, + 477 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We assume a measurement model in which images are corrupted by additive noise: $y = x + n$ , where $\\boldsymbol { x } \\in \\mathbb { R } ^ { N }$ is the original image, containing $N$ pixels, $n$ is an image of i.i.d. samples of Gaussian noise with variance $\\sigma ^ { 2 }$ , and $y$ is the noisy observation. The denoising problem consists of finding a function $f : \\mathbb { R } ^ { N } \\to \\mathbb { R } ^ { N }$ , that provides a good estimate of the original image, $x$ . Commonly, one minimizes the mean squared error : $\\begin{array} { r } { f = \\arg \\operatorname* { m i n } _ { g } E | | x - g ( y ) | | ^ { 2 } } \\end{array}$ , where the expectation is taken over some distribution over images, $x$ , as well as over the distribution of noise realizations. In deep learning, the denoising function $g$ is parameterized by the weights of the network, so the optimization is over these parameters. If the noise standard deviation, $\\sigma$ , is unknown, the expectation must also be taken over a distribution of $\\sigma$ . This problem is often called blind denoising in the literature. In this work, we study the generalization performance of CNNs across noise levels $\\sigma$ , i.e. when they are tested on noise levels not included in the training set. ", + "bbox": [ + 173, + 491, + 825, + 645 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Feedforward neural networks with rectified linear units (ReLUs) are piecewise affine: for a given activation pattern of the ReLUs, the effect of the network on the input is a cascade of linear transformations (convolutional or fully connected layers, $W _ { k }$ ), additive constants $( b _ { k } )$ , and pointwise multiplications by a binary mask corresponding to the fixed activation pattern $( R )$ . Since each of these is affine, the entire cascade implements a single affine transformation. For a fixed noisy input image $\\boldsymbol { y } \\in \\mathbb { R } ^ { N }$ with $N$ pixels, the function $f : \\mathbb { R } ^ { N } \\xrightarrow [ ] { } \\mathbb { R } ^ { N }$ computed by a denoising neural network may be written ", + "bbox": [ + 173, + 651, + 825, + 750 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c5e6148c05a20ec00ce479b53d9d14752e315c1dbd41a5a6cf86212abf8a3839.jpg", + "text": "$$\nf ( y ) = W _ { L } R ( W _ { L - 1 } . . . . R ( W _ { 1 } y + b _ { 1 } ) + . . . b _ { L - 1 } ) + b _ { L } = A _ { y } y + b _ { y } ,\n$$", + "text_format": "latex", + "bbox": [ + 276, + 753, + 720, + 771 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $A _ { y } \\in \\mathbb R ^ { N \\times N }$ is the Jacobian of $f ( \\cdot )$ evaluated at input $y$ , and $b _ { y } \\in \\mathbb { R } ^ { N }$ represents the net bias. The subscripts on $A _ { y }$ and $b _ { y }$ serve as a reminder that both depend on the ReLU activation patterns, which in turn depend on the input vector $y$ . ", + "bbox": [ + 176, + 776, + 823, + 820 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Based on equation 1 we can perform a first-order decomposition of the error or residual of the neural network for a specific input: $\\dot { y } - f ( y ) = ( I - A _ { y } ) y - b _ { y }$ . Figure 1 shows the magnitude of the residual and the constant, which is equal to the net bias $b _ { y }$ , for a range of noise levels. Over the training range, the net bias is small, implying that the linear term is responsible for most of the denoising (see Figures 9 and 10 for a visualization of both components). However, when the network is evaluated at noise levels outside of the training range, the norm of the bias increases dramatically, and the residual is significantly smaller than the noise, suggesting a form of overfitting. Indeed, network performance generalizes very poorly to noise levels outside the training range. This is illustrated for an example image in Figure 2, and demonstrated through extensive experiments in Section 5. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/aa7b1aeaed1b81a05c927bc5cac02088ee2ed9668f33db4475822b1668937454.jpg", + "image_caption": [ + "Figure 3: Comparison of the performance of a CNN and a BF-CNN with the same architecture for the experimental design described in Section 5. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR. Both networks are trained over a fixed ranges of noise levels indicated by a blue background. In all cases, the performance of BF-CNN generalizes robustly beyond the training range, while that of the CNN degrades significantly. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figures 11 and 12). " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 816, + 222 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 363, + 823, + 392 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 PROPOSED METHODOLOGY: BIAS-FREE NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 414, + 642, + 431 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Section 3 shows that CNNs overfit to the noise levels present in the training set, and that this is associated with wild fluctuations of the net bias $b _ { y }$ . This suggests that the overfitting might be ameliorated by removing additive (bias) terms from every stage of the network, resulting in a biasfree CNN (BF-CNN). Note that bias terms are also removed from the batch-normalization used during training. This simple change in the architecture has an interesting consequence. If the CNN has ReLU activations the denoising map is locally homogeneous, and consequently invariant to scaling: rescaling the input by a constant value simply rescales the output by the same amount, just as it would for a linear system. ", + "bbox": [ + 173, + 446, + 826, + 560 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 1. Let $f _ { \\mathrm { B F } } : \\mathbb { R } ^ { N } \\to \\mathbb { R } ^ { N }$ be a feedforward neural network with ReLU activation functions and no additive constant terms in any layer. For any input $y \\in \\mathbb R$ and any nonnegative constant $\\alpha$ , ", + "bbox": [ + 169, + 565, + 823, + 594 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/29bf7bea298b1b22bcf1972a5ebb75c98d3c1cb50bef8daa2095597762126414.jpg", + "text": "$$\nf _ { \\mathrm { B F } } ( \\alpha y ) = \\alpha f _ { \\mathrm { B F } } ( y ) .\n$$", + "text_format": "latex", + "bbox": [ + 426, + 604, + 573, + 622 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proof. We can write the action of a bias-free neural network with $L$ layers in terms of the weight matrix $W _ { i }$ , $1 \\leq i \\leq L$ , of each layer and a rectifying operator $\\mathcal { R }$ , which sets to zero any negative entries in its input. Multiplying by a nonnegative constant does not change the sign of the entries of a vector, so for any $z$ with the right dimension and any $\\alpha > 0$ $\\mathcal { R } ( \\alpha z ) = \\bar { \\alpha } \\mathcal { R } ( z )$ , which implies ", + "bbox": [ + 174, + 645, + 825, + 702 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ab4424fb6f2097baf2bed036f3c261a6c82bec3b14bbb03b50e7a4acf4418ef5.jpg", + "text": "$$\nf _ { \\mathrm { B F } } ( \\alpha y ) = W _ { L } \\mathcal { R } ( W _ { L - 1 } \\cdot \\cdot \\cdot \\mathcal { R } ( W _ { 1 } \\alpha y ) ) = \\alpha W _ { L } \\mathcal { R } ( W _ { L - 1 } \\cdot \\cdot \\cdot \\mathcal { R } ( W _ { 1 } y ) ) = \\alpha f _ { \\mathrm { B F } } ( y ) .\n$$", + "text_format": "latex", + "bbox": [ + 217, + 710, + 781, + 728 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that networks with nonzero net bias are not scaling invariant because scaling the input may change the activation pattern of the ReLUs. Scaling invariance is intuitively desireable for a denoising method operating on natural images; a rescaled image is still an image. Note that Lemma 1 holds for networks with skip connections where the feature maps are concatenated or added, because both of these operations are linear. ", + "bbox": [ + 173, + 776, + 825, + 847 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the following sections we demonstrate that removing all additive terms in CNN architectures has two important consequences: (1) the networks gain the ability to generalize to noise levels not encountered during training (as illustrated by Figure 2 the improvement is striking), and (2) the denoising mechanism can be analyzed locally via linear-algebraic tools that reveal intriguing ties to more traditional denoising methodology such as nonlinear filtering and sparsity-based techniques. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/b0a76b776318596f885de52b9a72ab7cecf15a0b17c307724c032d5509daf174.jpg", + "image_caption": [ + "Figure 4: Visualization of the linear weighting functions (rows of $A _ { y }$ in equation 4) of a BF-CNN for three example pixels of an input image, and three levels of noise. The images in the three rightmost columns show the weighting functions used to compute each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (note that some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. As the noise level $\\sigma$ increases, the spatial extent of the weight functions increases in order to average out the noise, while respecting boundaries between different regions in the image, which results in dramatically different functions for each pixel. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figure 13). " + ], + "image_footnote": [], + "bbox": [ + 186, + 98, + 818, + 395 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 BIAS-FREE NETWORKS GENERALIZE ACROSS NOISE LEVELS ", + "text_level": 1, + "bbox": [ + 176, + 563, + 718, + 579 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to evaluate the effect of removing the net bias in denoising CNNs, we compare several state-ofthe-art architectures to their bias-free counterparts, which are exactly the same except for the absence of any additive constants within the networks (note that this includes the batch-normalization additive parameter). These architectures include popular features of existing neural-network techniques in image processing: recurrence, multiscale filters, and skip connections. More specifically, we examine the following models (see Section A for additional details): ", + "bbox": [ + 174, + 595, + 826, + 679 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• DnCNN (Zhang et al., 2017): A feedforward CNN with 20 convolutional layers, each consisting of $3 \\times 3$ filters, 64 channels, batch normalization (Ioffe & Szegedy, 2015), a ReLU nonlinearity, and a skip connection from the initial layer to the final layer. Recurrent CNN: A recurrent architecture inspired by Zhang et al. (2018a) where the basic module is a CNN with 5 layers, $3 \\times 3$ filters and 64 channels in the intermediate layers. The order of the recurrence is 4. \n• UNet (Ronneberger et al., 2015): A multiscale architecture with 9 convolutional layers and skip connections between the different scales. \n• Simplified DenseNet: CNN with skip connections inspired by the DenseNet architecture (Huang et al., 2017; Zhang et al., 2018b). ", + "bbox": [ + 215, + 693, + 825, + 854 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We train each network to denoise images corrupted by i.i.d. Gaussian noise over a range of standard deviations (the training range of the network). We then evaluate the network for noise levels that are both within and beyond the training range. Our experiments are carried out on $1 8 0 \\times 1 8 0$ natural images from the Berkeley Segmentation Dataset (Martin et al., 2001) to be consistent with previous results (Schmidt & Roth, 2014; Chen & Pock, 2017; Zhang et al., 2017). Additional details about the dataset and training procedure are provided in Section B. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/3e83acb0b73db0ce69e3ecd87f63c557fe9868a70c632e4a15cf3d159d405582.jpg", + "image_caption": [ + "Figure 5: Analysis of the SVD of the Jacobian of a BF-CNN for ten natural images, corrupted by noise of standard deviation $\\sigma = 5 0$ . (a) Singular value distributions. For all images, a large proportion of the values are near zero, indicating (approximately) a projection onto a subspace (the signal subspace). (b) Histogram of dot products (cosine of angle) between the left and right singular vectors that lie within the signal subspaces. (c) Effective dimensionality of the signal subspaces (computed as sum of squared singular values) as a function of noise level. For comparison, the total dimensionality of the space is 1600 ( $4 0 \\times 4 0$ pixels). Average dimensionality (red curve) falls approximately as the inverse of $\\sigma$ (dashed curve). The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figure 17). " + ], + "image_footnote": [], + "bbox": [ + 184, + 102, + 813, + 279 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 440, + 823, + 469 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figures 3, 11 and 12 show our results. For a wide range of different training ranges, and for all architectures, we observe the same phenomenon: the performance of CNNs is good over the training range, but degrades dramatically at new noise levels; in stark contrast, the corresponding BF-CNNs provide strong denoising performance over noise levels outside the training range. This holds for both PSNR and the more perceptually-meaningful Structural Similarity Index (Wang et al., 2004) (see Figure 12). Figure 2 shows an example image, demonstrating visually the striking difference in generalization performance between a CNN and its corresponding BF-CNN. Our results provide strong evidence that removing net bias in CNN architectures results in effective generalization to noise levels out of the training range. ", + "bbox": [ + 173, + 476, + 825, + 603 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 REVEALING THE DENOISING MECHANISMS LEARNED BY BF-CNNS ", + "text_level": 1, + "bbox": [ + 171, + 621, + 776, + 638 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section we perform a local analysis of BF-CNN networks, which reveals the underlying denoising mechanisms learned from the data. A bias-free network is strictly linear, and its net action can be expressed as ", + "bbox": [ + 173, + 651, + 825, + 694 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/209abe3c5217780bc16ff986167540544a58a9145281baf22be9b553a50a6cba.jpg", + "text": "$$\nf _ { \\mathrm { B F } } ( y ) = W _ { L } R ( W _ { L - 1 } . . . R ( W _ { 1 } y ) ) = A _ { y } y ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 699, + 643, + 715 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $A _ { y }$ is the Jacobian of $f _ { \\mathrm { B F } } ( \\cdot )$ evaluated at $y$ . The Jacobian at a fixed input provides a local characterization of the denoising map. In order to study the map we perform a linear-algebraic analysis of the Jacobian. Our approach is similar in spirit to visualization approaches– proposed in the context of image classification– that differentiate neural-network functions with respect to their input (e.g. Simonyan et al. (2013); Montavon et al. (2017)). ", + "bbox": [ + 174, + 718, + 825, + 789 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.1 NONLINEAR ADAPTIVE FILTERING ", + "text_level": 1, + "bbox": [ + 176, + 805, + 450, + 820 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The linear representation of the denoising map given by equation 4 implies that the ith pixel of the output image is computed as an inner product between the $i$ th row of $A _ { y }$ , denoted $a _ { y } ( i )$ , and the input image: ", + "bbox": [ + 174, + 830, + 825, + 873 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/24b3708a65a53c4cc948fcf3256a79fc583a464b26157b94fe6f2c968a2044a6.jpg", + "text": "$$\nf _ { \\mathrm { B F } } ( y ) ( i ) = \\sum _ { j = 1 } ^ { N } A _ { y } ( i , j ) y ( j ) = a _ { y } ( i ) ^ { T } y .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 877, + 637, + 921 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/16b082547c2cb6eb78a62cd5e5653354fe3fe299de582ebb63c0f2eb9d9af4c3.jpg", + "image_caption": [ + "Figure 6: Visualization of left singular vectors of the Jacobian of a BF-CNN, evaluated on two different images (top and bottom rows), corrupted by noise with standard deviation $\\sigma = 5 0$ . The left column shows original (clean) images. The next three columns show singular vectors corresponding to non-negligible singular values. The vectors capture features from the clean image. The last three columns on the right show singular vectors corresponding to singular values that are almost equal to zero. These vectors are noisy and unstructured. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figure 16). " + ], + "image_footnote": [], + "bbox": [ + 176, + 103, + 820, + 237 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The vectors $a _ { y } ( i )$ can be interpreted as adaptive filters that produce an estimate of the denoised pixel via a weighted average of noisy pixels. Examination of these filters reveals their diversity, and their relationship to the underlying image content: they are adapted to the local features of the noisy image, averaging over homogeneous regions of the image without blurring across edges. This is shown for two separate examples and a range of noise levels in Figures 4, 13, 14 and 15 for the architectures described in Section 5. We observe that the equivalent filters of all architectures adapt to image structure. ", + "bbox": [ + 173, + 376, + 825, + 473 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Classical Wiener filtering (Wiener, 1950) denoises images by computing a local average dependent on the noise level. As the noise level increases, the averaging is carried out over a larger region. As illustrated by Figures 4, 13, 14 and 15, the equivalent filters of BF-CNNs also display this behavior. The crucial difference is that the filters are adaptive. The BF-CNNs learn such filters implicitly from the data, in the spirit of modern nonlinear spatially-varying filtering techniques designed to preserve fine-scale details such as edges (e.g. Tomasi & Manduchi (1998), see also Milanfar (2012) for a comprehensive review, and Choi et al. (2018) for a recent learning-based approach). ", + "bbox": [ + 173, + 479, + 825, + 578 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.2 PROJECTION ONTO ADAPTIVE LOW-DIMENSIONAL SUBSPACES ", + "text_level": 1, + "bbox": [ + 176, + 597, + 643, + 611 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The local linear structure of a BF-CNN facilitates analysis of its functional capabilities via the singular value decomposition (SVD). For a given input $y$ , we compute the SVD of the Jacobian matrix: $\\begin{array} { r } { A _ { y } = U S V ^ { T } } \\end{array}$ , with $U$ and $V$ orthogonal matrices, and $S$ a diagonal matrix. We can decompose the effect of the network on its input in terms of the left singular vectors $\\{ U _ { 1 } , U _ { 2 } \\ldots , U _ { N } \\}$ (columns of $U _ { . }$ ), the singular values $\\lbrace s _ { 1 } , s _ { 2 } \\ldots , s _ { N } \\rbrace$ (diagonal elements of $S$ ), and the right singular vectors $\\{ V _ { 1 } , V _ { 2 } , \\ldots V _ { N } \\}$ (columns of $V$ ): ", + "bbox": [ + 173, + 621, + 825, + 707 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/22943eddb0a30dbc8116a9cd3581718ad87d4ac41b3583de84807fec9e4d450e.jpg", + "text": "$$\nf _ { \\mathrm { B F } } ( y ) = A _ { y } y = U S V ^ { T } y = \\sum _ { i = 1 } ^ { N } s _ { i } ( V _ { i } ^ { T } y ) U _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 713, + 650, + 757 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The output is a linear combination of the left singular vectors, each weighted by the projection of the input onto the corresponding right singular vector, and scaled by the corresponding singular value. ", + "bbox": [ + 173, + 762, + 825, + 792 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Analyzing the SVD of a BF-CNN on a set of ten natural images reveals that most singular values are very close to zero (Figure 5a). The network is thus discarding all but a very low-dimensional portion of the input image. We also observe that the left and right singular vectors corresponding to the singular values with non-negligible amplitudes are approximately the same (Figure 5b). This means that the Jacobian is (approximately) symmetric, and we can interpret the action of the network as projecting the noisy signal onto a low-dimensional subspace, as is done in wavelet thresholding schemes. This is confirmed by visualizing the singular vectors as images (Figure 6). The singular vectors corresponding to non-negligible singular values are seen to capture features of the input image; those corresponding to near-zero singular values are unstructured. The BF-CNN therefore implements an approximate projection onto an adaptive signal subspace that preserves image structure, while suppressing the noise. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/bfd30d8deaf464b4270100538b296bae32b80a0126072796e84c43d2074f0a75.jpg", + "image_caption": [ + "Figure 7: Signal subspace properties. Left: Signal subspace, computed from Jacobian of a BF-CNN evaluated at a particular noise level, contains the clean image. Specifically, the fraction of squared $\\ell _ { 2 }$ norm preserved by projection onto the subspace is nearly one as $\\sigma$ grows from 10 to 100 (relative to the image pixels, which lie in the range [0, 255]). Results are averaged over 50 example clean images. Right: Signal subspaces at different noise levels are nested. The subspace axes for a higher noise level lie largely within the subspace obtained for the lowest noise level $\\sigma = 1 0$ ), as measured by the sum of squares of their projected norms. Results are shown for 10 example clean images. " + ], + "image_footnote": [], + "bbox": [ + 254, + 102, + 743, + 244 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 381, + 825, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We can define an \"effective dimensionality\" of the signal subspace as $d : = \\textstyle \\sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 }$ , the amount of variance captured by applying the linear map to an $N$ -dimensional Gaussian noise vector with variance $\\sigma ^ { 2 }$ , normalized by the noise variance. The remaining variance equals ", + "bbox": [ + 174, + 417, + 823, + 462 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/ba395fb76f290ea6a010153ba116c63e975fd4b4421f3ae7f6da05c8adbf4113.jpg", + "text": "$$\nE _ { n } | | A _ { y } n | | ^ { 2 } = E _ { n } | | U _ { y } S _ { y } V _ { y } ^ { T } n | | ^ { 2 } = E _ { n } | | S _ { y } n | | ^ { 2 } = E _ { n } \\sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } n _ { i } ^ { 2 } = \\sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } E _ { n } ( n _ { i } ^ { 2 } ) \\approx \\sigma ^ { 2 } \\sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 187, + 467, + 805, + 511 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$E _ { n }$ $n$ $\\begin{array} { r } { d = E _ { n } \\| \\boldsymbol { A } _ { y } n \\| ^ { 2 } / \\sigma ^ { 2 } = \\sum _ { i = 1 } ^ { N } s _ { i } ^ { 2 } . } \\end{array}$ ", + "bbox": [ + 176, + 518, + 735, + 536 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "When we examine the preserved signal subspace, we find that the clean image lies almost completely within it. For inputs of the form $y : = x + n$ (where $x$ is the clean image and $n$ the noise), we find that the subspace spanned by the singular vectors up to dimension $d$ contains $x$ almost entirely, in the sense that projecting $x$ onto the subspace preserves most of its energy. This holds for the whole range of noise levels over which the network is trained (Figure 7). ", + "bbox": [ + 173, + 541, + 825, + 612 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also find that for any given clean image, the effective dimensionality of the signal subspace $( d )$ decreases systematically with noise level (Figure 5c). At lower noise levels the network detects a richer set of image features, and constructs a larger signal subspace to capture and preserve them. Empirically, we found that (on average) $d$ is approximately proportional to $\\frac { 1 } { \\sigma }$ (see dashed line in Figure 5c). These signal subspaces are nested: the subspaces corresponding to lower noise levels contain more than $9 5 \\%$ of the subspace axes corresponding to higher noise levels (Figure 7). ", + "bbox": [ + 173, + 617, + 826, + 702 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, we note that this behavior of the signal subspace dimensionality, combined with the fact that it contains the clean image, explains the observed denoising performance across different noise levels (Figure 3). Specifically, if we assume $d \\approx \\alpha / \\sigma$ , the mean squared error is proportional to $\\sigma$ : ", + "bbox": [ + 174, + 708, + 825, + 751 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/2c507da274c32e04771c697e3c83ea1c6ac9df8beb6eca8e071f0c76a0f6de33.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { M S E } = E _ { n } \\vert \\vert A _ { y } ( x + n ) - x \\vert \\vert ^ { 2 } } \\\\ & { ~ \\approx E _ { n } \\vert \\vert A _ { y } n \\vert \\vert ^ { 2 } } \\\\ & { ~ \\approx \\sigma ^ { 2 } d } \\\\ & { ~ \\approx \\alpha \\sigma } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 393, + 757, + 602, + 834 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Note that this result runs contrary to the intuitive expectation that MSE should be proportional to the noise variance, which would be the case if the denoiser operated by projecting onto a fixed subspace. The scaling of MSE with the square root of the noise variance implies that the PSNR of the denoised image should be a linear function of the input PSNR, with a slope of $1 / 2$ , consistent with the empirical results shown in Figure 3. Note that this behavior holds even when the networks are trained only on modest levels of noise (e.g., $\\sigma \\in [ 0 , 1 0 ] \\rangle$ . ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 310, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we show that removing constant terms from CNN architectures ensures strong generalization across noise levels, and also provides interpretability of the denoising method via linear-algebra techniques. We provide insights into the relationship between bias and generalization through a set of observations. Theoretically, we argue that if the denoising network operates by projecting the noisy observation onto a linear space of “clean” images, then that space should include all rescalings of those images, and thus, the origin. This property can be guaranteed by eliminating bias from the network. Empirically, in networks that allow bias, the net bias of the trained network is quite small within the training range. However, outside the training range the net bias grows dramatically resulting in poor performance, which suggests that the bias may be the cause of the failure to generalize. In addition, when we remove bias from the architecture, we preserve performance within the training range, but achieve near-perfect generalization, even to noise levels more than $1 0 \\mathrm { x }$ those in the training range. These observations do not fully elucidate how our network achieves its remarkable generalization- only that bias prevents that generalization, and its removal allows it. ", + "bbox": [ + 174, + 133, + 825, + 314 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "It is of interest to examine whether bias removal can facilitate generalization in noise distributions beyond Gaussian, as well as other image-processing tasks, such as image restoration and image compression. We have trained bias-free networks on uniform noise and found that they generalize outside the training range. In fact, bias-free networks trained for Gaussian noise generalize well when tested on uniform noise (Figures 18 and 19). In addition, we have applied our methodology to image restoration (simultaneous deblurring and denoising). Preliminary results indicate that bias-free networks generalize across noise levels for a fixed blur level, whereas networks with bias do not (Figure 20). An interesting question for future research is whether it is possible to achieve generalization across blur levels. Our initial results indicate that removing bias is not sufficient to achieve this. ", + "bbox": [ + 174, + 321, + 825, + 459 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, our linear-algebraic analysis uncovers interesting aspects of the denoising map, but these interpretations are very local: small changes in the input image change the activation patterns of the network, resulting in a change in the corresponding linear mapping. Extending the analysis to reveal global characteristics of the neural-network functionality is a challenging direction for future research. ", + "bbox": [ + 174, + 467, + 825, + 536 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 559, + 367, + 573 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work was partially supported by the Howard Hughes Medical Institute (HHMI). 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", + "bbox": [ + 176, + 635, + 825, + 665 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Carlo Tomasi and Roberto Manduchi. Bilateral filtering for gray and color images. In ICCV, volume 98, 1998. ", + "bbox": [ + 173, + 675, + 825, + 704 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Zhou Wang, Alan C Bovik, Hamid R Sheikh, Eero P Simoncelli, et al. Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Processing, 13(4):600–612, 2004. ", + "bbox": [ + 171, + 714, + 825, + 746 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Norbert Wiener. Extrapolation, interpolation, and smoothing of stationary time series: with engineering applications. Technology Press, 1950. ", + "bbox": [ + 171, + 755, + 825, + 785 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Kai Zhang, Wangmeng Zuo, Yunjin Chen, Deyu Meng, and Lei Zhang. Beyond a gaussian denoiser: Residual learning of deep CNN for image denoising. IEEE Trans. Image Processing, 26(7): 3142–3155, 2017. ", + "bbox": [ + 173, + 794, + 826, + 838 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Xiaoshuai Zhang, Yiping Lu, Jiaying Liu, and Bin Dong. Dynamically unfolding recurrent restorer: A moving endpoint control method for image restoration. arXiv preprint arXiv:1805.07709, 2018a. ", + "bbox": [ + 174, + 848, + 825, + 877 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Yulun Zhang, Yapeng Tian, Yu Kong, Bineng Zhong, and Yun Fu. Residual dense network for image restoration. CoRR, abs/1812.10477, 2018b. URL http://arxiv.org/abs/1812.10477. ", + "bbox": [ + 174, + 887, + 825, + 917 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A DESCRIPTION OF DENOISING ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 596, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In this section we describe the denoising architectures used for our computational experiments in more detail. ", + "bbox": [ + 174, + 133, + 823, + 161 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.1 DNCNN ", + "text_level": 1, + "bbox": [ + 176, + 178, + 279, + 193 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We implement BF-DnCNN based on the architecture of the Denoising CNN (DnCNN) (Zhang et al., 2017). DnCNN consists of 20 convolutional layers, each consisting of $3 \\times 3$ filters and 64 channels, batch normalization (Ioffe & Szegedy, 2015), and a ReLU nonlinearity. It has a skip connection from the initial layer to the final layer, which has no nonlinear units. To construct a bias-free DnCNN (BF-DnCNN) we remove all sources of additive bias, including the mean parameter of the batch-normalization in every layer (note however that the scaling parameter is preserved). ", + "bbox": [ + 174, + 204, + 825, + 287 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.2 RECURRENT CNN", + "text_level": 1, + "bbox": [ + 176, + 304, + 346, + 319 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Inspired by Zhang et al. (2018a), we consider a recurrent framework that produces a denoised image estimate of the form $\\hat { x } _ { t } = f ( \\hat { x } _ { t - 1 } , y _ { \\mathrm { n o i s y } } )$ , at time $t$ where $f$ is a neural network. We use a 5-layer fully convolutional network with $3 \\times 3$ filters in all layers and 64 channels in each intermediate layer to implement $f$ . We initialize the denoised estimate as the noisy image, i.e ${ \\hat { x } } _ { 0 } : = y _ { \\mathrm { n o i s y } }$ . For the version of the network with net bias, we add trainable additive constants to every filter in all but the last layer. During training, we run the recurrence for a maximum of $T$ times, sampling $T$ uniformly at random from $\\{ 1 , 2 , 3 , 4 \\}$ for each mini-batch. At test time we fix $T = 4$ . ", + "bbox": [ + 174, + 330, + 825, + 429 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.3 UNET ", + "text_level": 1, + "bbox": [ + 174, + 445, + 261, + 459 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Our UNet model (Ronneberger et al., 2015) has the following layers: ", + "bbox": [ + 176, + 470, + 624, + 486 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "1. conv1 - Takes in input image and maps to 32 channels with $5 \\times 5$ convolutional kernels. \n2. conv2 - Input: 32 channels. Output: 32 channels. $3 \\times 3$ convolutional kernels. \n3. conv3 - Input: 32 channels. Output: 64 channels. $3 \\times 3$ convolutional kernels with stride 2. \n4. conv4- Input: 64 channels. Output: 64 channels. $3 \\times 3$ convolutional kernels. \n5. conv5- Input: 64 channels. Output: 64 channels. $3 \\times 3$ convolutional kernels with dilation factor of 2. \n6. conv6- Input: 64 channels. Output: 64 channels. $3 \\times 3$ convolutional kernels with dilation factor of 4. \n7. conv7- Transpose Convolution layer. Input: 64 channels. Output: 64 channels. $4 \\times 4$ filters with stride 2. \n8. conv8- Input: 96 channels. Output: 64 channels. $3 \\times 3$ convolutional kernels. The input to this layer is the concatenation of the outputs of layer conv7 and conv2. \n9. conv9- Input: 32 channels. Output: 1 channels. $5 \\times 5$ convolutional kernels. ", + "bbox": [ + 210, + 496, + 826, + 717 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The structure is the same as in Zhang et al. (2018a), but without recurrence. For the version with bias, we add trainable additive constants to all the layers other than conv9. This configuration of UNet assumes even width and height, so we remove one row or column from images in with odd height or width. ", + "bbox": [ + 174, + 727, + 825, + 782 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.4 SIMPLIFIED DENSENET ", + "text_level": 1, + "bbox": [ + 176, + 800, + 382, + 814 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Our simplified version of the DenseNet architecture (Huang et al., 2017) has 4 blocks in total. Each block is a fully convolutional 5-layer CNN with $3 \\times 3$ filters and 64 channels in the intermediate layers with ReLU nonlinearity. The first three blocks have an output layer with 64 channels while the last block has an output layer with only one channel. The output of the $i ^ { t h }$ block is concatenated with the input noisy image and then fed to the $( i + 1 ) ^ { t h }$ block, so the last three blocks have 65 input channels. In the version of the network with bias, we add trainable additive parameters to all the layers except for the last layer in the final block. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B DATASETS AND TRAINING PROCEDURE ", + "text_level": 1, + "bbox": [ + 174, + 102, + 532, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Our experiments are carried out on $1 8 0 \\times 1 8 0$ natural images from the Berkeley Segmentation Dataset (Martin et al., 2001). We use a training set of 400 images. The training set is augmented via downsampling, random flips, and random rotations of patches in these images (Zhang et al., 2017). A test set containing 68 images is used for evaluation. We train the DnCNN and it’s bias free model on patches of size $5 0 \\times 5 0$ , which yields a total of 541,600 clean training patches. For the remaining architectures, we use patches of size $1 2 8 \\times 1 2 8$ for a total of 22,400 training patches. ", + "bbox": [ + 174, + 133, + 825, + 217 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We train DnCNN and its bias-free counterpart using the Adam Optimizer (Kingma & Ba, 2014) over 70 epochs with an initial learning rate of $\\mathrm { 1 0 ^ { - 3 } }$ and a decay factor of 0.5 at the $\\mathbf { \\bar { 5 } 0 } ^ { t h }$ and $6 0 ^ { t h }$ epochs, with no early stopping. We train the other models using the Adam optimizer with an initial learning rate of $1 0 ^ { - 3 }$ and train for 50 epochs with a learning rate schedule which decreases by a factor of 0.25 if the validation PSNR decreases from one epoch to the next. We use early stopping and select the model with the best validation PSNR. ", + "bbox": [ + 174, + 223, + 825, + 308 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 329, + 397, + 344 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In this section we report additional results of our computational experiments: ", + "bbox": [ + 173, + 359, + 678, + 375 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• Figure 8 shows the first-order analysis of the residual of the different architectures described in Section A, except for DnCNN which is shown in Figure 1. \n• Figures 9 and 10 visualize the linear and net bias terms in the first-order decomposition of an example image at different noise levels. \n• Figure 11 shows the PSNR results for the experiments described in Section 5. \n• Figure 12 shows the SSIM results for the experiments described in Section 5. \n• Figures 13, 14 and 15 show the equivalent filters at several pixels of two example images for different architectures (see Section 6.1). \n• Figure 16 shows the singular vectors of the Jacobian of different BF-CNNs (see Section 6.2). Figure 17 shows the singular values of the Jacobian of different BF-CNNs (see Section 6.2). \nFigure 18 and 19 shows that networks trained on noise samples drawn from Gaussian distribution with 0 mean generalizes to noise drawn from uniform distribution with 0 mean during test time. Experiments follow the procedure described in Section 5 except that the networks are evaluated on a different noise distribution during the test time. \nFigure 20 shows the application of BF-CNN and CNN to the task of image restoration, where the image is corrupted with both noise and blur at the same time. We show that BF-CNNs can generalize outside the training range for noise levels for a fixed blur level, but do not outperform CNN when generalizing to unseen blur levels. ", + "bbox": [ + 215, + 386, + 826, + 678 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/99874d0030f5b131a9708e7019ffd08fffde12f32af9cb61e5a9bf661839aee7.jpg", + "image_caption": [ + "Figure 8: First-order analysis of the residual of Recurrent-CNN (Section A.2), UNet (Section A.3) and DenseNet (Section A.4) as a function of noise level. The plots show the magnitudes of the residual and the net bias averaged over 68 images in Set68 test set of Berkeley Segmentation Dataset (Martin et al., 2001) for networks trained over different training ranges. The range of noises used for training is highlighted in gray. (left) When the network is trained over the full range of noise levels $\\mathit { \\sigma } _ { \\mathcal { \\sigma } } \\in [ 0 , 1 0 0 ] $ ) the net bias is small, growing slightly as the noise increases. (middle and right) When the network is trained over the a smaller range $( \\sigma \\in [ 0 , 5 5 ]$ and $\\sigma \\in [ 0 , 3 0 ] )$ , the net bias grows explosively for noise levels outside the training range. This coincides with the dramatic drop in performance due to overfitting, reflected in the difference between the residual and the true noise. " + ], + "image_footnote": [], + "bbox": [ + 223, + 305, + 795, + 577 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/e5fc7c006db5b0e106b6a9a140bd37fa5e52057dd779277911dd6a96245eb993.jpg", + "image_caption": [ + "Figure 9: Visualization of the decomposition of output of DnCNN trained for noise range [0, 55] into linear part and net bias. The noise level $\\sigma = 7 0$ (highlighted by $^ *$ ) is outside the training range. Over the training range, the net bias is small, and the linear part is responsible for most of the denoising effort. However, when the network is evaluated out of the training range, the contribution of the bias increases dramatically, which coincides with a significant drop in denoising performance. " + ], + "image_footnote": [], + "bbox": [ + 181, + 271, + 843, + 666 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/f820b3943690865fa7a59dcb037d199c6ab17fcd38920470b2c2338db528af7c.jpg", + "image_caption": [ + "Figure 10: Visualization of the decomposition of output of Recurrent-CNN (Section A.2, UNet (Section A.3) and DenseNet (Section A.4) trained for noise range [0, 55] into linear part and net bias. The noise level $\\sigma = 9 0$ (highlighted by $^ *$ ) is outside the training range. Over the training range, the net bias is small, and the linear part is responsible for most of the denoising effort. However, when the network is evaluated out of the training range, the contribution of the bias increases dramatically, which coincides with a significant drop in denoising performance. " + ], + "image_footnote": [], + "bbox": [ + 199, + 172, + 846, + 760 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/89814a7a607446d1ce837255a96e5cd0f61ec833e6ce82fd547639b8347f107f.jpg", + "image_caption": [ + "Figure 11: Comparisons of architectures with (red curves) and without (blue curves) a net bias for the experimental design described in Section 5. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels. (a) Deep Convolutional Neural Network, DnCNN (Zhang et al., 2017). (b) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (c) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (d) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). " + ], + "image_footnote": [], + "bbox": [ + 174, + 175, + 820, + 306 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/22a873533c511400a20f0be326a3c082c3f6113c21e8128eaf8fa4792abf9477.jpg", + "image_caption": [ + "Figure 12: Comparisons of architectures with (red curves) and without (blue curves) a net bias for the experimental design described in Section 5. The performance is quantified by the SSIM of the denoised image as a function of the input SSIM of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels. (a) Deep Convolutional Neural Network, DnCNN (Zhang et al., 2017). (b) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (c) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (d) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). " + ], + "image_footnote": [], + "bbox": [ + 174, + 584, + 821, + 724 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/0bc851c120b9b9c6a975d32b70f50b3dfa9c9a1e6b3ccdaade89f7717b0f53e4.jpg", + "image_caption": [ + "Figure 13: Visualization of the linear weighting functions (rows of $A _ { y }$ ) of Bias-Free Recurrent-CNN (top 2 rows) (Section A.2), Bias-Free UNet (next 2 rows) (Section A.3) and Bias-Free DenseNet (bottom 2 rows) (Section A.4) for three example pixels of a noisy input image (left). The next image is the denoised output. The three images on the right show the linear weighting functions corresponding to each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (although some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. Each row corresponds to a noisy input with increasing $\\sigma$ and the filters adapt by averaging over a larger region. " + ], + "image_footnote": [], + "bbox": [ + 214, + 141, + 828, + 750 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/94994115094756ebf4076ef017149f91680027ba3e68ecbe2ffebf6fca71a4a1.jpg", + "image_caption": [ + "Figure 14: Visualization of the linear weighting functions (rows of $A _ { y . }$ ) of a BF-DnCNN for three example pixels of a noisy input image (left). The next image is the denoised output. The three images on the right show the linear weighting functions corresponding to each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (although some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. Each row corresponds to a noisy input with increasing $\\sigma$ and the filters adapt by averaging over a larger region. " + ], + "image_footnote": [], + "bbox": [ + 196, + 253, + 810, + 659 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/d3aa2b6f29e268f36288778aef4914b553be2fe8dd24c956bb9845a2e5292297.jpg", + "image_caption": [ + "Figure 15: Visualization of the linear weighting functions (rows of $A _ { y }$ ) of Bias-Free Recurrent-CNN (top 2 rows) (Section A.2), Bias-Free UNet (next 2 rows) (Section A.3) and Bias-Free DenseNet (bottom 2 rows) (Section A.4) for three example pixels of a noisy input image (left). The next image is the denoised output. The three images on the right show the linear weighting functions corresponding to each of the indicated pixels (red squares). All weighting functions sum to one, and thus compute a local average (although some weights are negative, indicated in red). Their shapes vary substantially, and are adapted to the underlying image content. Each row corresponds to a noisy input with increasing $\\sigma$ and the filters adapt by averaging over a larger region. " + ], + "image_footnote": [], + "bbox": [ + 215, + 142, + 828, + 750 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/44fb6f8d36b9f7ad7e6e80a4141748b5ec9bb00d0167b3f68a530ef80daffe9a.jpg", + "image_caption": [ + "Figure 16: Visualization of left singular vectors of the Jacobian of a BF Recurrent CNN (top 2 rows), BF UNet (next 2 rows) and BF DenseNet (bottom 2 rows) evaluated on three different images, corrupted by noise with standard deviation $\\sigma = 2 5$ . The left column shows original (clean) images. The next three columns show singular vectors corresponding to non-negligible singular values. The vectors capture features from the clean image. The last three columns on the right show singular vectors corresponding to singular values that are almost equal to zero. These vectors are noisy and unstructured. " + ], + "image_footnote": [], + "bbox": [ + 173, + 117, + 825, + 534 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/460223787582bc2925179c1185aac04d273d98e03459b69d964ece8ece43bc00.jpg", + "image_caption": [ + "Figure 17: Analysis of the SVD of the Jacobian of BF-CNN for ten natural images, corrupted by noise of standard deviation $\\sigma = 5 0$ . For all images, a large proportion of the singular values are near zero, indicating (approximately) a projection onto a subspace (the signal subspace). (a) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (b) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (c) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). " + ], + "image_footnote": [], + "bbox": [ + 173, + 680, + 825, + 810 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/e0c2db51a56a6c78406c9cd03e489455d59948648fffe9de854084945110df5a.jpg", + "image_caption": [ + "Figure 18: Comparison of the performance of a CNN and a BF-CNN with the same architecture for the experimental design described in Section 5. The networks are trained using i.i.d. Gaussian noise but evaluated on noise drawn i.i.d. from a uniform distribution with mean 0. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels, i.e. they are able to generalize across the two different noise distributions. The CNN used for this example is DnCNN (Zhang et al., 2017); using alternative architectures yields similar results (see Figures 19). " + ], + "image_footnote": [], + "bbox": [ + 176, + 171, + 816, + 291 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/4e3fbf39bc440f03916b2ef2f01bcd4439e441c5544651506c506e585b668fd8.jpg", + "image_caption": [ + "Figure 19: Comparisons of architectures with (red curves) and without (blue curves) a net bias for the experimental design described in Section 5. The networks are trained using i.i.d. Gaussian noise but evaluated on noise drawn i.i.d. from a uniform distribution with mean 0. The performance is quantified by the PSNR of the denoised image as a function of the input PSNR of the noisy image. All the architectures with bias perform poorly out of their training range, whereas the bias-free versions all achieve excellent generalization across noise levels, i.e. they are able to generalize across the two different noise distributions. (a) Deep Convolutional Neural Network, DnCNN (Zhang et al., 2017). (b) Recurrent architecture inspired by DURR (Zhang et al., 2018a). (c) Multiscale architecture inspired by the UNet (Ronneberger et al., 2015). (d) Architecture with multiple skip connections inspired by the DenseNet (Huang et al., 2017). " + ], + "image_footnote": [], + "bbox": [ + 174, + 568, + 820, + 699 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/a79f7d18cb7a353ab54a56266e0ade542a4c3778358b1ec216d285a54d9ce429.jpg", + "image_caption": [ + "Figure 20: Comparison of the performance of DnCNN and a corresponding BF-CNN for image restoration. Training is carried out on data corrupted with Gaussian noise $\\sigma _ { \\mathrm { n o i s e } } \\in [ 0 , 5 5 ]$ and Gaussian blur $\\sigma _ { \\mathrm { b l u r } } \\in [ 0 , 4 ]$ . Performance is measured on test data for inside and outside the training ranges. Left: The difference in performance measured in $\\Delta \\mathrm { P S N R } = \\mathrm { P S N R } _ { \\mathrm { B F - C N N } } - \\mathrm { P S N R } _ { \\mathrm { D n C N N } }$ . The training region is illustrated by the rectangular boundary. Bias-free network generalizes across noise levels for each fixed blur levels, whereas DnCNN does not. However, BF-CNN does not generalize across blur levels. Right: A horizontal slice of the left plot for a fixed blur level of $\\sigma _ { \\mathrm { { b l u r } } } = 2 $ . BF-CNN generalizes robustly beyond the training range, while the performance of DnCNN degrades significantly. " + ], + "image_footnote": [], + "bbox": [ + 207, + 366, + 746, + 529 + ], + "page_idx": 21 + } +] \ No newline at end of file diff --git a/parse/train/HkCjNI5ex/HkCjNI5ex.md b/parse/train/HkCjNI5ex/HkCjNI5ex.md new file mode 100644 index 0000000000000000000000000000000000000000..fa404cf9e669d2226307a3e688d8851069b09606 --- /dev/null +++ b/parse/train/HkCjNI5ex/HkCjNI5ex.md @@ -0,0 +1,275 @@ +# REGULARIZING NEURAL NETWORKS BY PENALIZING CONFIDENT OUTPUT DISTRIBUTIONS + +Gabriel Pereyra ∗ † Google Brain pereyra@google.com + +George Tucker ∗ † Google Brain gjt@google.com + +Jan Chorowski +Google Brain +chorowski@google.com +Łukasz Kaiser +Google Brain +lukaszkaiser@google.com + +Geoffrey Hinton University of Toronto & Google Brain geoffhinton@google.com + +# ABSTRACT + +We systematically explore regularizing neural networks by penalizing low entropy output distributions. We show that penalizing low entropy output distributions, which has been shown to improve exploration in reinforcement learning, acts as a strong regularizer in supervised learning. Furthermore, we connect a maximum entropy based confidence penalty to label smoothing through the direction of the KL divergence. We exhaustively evaluate the proposed confidence penalty and label smoothing on 6 common benchmarks: image classification (MNIST and Cifar-10), language modeling (Penn Treebank), machine translation (WMT’14 English-to-German), and speech recognition (TIMIT and WSJ). We find that both label smoothing and the confidence penalty improve state-of-the-art models across benchmarks without modifying existing hyperparameters, suggesting the wide applicability of these regularizers. + +# 1 INTRODUCTION + +Large neural networks with millions of parameters achieve strong performance on image classification (Szegedy et al., 2015a), machine translation (Wu et al., 2016), language modeling (Jozefowicz et al., 2016), and speech recognition (Graves et al., 2013). However, despite using large datasets, neural networks are still prone to overfitting. Numerous techniques have been proposed to prevent overfitting, including early stopping, L1/L2 regularization (weight decay), dropout (Srivastava et al., 2014), and batch normalization (Ioffe & Szegedy, 2015). These techniques, along with most other forms of regularization, act on the hidden activations or weights of a neural network. Alternatively, regularizing the output distribution of large, deep neural networks has largely been unexplored. + +To motivate output regularizers, we can view the knowledge of a model as the conditional distribution it produces over outputs given an input (Hinton et al., 2015) as opposed to the learned values of its parameters. Given this functional view of knowledge, the probabilities assigned to class labels that are incorrect (according to the training data) are part of the knowledge of the network. For example, when shown an image of a BMW, a network that assigns a probability of $1 0 ^ { - 3 }$ to “Audi” and $1 \bar { 0 } ^ { - 9 }$ to “carrot” is clearly better than a network that assigns $1 0 ^ { - 9 }$ to “Audi” and $1 0 ^ { - 3 }$ to carrot, all else being equal. One reason it is better is that the probabilities assigned to incorrect classes are an indication of how the network generalizes. Distillation (Hinton et al., 2015; Bucilu et al., 2006) exploits this fact by explicitly training a small network to assign the same probabilities to incorrect classes as a large network or ensemble of networks that generalizes well. Further, by operating on the output distribution that has a natural scale rather than on internal weights, whose significance depends on the values of the other weights, output regularization has the property that it is invariant to the parameterization of the underlying neural network. + +In this paper, we systematically evaluated two output regularizers: a maximum entropy based confidence penalty and label smoothing (uniform and unigram) for large, deep neural networks on 6 common benchmarks: image classification (MNIST and Cifar-10), language modeling (Penn Treebank), machine translation (WMT’14 English-to-German), and speech recognition (TIMIT and WSJ). We find that both label smoothing and the confidence penalty improve state-of-the-art models across benchmarks without modifying existing hyperparameters. + +# 2 RELATED WORK + +The maximum entropy principle (Jaynes, 1957) has a long history with deep connections to many areas of machine learning including unsupervised learning, supervised learning, and reinforcement learning. In supervised learning, we can search for the model with maximum entropy subject to constraints on empirical statistics, which naturally gives rise to maximum likelihood in log-linear models (see (Berger et al., 1996) for a review). Deterministic annealing Rose (1998) is a general approach for optimization that is widely applicable, avoids local minima, and can minimize discrete objectives, and it can be derived from the maximum entropy principle. Closely related to our work, Miller et al. (1996) apply deterministic annealing to train multilayer perceptrons, where an entropy based regularizer is introduced and slowly annealed. However, their focus is avoiding poor initialization and local minima, and while they find that deterministic annealing helps, the improvement diminishes quickly as the number of hidden units exceeds eight. + +In reinforcement learning, encouraging the policy to have an output distribution with high entropy has been used to improve exploration (Williams & Peng, 1991). This prevents the policy from converging early and leads to improved performance (Mnih et al., 2016). Penalizing low entropy has also been used when combining reinforcement learning and supervised learning to train a neural speech recognition model to learn when to emit tokens (Luo et al., 2016). When learning to emit, the entropy of the emission policy was added to the training objective and was annealed throughout training. Indeed, in recent work on reward augmented maximum likelihood (Norouzi et al., 2016), this entropy augmented reinforcement learning objective played a direct role in linking maximum likelihood and reinforcement learning objectives. + +Penalizing the entropy of a network’s output distribution has not been evaluated for large deep neural networks in supervised learning, but a closely related idea, label smoothing regularization, has been shown to improve generalization (Szegedy et al., 2015b). Label smoothing regularization estimates the marginalized effect of label-dropout during training, reducing overfitting by preventing a network from assigning full probability to each training example and maintaining a reasonable ratio between the logits of the incorrect classes. Simply adding label noise has also been shown to be effective at regularizing neural networks (Xie et al., 2016). Instead of smoothing the labels with a uniform distribution, as in label smoothing, we can smooth the labels with a teacher model (Hinton et al., 2015) or the model’s own distribution (Reed et al., 2014). Distillation and self-distillation both regularize a network by incorporating information about the ratios between incorrect classes. + +Virtual adversarial training (VAT) (Miyato et al., 2015) is another promising smoothing regularizer. However, we did not compare to VAT because it has multiple hyperparameters and the approximated gradient of the local distributional smoothness can be computed with no more than three pairs of forward and back propagations, which is significantly more computation in grid-searching and training than the other approaches we compared to. + +# 3 DIRECTLY PENALIZING CONFIDENCE + +Confident predictions correspond to output distributions that have low entropy. A network is overconfident when it places all probability on a single class in the training set, which is often a symptom of overfitting (Szegedy et al., 2015b). The confidence penalty constitutes a regularization term that prevents these peaked distributions, leading to better generalization. + +A neural network produces a conditional distribution $p _ { \theta } ( \pmb { y } | \pmb { x } )$ over classes $\textbf { { y } }$ given an input $_ { \textbf { \em x } }$ through a softmax function. The entropy of this conditional distribution is given by + +![](images/fc328408db4d728f25de361778b52b78b6c3e9efdf3e65df02c6726c9d1e4d89.jpg) +Figure 1: Distribution of the magnitude of softmax probabilities on the MNIST validation set. A fully-connected, 2-layer, 1024-unit neural network was trained with dropout (left), label smoothing (center), and the confidence penalty (right). Dropout leads to a softmax distribution where probabilities are either 0 or 1. By contrast, both label smoothing and the confidence penalty lead to smoother output distributions, which results in better generalization. + +$$ +H ( p _ { \theta } ( \pmb { y } | \pmb { x } ) ) = - \sum _ { i } p _ { \theta } ( \pmb { y } _ { i } | \pmb { x } ) \log ( p _ { \theta } ( \pmb { y } _ { i } | \pmb { x } ) ) . +$$ + +To penalize confident output distributions, we add the negative entropy to the negative log-likelihood during training + +$$ +\mathcal { L } ( \theta ) = - \sum \log p _ { \theta } ( \pmb { y } | \pmb { x } ) - \beta H ( p _ { \theta } ( \pmb { y } | \pmb { x } ) ) , +$$ + +where $\beta$ controls the strength of the confidence penalty. Notably, the gradient of the entropy term with respect to the logits is simple to compute. Denoting the ith logit by $z _ { i }$ , then + +$$ +\frac { \partial H ( p _ { \theta } ) } { \partial z _ { i } } = p _ { \theta } ( \pmb { y } _ { i } | \pmb { x } ) \left( - \log p _ { \theta } ( \pmb { y } _ { i } | \pmb { x } ) - H ( p _ { \theta } ) \right) , +$$ + +which is the weighted deviation from the mean. + +# 3.1 ANNEALING AND THRESHOLDING THE CONFIDENCE PENALTY + +In reinforcement learning, penalizing low entropy distributions prevents a policy network from converging early and encourages exploration. However, in supervised learning, we typically want quick convergence, while preventing overfitting near the end of training, suggesting a confidence penalty that is weak at the beginning of training and strong near convergence. A simple way to achieve this is to anneal the confidence penalty. + +Another way to strengthen the confidence penalty as training progresses is to only penalize output distributions when they are below a certain entropy threshold. We can achieve this by adding a hinge loss to the confidence penalty, leading to an objective of the form + +$$ +\mathcal { L } ( \theta ) = - \sum \log p _ { \theta } ( \pmb { y } | \pmb { x } ) - \beta \operatorname* { m a x } ( 0 , \Gamma - H ( p _ { \theta } ( \pmb { y } | \pmb { x } ) ) , +$$ + +where $\Gamma$ is the entropy threshold below which we begin applying the confidence penalty. + +Initial experiments suggest that thresholding the confidence penalty leads to faster convergence at the cost of introducing an extra hyper-parameter. For the majority of our experiments, we were able to achieve comparable performance without using the thresholded version. For the sake of simplicity, we focus on the single hyper-parameter version in our experiments. + +# 3.2 CONNECTION TO LABEL SMOOTHING + +Label smoothing estimates the marginalized effect of label noise during training. When the prior label distribution is uniform, label smoothing is equivalent to adding the KL divergence between the uniform distribution $u$ and the network’s predicted distribution $p _ { \theta }$ to the negative log-likelihood + +$$ +\mathcal { L } ( \theta ) = - \sum \log p _ { \theta } ( \pmb { y } | \pmb { x } ) - D _ { K L } ( \boldsymbol { u } | | p _ { \theta } ( \pmb { y } | \pmb { x } ) ) . +$$ + +By reversing the direction of the KL divergence, $D _ { K L } ( p _ { \boldsymbol { \theta } } ( \pmb { y } | \pmb { x } ) \| \boldsymbol { u } )$ , we recover the confidence penalty. This interpretation suggests further confidence regularizers that use alternative target distributions instead of the uniform distribution. We leave the exploration of these regularizers to future work. + +# 4 EXPERIMENTS + +We evaluated the confidence penalty and label smoothing on MNIST and CIFAR-10 for image classification, Penn Treebank for language modeling, WMT’14 English-to-German for machine translation, and TIMIT and WSJ for speech recognition. All models were implemented using TensorFlow (Abadi et al., 2016) and trained on NVIDIA Tesla K40 or K80 GPUs. + +# 4.1 IMAGE CLASSIFICATION + +# 4.1.1 MNIST + +As a preliminary experiment, we evaluated the approaches on the standard MNIST digit recognition task. We used the standard split into 60k training images and 10k testing images. We use the last 10k images of the training set as a held-out validation set for hyper-parameter tuning and then retrained the models on the entire dataset with the best configuration. + +We trained fully-connected, ReLu activation neural networks with 1024 units per layer and two hidden layers. Weights were initialized from a normal distribution with standard deviation 0.01. Models were optimized with stochastic gradient descent with a constant learning rate 0.05 (except for dropout where we set the learning rate to 0.001). + +For label smoothing, we varied the smoothing parameter in the range [0.05, 0.1, 0.2, 0.3, 0.4, 0.5], and found 0.1 to work best for both methods. For the confidence penalty, we varied the weight values over [0.1, 0.3, 0.5, 1.0, 2.0, 4.0, 8.0] and found a confidence penalty weight of 1.0 to work best. + +We also plotted the norm of the gradient as training progressed in Figure 2. We observed that label smoothing and confidence penalty had smaller gradient norms and converged more quickly than models regularized with dropout. If the output distributions is peaked on a misclassified example, the model receives a large gradient. This may explain why the regularized models have smaller gradient norms. + +Table 1: Test error $( \% )$ for permutation-invariant MNIST. + +
ModelLayersSizeTest
Wan et al. (2013) - Unregularized28001.40%
Srivastava et al. (2014) - Dropout310241.25%
Wan et al. (2013) - DropConnect28001.20%
Srivastava et al. (2014) - MaxNorm + Dropout281920.95%
Dropout210241.28 ± 0.06%
Label Smoothing210241.23 ± 0.06%
Confidence Penalty210241.17 ± 0.06%
+ +# 4.1.2 CIFAR-10 + +CIFAR-10 is an image classification dataset consisting of $3 2 \mathbf { x } 3 2 \mathbf { x } 3$ RGB images of 10 classes. The dataset is split into $5 0 \mathrm { k }$ training images and $1 0 \mathrm { k }$ testing images. We use the last $5 \mathrm { k }$ images of the training set as a held-out validation set for hyper-parameter tuning, as is common practice. + +For our experiments, we used a densely connected convolutional neural network, which represents the current state-of-the-art on CIFAR-10 (Huang et al., 2016a). We use the small configuration from (Huang et al., 2016a), which consists of 40-layers, with a growth rate of 12. All models were trained for 300 epochs, with a batch-size of 50 and a learning rate 0.1. The learning rate was reduced by a factor of 10 at 150 and 225 epochs. We present results for training without data-augmentation. We found that the confidence penalty did not lead to improved performance when training with data augmentation, however neither did other regularization techniques, including dropout. + +For our final test scores, we trained on the entire training set. For label smoothing, we tried smoothing parameter values of [0.05, 0.1, 0.2, 0.3, 0.4, 0.5], and found 0.1 to work best. For the confidence penalty, we performed a grid search over confidence penalty weight values of [0.1, 0.25, 0.5, 1.0, 1.5] and found a confidence penalty weight of 0.1 to work best. + +Table 2: Test error $\%$ ) on Cifar-10 without data augmentation. + +
ModelLayersParametersTest
He et al. (2015) - Residual CNN1101.7M13.63%
Huang et al. (2016b) - Stochastic Depth Residual CNN1101.7M11.66%
Larsson et al. (2016) - Fractal CNN2138.6M10.18%
Larsson et al. (2016) - Fractal CNN (Dropout)2138.6M7.33%
Huang et al. (2O16a) - Densely Connected CNN401.0M7.00%
Huang et al. (2O16a) - Densely Connected CNN1007.0M5.77%
Densely Connected CNN (Dropout)401.0M7.04%
Densely Connected CNN (Dropout + Label Smoothing)401.0M6.89%
Densely Connected CNN (Dropout + Confidence Penalty)401.0M6.77%
+ +# 4.2 LANGUAGE MODELING + +For language modeling, we found that confidence penalty significantly outperforms label noise and label smoothing. We performed word-level language modeling experiments using the Penn Treebank dataset (PTB) (Marcus et al., 1993). We used the hyper-parameter settings from the large configuration in (Zaremba et al., 2014). Briefly, we used a 2-layer, 1500-unit LSTM, with $65 \%$ dropout applied on all non-recurrent connections. We trained using stochastic gradient descent for 55 epochs, decaying the learning rate by 1.15 after 14 epochs, and clipped the norm of the gradients when they were larger than 10. + +For label noise and label smoothing, we performed a grid search over noise and smoothing values of [0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5]. For label noise, we found 0.1 to work best. For label smoothing, we found 0.1 to work best. For the confidence penalty, we performed a grid search over confidence penalty weight values of [0.1, 0.5, 1.0, 2.0, 3.0]. We found a confidence penalty weight of 2.0 to work best, which led to an improvement of 3.7 perplexity points over the baseline. + +For reference, we also include results of the existing state-of-the-art models for the word-level language modeling task on PTB. Variational dropout (Gal, 2015) applies a fixed dropout mask (stochastic for each sample) at each time-step, instead of resampling at each time-step as in traditional dropout. Note, that we do not include the variational dropout results that use Monte Carlo (MC) model averaging, which achieves lower perplexity on the test set but requires 1000 model evaluations, which are then averaged. Recurrent highway networks (Zilly et al., 2016) currently represent the state-of-the-art performance on PTB. + +Table 3: Validation and test perplexity for word-level Penn Treebank. + +
ModelParametersValidationTest
Zaremba et al. (2014) - Regularized LSTM Gal (2015) - Variational LSTM66M82.278.4
Press& Wolf(2016) - TiedVariationalLSTM66M77.975.2
51M79.675.0
Merity et al. (2016) - Pointer Sentinel LSTM21M72.470.9
Zilly et al. (2016) - Variational RHN32M71.268.5
Zilly et al. (2016) - Tied Variational RHN24M68.166.0
Regularized LSTM (label noise)66M79.777.7
Regularized LSTM (label smoothing)66M78.976.6
Regularized LSTM (unigram smoothing)66M79.176.3
Regularized LSTM (confidence penalty)66M77.874.7
+ +# 4.3 MACHINE TRANSLATION + +For machine translation, we evaluated the confidence penalty on the WMT’14 English-to-German translation task using Google’s production-level translation system Wu et al. (2016). The training set consists of 5M sentence pairs, and we used newstest2012 and newtests2013 for validation and newstest2014 for testing. We report tokenized BLEU scores as computed by the multi-bleu.perl script from the Moses translation machine translation package. + +Our model was an 8-layer sequence-to-sequence model with attention (Bahdanau et al., 2014). The first encoder was a bidirectional LSTM, the remaining encoder and decoder layers were unidirectional LSTMs, and the attention network was a single layer feed-forward network. Each layer had 512 units (compared to 1024 in (Wu et al., 2016)). The model was trained using 12 replicas running concurrently with asynchronous updates. Dropout of $30 \%$ was applied as described in (Zaremba et al., 2014). Optimization used a mix of Adam and SGD with gradient clipping. Unlike (Wu et al., 2016), we did not use reinforcement learning to fine-tune our model. We used a beam size of 12 during decoding. For more details, see (Wu et al., 2016). + +For label smoothing, we performed a grid search over values $[ 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 ]$ and found 0.1 to work best for both uniform and unigram smoothing. For the confidence penalty, we searched over values of [0.5, 2.5, 4.5] and found a value of 2.5 to work best . For machine translation, we found label smoothing slightly outperformed confidence penalty. When applied without dropout, both lead to an improvement of just over 1 BLEU point (dropout leads to an improvement of just over 2 BLEU points). However, when combined with dropout, the effect of both regularizers was diminished. + +
ModelParametersValidationTest
Buck et al. (2014) - PBMT20.7
Cho et al. (2015) - RNNSearch16.9
Zhou et al. (2016) - Deep-Att20.6
Luong et al. (2015) - P-Attention164M20.9
Wu et al. (2016) - WPM-16K167M24.4
Wu et al. (2016) - WPM-32K278M24.6
WPM-32K (without dropout)94M22.3321.24
WPM-32K (label smoothing)94M23.8522.42
WPM-32K (confidence penalty)94M23.2522.52
WPM-32K (dropout)94M24.1 ±0.123.41 ± 0.04
WPM-32K (dropout + label smoothing)94M24.3 ± 0.123.52 ±0.03
WPM-32K (dropout + unigram smoothing)94M24.3 ± 0.123.57 ± 0.02
WPM-32K (dropout + confidence penalty)94M24.3 ± 0.123.4 ± 0.1
+ +Table 4: Validation and test BLEU for WMT’14 English-to-German. For the last four model configurations, we report the mean and standard error of the mean (SEM) over 5 random initializations. + +# 4.4 SPEECH RECOGNITION + +# 4.4.1 TIMIT + +In the TIMIT corpus, the training set consists of 3512 utterances, the validation set consists of 184 utterances and the test set consists of 192 utterances. All 61 phonemes were used during training and decoding, and during scoring, these 61 phonemes were reduced to 39 to compute the phoneme error rate (PER). + +As our base model, we used a sequence-to-sequence model with attention. The encoder consisted of 3 bidirectional LSTM layers, the decoder consisted of a single unidirectional LSTM layer, and the attention network consisted of a single layer feed-forward network. All layers consisted of 256 units. Dropout of $15 \%$ was applied as described in Zaremba et al. (2014). We trained the model with asynchronous SGD with 5 replicas. We used a batch size of 32, a learning rate of 0.01, and momentum of 0.9. Gradients were clipped at 5.0. For more details, see Norouzi et al. (2016). + +For label smoothing, we performed a grid search over values [0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8] and found 0.2 to work best. For the confidence penalty, we performed a grid search over values of $[ 0 . 1 2 5 , 0 . 2 5 , 0 . 5 , 1 . 0 , 2 . 0 , 4 . 0 , 8 . 0 ]$ and found a value of 1.0 to work best. Label smoothing led to an absolute improvement over the dropout baseline of $1 . 6 \%$ , while the confidence penalty led to an absolute improvement of $1 . 2 \%$ . + +Table 5: Validation and test phoneme error rates (PER) for TIMIT. We report the mean and SEM over 5 random initializations. + +
ModelParametersValidationTest
Mohamed et al. (2012) - DNN-HMM120.7
Norouzi et al. (2016) - RML6.5M18.019.9
Graves et al. (2006) - CTC6.8M-18.4
Graves et al. (2013) - RNN Transducer4.3M117.7
T6th (2014) - CNN113.916.7
Dropout6.5M21.0 ±0.123.2 ± 0.4
Dropout + Label Smoothing6.5M19.3 ± 0.121.6 ± 0.2
Dropout + Confidence Penalty6.5M19.9 ± 0.222.0± 0.4
+ +# 4.4.2 WALL STREET JOURNAL + +For the WSJ corpus we used attention-based sequence-to-sequence networks that directly predicted characters. We used the SI284 subset for training, DEV93 for validation, and EVAL92 for testing. We used 240-dimensional vectors consisting of 80-bin filterbank features augmented with their deltas and delta-deltas with per-speaker normalized mean and variances computed with Kaldi Povey et al. (2011). We did not use text-only data or separate language models during decoding. + +Network architecture details were as follows. The encoder of the network consisted of 4 bidirectional LSTM layers each having 256 units, interleaved with 3 time-subsampling layers, configured to drop every second frame (Bahdanau et al., 2016; Chan et al., 2015). The decoder used a single LSTM layer with 256 units. The attention vectors were computed with a single layer feedforward network having 64 hidden units and the convolutional filters as described in Chorowski et al. (2015). Weights were initialized from a uniform distribution $[ - 0 . 0 7 5 , 0 . 0 7 5 ]$ . All models used weight decay of $1 \bar { 0 ^ { - 6 } }$ , additive Gaussian weight noise with standard deviation 0.075, applied after 20K steps, and were trained for 650K steps. We used the ADAM optimizer asynchronously over 8 GPUs. We used a learning rate of $1 0 ^ { - 3 }$ , which was reduced to $1 0 ^ { - 4 }$ after $4 0 0 \mathrm { K }$ and $1 0 ^ { - 5 }$ after 500K steps. + +We tested three methods of increasing the entropy of outputs: the confidence penalty and two variants of label smoothing: uniform and unigram. All resulted in improved Word Error Rates (WER), however the unigram smoothing resulted in the greatest WER reduction, and we found it to be least sensitive to its hyperparameter (the smoothing value). Furthermore, uniform smoothing and confidence penalty required masking network outputs corresponding to tokens that never appeared as labels, such as the start-of-sequence token. + +Table 6 compares the performance of the regularized networks with several recent results. We observe that the benefits of label smoothing (WER reduction from 14.2 to 11) improve over the recently proposed Latent Sequence Decompositions (LSD) method (Chan et al., 2016) which reduces the WER from 14.7 to 12.9 by extending the space of output tokens to dynamically chosen character n-grams. + +Table 6: Validation and test word error rates (WER) for WSJ. For Baseline, Uniform Label Smoothing and Confidence Penalty we report the average over two runs. For the best setting (Unigram Label Smoothing), we report the average over 6 runs together with the standard deviation. + +
ModelParametersValidationTest
Graves & Jaitly (2014) - CTC26.5M27.3
Bahdanau et al. (2016) - seq2seq Chan et al. (2016) - Baseline5.7M18.6
Chan et al. (2016) - LSD5.1M14.7
5.9M112.9
Baseline6.6M17.914.2
Uniform Label Smoothing6.6M14.711.3
Unigram Label Smoothing6.6M14.0 ± 0.2511.0± 0.35
Confidence Penalty6.6M17.212.7
+ +# 5 CONCLUSION + +Motivated by recent successes of output regularizers (Szegedy et al., 2015b; Xie et al., 2016), we conduct a systematic evaluation of two output regularizers: the confidence penalty and label smoothing. We show that this form of regularization, which has been shown to improve exploration in reinforcement learning, also acts as a strong regularizer in supervised learning. We find that both the confidence penalty and label smoothing improve a wide range of state-of-the-art models, without the need to modify hyper-parameters. + +# ACKNOWLEDGMENTS + +We would like to thank Sergey Ioffe, Alex Alemi and Navdeep Jaitly for helpful discussions. We would also like to thank Prajit Ramachandran, Barret Zoph, Mohammad Norouzi, and Yonghui Wu for technical help with the various models used in our experiments. 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Recurrent neural network regularization. arXiv preprint arXiv:1409.2329, 2014. + +Jie Zhou, Ying Cao, Xuguang Wang, Peng Li, and Wei Xu. Deep recurrent models with fast-forward connections for neural machine translation. arXiv preprint arXiv:1606.04199, 2016. + +Julian Georg Zilly, Rupesh Kumar Srivastava, Jan Koutn´ık, and Jurgen Schmidhuber. Recurrent ¨ highway networks. arXiv preprint arXiv:1607.03474, 2016. + +# APPENDIX + +# 6 GRADIENT NORMS + +![](images/65a42cb6dcec02738c6b4d251b50c0e3e158940e82aa51754f8d301d489346b9.jpg) +Figure 2: Norm of the gradient as training proceeds on the MNIST dataset. We plot the norm of the gradient while training with confidence penalty, dropout, label smoothing, and without regularization. We use early stopping on the validation set, which explains the difference in training steps between methods. Both confidence penalty and label smoothing result in smaller gradient norm. \ No newline at end of file diff --git a/parse/train/HkCjNI5ex/HkCjNI5ex_content_list.json b/parse/train/HkCjNI5ex/HkCjNI5ex_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..1e0d7ee8fd5b0d22a99c0622f01f812fccaeb130 --- /dev/null +++ b/parse/train/HkCjNI5ex/HkCjNI5ex_content_list.json @@ -0,0 +1,1413 @@ +[ + { + "type": "text", + "text": "REGULARIZING NEURAL NETWORKS BY PENALIZING CONFIDENT OUTPUT DISTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Gabriel Pereyra ∗ † Google Brain pereyra@google.com ", + "bbox": [ + 183, + 169, + 362, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "George Tucker ∗ † Google Brain gjt@google.com ", + "bbox": [ + 419, + 169, + 560, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jan Chorowski \nGoogle Brain \nchorowski@google.com \nŁukasz Kaiser \nGoogle Brain \nlukaszkaiser@google.com ", + "bbox": [ + 614, + 170, + 815, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 183, + 233, + 411, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Geoffrey Hinton University of Toronto & Google Brain geoffhinton@google.com ", + "bbox": [ + 495, + 233, + 746, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 311, + 544, + 327 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We systematically explore regularizing neural networks by penalizing low entropy output distributions. We show that penalizing low entropy output distributions, which has been shown to improve exploration in reinforcement learning, acts as a strong regularizer in supervised learning. Furthermore, we connect a maximum entropy based confidence penalty to label smoothing through the direction of the KL divergence. We exhaustively evaluate the proposed confidence penalty and label smoothing on 6 common benchmarks: image classification (MNIST and Cifar-10), language modeling (Penn Treebank), machine translation (WMT’14 English-to-German), and speech recognition (TIMIT and WSJ). We find that both label smoothing and the confidence penalty improve state-of-the-art models across benchmarks without modifying existing hyperparameters, suggesting the wide applicability of these regularizers. ", + "bbox": [ + 233, + 347, + 764, + 513 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 546, + 336, + 563 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Large neural networks with millions of parameters achieve strong performance on image classification (Szegedy et al., 2015a), machine translation (Wu et al., 2016), language modeling (Jozefowicz et al., 2016), and speech recognition (Graves et al., 2013). However, despite using large datasets, neural networks are still prone to overfitting. Numerous techniques have been proposed to prevent overfitting, including early stopping, L1/L2 regularization (weight decay), dropout (Srivastava et al., 2014), and batch normalization (Ioffe & Szegedy, 2015). These techniques, along with most other forms of regularization, act on the hidden activations or weights of a neural network. Alternatively, regularizing the output distribution of large, deep neural networks has largely been unexplored. ", + "bbox": [ + 174, + 580, + 825, + 693 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To motivate output regularizers, we can view the knowledge of a model as the conditional distribution it produces over outputs given an input (Hinton et al., 2015) as opposed to the learned values of its parameters. Given this functional view of knowledge, the probabilities assigned to class labels that are incorrect (according to the training data) are part of the knowledge of the network. For example, when shown an image of a BMW, a network that assigns a probability of $1 0 ^ { - 3 }$ to “Audi” and $1 \\bar { 0 } ^ { - 9 }$ to “carrot” is clearly better than a network that assigns $1 0 ^ { - 9 }$ to “Audi” and $1 0 ^ { - 3 }$ to carrot, all else being equal. One reason it is better is that the probabilities assigned to incorrect classes are an indication of how the network generalizes. Distillation (Hinton et al., 2015; Bucilu et al., 2006) exploits this fact by explicitly training a small network to assign the same probabilities to incorrect classes as a large network or ensemble of networks that generalizes well. Further, by operating on the output distribution that has a natural scale rather than on internal weights, whose significance depends on the values of the other weights, output regularization has the property that it is invariant to the parameterization of the underlying neural network. ", + "bbox": [ + 174, + 699, + 825, + 878 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we systematically evaluated two output regularizers: a maximum entropy based confidence penalty and label smoothing (uniform and unigram) for large, deep neural networks on 6 common benchmarks: image classification (MNIST and Cifar-10), language modeling (Penn Treebank), machine translation (WMT’14 English-to-German), and speech recognition (TIMIT and WSJ). We find that both label smoothing and the confidence penalty improve state-of-the-art models across benchmarks without modifying existing hyperparameters. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 218, + 343, + 234 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The maximum entropy principle (Jaynes, 1957) has a long history with deep connections to many areas of machine learning including unsupervised learning, supervised learning, and reinforcement learning. In supervised learning, we can search for the model with maximum entropy subject to constraints on empirical statistics, which naturally gives rise to maximum likelihood in log-linear models (see (Berger et al., 1996) for a review). Deterministic annealing Rose (1998) is a general approach for optimization that is widely applicable, avoids local minima, and can minimize discrete objectives, and it can be derived from the maximum entropy principle. Closely related to our work, Miller et al. (1996) apply deterministic annealing to train multilayer perceptrons, where an entropy based regularizer is introduced and slowly annealed. However, their focus is avoiding poor initialization and local minima, and while they find that deterministic annealing helps, the improvement diminishes quickly as the number of hidden units exceeds eight. ", + "bbox": [ + 174, + 256, + 825, + 409 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In reinforcement learning, encouraging the policy to have an output distribution with high entropy has been used to improve exploration (Williams & Peng, 1991). This prevents the policy from converging early and leads to improved performance (Mnih et al., 2016). Penalizing low entropy has also been used when combining reinforcement learning and supervised learning to train a neural speech recognition model to learn when to emit tokens (Luo et al., 2016). When learning to emit, the entropy of the emission policy was added to the training objective and was annealed throughout training. Indeed, in recent work on reward augmented maximum likelihood (Norouzi et al., 2016), this entropy augmented reinforcement learning objective played a direct role in linking maximum likelihood and reinforcement learning objectives. ", + "bbox": [ + 174, + 416, + 825, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Penalizing the entropy of a network’s output distribution has not been evaluated for large deep neural networks in supervised learning, but a closely related idea, label smoothing regularization, has been shown to improve generalization (Szegedy et al., 2015b). Label smoothing regularization estimates the marginalized effect of label-dropout during training, reducing overfitting by preventing a network from assigning full probability to each training example and maintaining a reasonable ratio between the logits of the incorrect classes. Simply adding label noise has also been shown to be effective at regularizing neural networks (Xie et al., 2016). Instead of smoothing the labels with a uniform distribution, as in label smoothing, we can smooth the labels with a teacher model (Hinton et al., 2015) or the model’s own distribution (Reed et al., 2014). Distillation and self-distillation both regularize a network by incorporating information about the ratios between incorrect classes. ", + "bbox": [ + 174, + 547, + 825, + 686 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Virtual adversarial training (VAT) (Miyato et al., 2015) is another promising smoothing regularizer. However, we did not compare to VAT because it has multiple hyperparameters and the approximated gradient of the local distributional smoothness can be computed with no more than three pairs of forward and back propagations, which is significantly more computation in grid-searching and training than the other approaches we compared to. ", + "bbox": [ + 174, + 694, + 823, + 763 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 DIRECTLY PENALIZING CONFIDENCE ", + "text_level": 1, + "bbox": [ + 176, + 795, + 516, + 810 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Confident predictions correspond to output distributions that have low entropy. A network is overconfident when it places all probability on a single class in the training set, which is often a symptom of overfitting (Szegedy et al., 2015b). The confidence penalty constitutes a regularization term that prevents these peaked distributions, leading to better generalization. ", + "bbox": [ + 174, + 832, + 823, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A neural network produces a conditional distribution $p _ { \\theta } ( \\pmb { y } | \\pmb { x } )$ over classes $\\textbf { { y } }$ given an input $_ { \\textbf { \\em x } }$ through a softmax function. The entropy of this conditional distribution is given by ", + "bbox": [ + 173, + 895, + 820, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/fc328408db4d728f25de361778b52b78b6c3e9efdf3e65df02c6726c9d1e4d89.jpg", + "image_caption": [ + "Figure 1: Distribution of the magnitude of softmax probabilities on the MNIST validation set. A fully-connected, 2-layer, 1024-unit neural network was trained with dropout (left), label smoothing (center), and the confidence penalty (right). Dropout leads to a softmax distribution where probabilities are either 0 or 1. By contrast, both label smoothing and the confidence penalty lead to smoother output distributions, which results in better generalization. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 820, + 224 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1d4830e290b7cf23e234f6c30a65a2f92c4ad78394d92bb5308c65c4683c44aa.jpg", + "text": "$$\nH ( p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) ) = - \\sum _ { i } p _ { \\theta } ( \\pmb { y } _ { i } | \\pmb { x } ) \\log ( p _ { \\theta } ( \\pmb { y } _ { i } | \\pmb { x } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 357, + 651, + 391 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To penalize confident output distributions, we add the negative entropy to the negative log-likelihood during training ", + "bbox": [ + 173, + 406, + 823, + 436 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b5f27061b4ebbc49b4c0534bbfeb81de5b85db2bbc31f37d9732102f1143d47d.jpg", + "text": "$$\n\\mathcal { L } ( \\theta ) = - \\sum \\log p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) - \\beta H ( p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 457, + 643, + 483 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\beta$ controls the strength of the confidence penalty. Notably, the gradient of the entropy term with respect to the logits is simple to compute. Denoting the ith logit by $z _ { i }$ , then ", + "bbox": [ + 173, + 498, + 826, + 529 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/754ccfbe0b3eaadaabbafe740e4e27c171f0d2bab743190358c90025dcb793b2.jpg", + "text": "$$\n\\frac { \\partial H ( p _ { \\theta } ) } { \\partial z _ { i } } = p _ { \\theta } ( \\pmb { y } _ { i } | \\pmb { x } ) \\left( - \\log p _ { \\theta } ( \\pmb { y } _ { i } | \\pmb { x } ) - H ( p _ { \\theta } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 336, + 550, + 658, + 583 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "which is the weighted deviation from the mean. ", + "bbox": [ + 174, + 599, + 485, + 614 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 ANNEALING AND THRESHOLDING THE CONFIDENCE PENALTY ", + "text_level": 1, + "bbox": [ + 174, + 636, + 648, + 651 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In reinforcement learning, penalizing low entropy distributions prevents a policy network from converging early and encourages exploration. However, in supervised learning, we typically want quick convergence, while preventing overfitting near the end of training, suggesting a confidence penalty that is weak at the beginning of training and strong near convergence. A simple way to achieve this is to anneal the confidence penalty. ", + "bbox": [ + 173, + 662, + 825, + 734 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Another way to strengthen the confidence penalty as training progresses is to only penalize output distributions when they are below a certain entropy threshold. We can achieve this by adding a hinge loss to the confidence penalty, leading to an objective of the form ", + "bbox": [ + 174, + 741, + 825, + 784 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8349565069d3d69e3f39d0e7ee782f88340943a4fe373b0968ab44651ed814d2.jpg", + "text": "$$\n\\mathcal { L } ( \\theta ) = - \\sum \\log p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) - \\beta \\operatorname* { m a x } ( 0 , \\Gamma - H ( p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 308, + 805, + 686, + 830 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\Gamma$ is the entropy threshold below which we begin applying the confidence penalty. ", + "bbox": [ + 169, + 845, + 753, + 862 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Initial experiments suggest that thresholding the confidence penalty leads to faster convergence at the cost of introducing an extra hyper-parameter. For the majority of our experiments, we were able to achieve comparable performance without using the thresholded version. For the sake of simplicity, we focus on the single hyper-parameter version in our experiments. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 CONNECTION TO LABEL SMOOTHING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 475, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Label smoothing estimates the marginalized effect of label noise during training. When the prior label distribution is uniform, label smoothing is equivalent to adding the KL divergence between the uniform distribution $u$ and the network’s predicted distribution $p _ { \\theta }$ to the negative log-likelihood ", + "bbox": [ + 174, + 131, + 825, + 174 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/132b15f12dd90014f33c96590c07d57b393048f98682771a5a5d05c9d4db5d28.jpg", + "text": "$$\n\\mathcal { L } ( \\theta ) = - \\sum \\log p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) - D _ { K L } ( \\boldsymbol { u } | | p _ { \\theta } ( \\pmb { y } | \\pmb { x } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 194, + 658, + 219 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "By reversing the direction of the KL divergence, $D _ { K L } ( p _ { \\boldsymbol { \\theta } } ( \\pmb { y } | \\pmb { x } ) \\| \\boldsymbol { u } )$ , we recover the confidence penalty. This interpretation suggests further confidence regularizers that use alternative target distributions instead of the uniform distribution. We leave the exploration of these regularizers to future work. ", + "bbox": [ + 174, + 233, + 825, + 290 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 314, + 326, + 330 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We evaluated the confidence penalty and label smoothing on MNIST and CIFAR-10 for image classification, Penn Treebank for language modeling, WMT’14 English-to-German for machine translation, and TIMIT and WSJ for speech recognition. All models were implemented using TensorFlow (Abadi et al., 2016) and trained on NVIDIA Tesla K40 or K80 GPUs. ", + "bbox": [ + 174, + 347, + 825, + 404 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 IMAGE CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 424, + 383, + 439 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1.1 MNIST ", + "text_level": 1, + "bbox": [ + 174, + 450, + 284, + 465 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As a preliminary experiment, we evaluated the approaches on the standard MNIST digit recognition task. We used the standard split into 60k training images and 10k testing images. We use the last 10k images of the training set as a held-out validation set for hyper-parameter tuning and then retrained the models on the entire dataset with the best configuration. ", + "bbox": [ + 174, + 477, + 825, + 534 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We trained fully-connected, ReLu activation neural networks with 1024 units per layer and two hidden layers. Weights were initialized from a normal distribution with standard deviation 0.01. Models were optimized with stochastic gradient descent with a constant learning rate 0.05 (except for dropout where we set the learning rate to 0.001). ", + "bbox": [ + 174, + 540, + 825, + 597 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For label smoothing, we varied the smoothing parameter in the range [0.05, 0.1, 0.2, 0.3, 0.4, 0.5], and found 0.1 to work best for both methods. For the confidence penalty, we varied the weight values over [0.1, 0.3, 0.5, 1.0, 2.0, 4.0, 8.0] and found a confidence penalty weight of 1.0 to work best. ", + "bbox": [ + 174, + 603, + 823, + 659 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We also plotted the norm of the gradient as training progressed in Figure 2. We observed that label smoothing and confidence penalty had smaller gradient norms and converged more quickly than models regularized with dropout. If the output distributions is peaked on a misclassified example, the model receives a large gradient. This may explain why the regularized models have smaller gradient norms. ", + "bbox": [ + 174, + 665, + 825, + 736 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/839b95c5f60737d32a9de3d917b203981009f66074dc660cb42e9694ca1ca364.jpg", + "table_caption": [ + "Table 1: Test error $( \\% )$ for permutation-invariant MNIST." + ], + "table_footnote": [], + "table_body": "
ModelLayersSizeTest
Wan et al. (2013) - Unregularized28001.40%
Srivastava et al. (2014) - Dropout310241.25%
Wan et al. (2013) - DropConnect28001.20%
Srivastava et al. (2014) - MaxNorm + Dropout281920.95%
Dropout210241.28 ± 0.06%
Label Smoothing210241.23 ± 0.06%
Confidence Penalty210241.17 ± 0.06%
", + "bbox": [ + 215, + 752, + 781, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1.2 CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 303, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "CIFAR-10 is an image classification dataset consisting of $3 2 \\mathbf { x } 3 2 \\mathbf { x } 3$ RGB images of 10 classes. The dataset is split into $5 0 \\mathrm { k }$ training images and $1 0 \\mathrm { k }$ testing images. We use the last $5 \\mathrm { k }$ images of the training set as a held-out validation set for hyper-parameter tuning, as is common practice. ", + "bbox": [ + 174, + 135, + 825, + 176 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For our experiments, we used a densely connected convolutional neural network, which represents the current state-of-the-art on CIFAR-10 (Huang et al., 2016a). We use the small configuration from (Huang et al., 2016a), which consists of 40-layers, with a growth rate of 12. All models were trained for 300 epochs, with a batch-size of 50 and a learning rate 0.1. The learning rate was reduced by a factor of 10 at 150 and 225 epochs. We present results for training without data-augmentation. We found that the confidence penalty did not lead to improved performance when training with data augmentation, however neither did other regularization techniques, including dropout. ", + "bbox": [ + 174, + 184, + 825, + 281 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For our final test scores, we trained on the entire training set. For label smoothing, we tried smoothing parameter values of [0.05, 0.1, 0.2, 0.3, 0.4, 0.5], and found 0.1 to work best. For the confidence penalty, we performed a grid search over confidence penalty weight values of [0.1, 0.25, 0.5, 1.0, 1.5] and found a confidence penalty weight of 0.1 to work best. ", + "bbox": [ + 174, + 287, + 825, + 344 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/820a3c46dcfd4959ecfe5b84c07253980e438aac0265ea253f77474474e5c1ea.jpg", + "table_caption": [ + "Table 2: Test error $\\%$ ) on Cifar-10 without data augmentation. " + ], + "table_footnote": [], + "table_body": "
ModelLayersParametersTest
He et al. (2015) - Residual CNN1101.7M13.63%
Huang et al. (2016b) - Stochastic Depth Residual CNN1101.7M11.66%
Larsson et al. (2016) - Fractal CNN2138.6M10.18%
Larsson et al. (2016) - Fractal CNN (Dropout)2138.6M7.33%
Huang et al. (2O16a) - Densely Connected CNN401.0M7.00%
Huang et al. (2O16a) - Densely Connected CNN1007.0M5.77%
Densely Connected CNN (Dropout)401.0M7.04%
Densely Connected CNN (Dropout + Label Smoothing)401.0M6.89%
Densely Connected CNN (Dropout + Confidence Penalty)401.0M6.77%
", + "bbox": [ + 178, + 371, + 820, + 532 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 LANGUAGE MODELING ", + "text_level": 1, + "bbox": [ + 176, + 613, + 377, + 627 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For language modeling, we found that confidence penalty significantly outperforms label noise and label smoothing. We performed word-level language modeling experiments using the Penn Treebank dataset (PTB) (Marcus et al., 1993). We used the hyper-parameter settings from the large configuration in (Zaremba et al., 2014). Briefly, we used a 2-layer, 1500-unit LSTM, with $65 \\%$ dropout applied on all non-recurrent connections. We trained using stochastic gradient descent for 55 epochs, decaying the learning rate by 1.15 after 14 epochs, and clipped the norm of the gradients when they were larger than 10. ", + "bbox": [ + 174, + 645, + 825, + 742 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For label noise and label smoothing, we performed a grid search over noise and smoothing values of [0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5]. For label noise, we found 0.1 to work best. For label smoothing, we found 0.1 to work best. For the confidence penalty, we performed a grid search over confidence penalty weight values of [0.1, 0.5, 1.0, 2.0, 3.0]. We found a confidence penalty weight of 2.0 to work best, which led to an improvement of 3.7 perplexity points over the baseline. ", + "bbox": [ + 174, + 750, + 825, + 820 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For reference, we also include results of the existing state-of-the-art models for the word-level language modeling task on PTB. Variational dropout (Gal, 2015) applies a fixed dropout mask (stochastic for each sample) at each time-step, instead of resampling at each time-step as in traditional dropout. Note, that we do not include the variational dropout results that use Monte Carlo (MC) model averaging, which achieves lower perplexity on the test set but requires 1000 model evaluations, which are then averaged. Recurrent highway networks (Zilly et al., 2016) currently represent the state-of-the-art performance on PTB. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/f770da827ec692ae26f593b4ac543bbc8df419d37ab0e3f3a2579dbccbc6456c.jpg", + "table_caption": [ + "Table 3: Validation and test perplexity for word-level Penn Treebank. " + ], + "table_footnote": [], + "table_body": "
ModelParametersValidationTest
Zaremba et al. (2014) - Regularized LSTM Gal (2015) - Variational LSTM66M82.278.4
Press& Wolf(2016) - TiedVariationalLSTM66M77.975.2
51M79.675.0
Merity et al. (2016) - Pointer Sentinel LSTM21M72.470.9
Zilly et al. (2016) - Variational RHN32M71.268.5
Zilly et al. (2016) - Tied Variational RHN24M68.166.0
Regularized LSTM (label noise)66M79.777.7
Regularized LSTM (label smoothing)66M78.976.6
Regularized LSTM (unigram smoothing)66M79.176.3
Regularized LSTM (confidence penalty)66M77.874.7
", + "bbox": [ + 217, + 101, + 781, + 276 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 MACHINE TRANSLATION ", + "text_level": 1, + "bbox": [ + 174, + 327, + 388, + 342 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For machine translation, we evaluated the confidence penalty on the WMT’14 English-to-German translation task using Google’s production-level translation system Wu et al. (2016). The training set consists of 5M sentence pairs, and we used newstest2012 and newtests2013 for validation and newstest2014 for testing. We report tokenized BLEU scores as computed by the multi-bleu.perl script from the Moses translation machine translation package. ", + "bbox": [ + 174, + 353, + 825, + 424 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our model was an 8-layer sequence-to-sequence model with attention (Bahdanau et al., 2014). The first encoder was a bidirectional LSTM, the remaining encoder and decoder layers were unidirectional LSTMs, and the attention network was a single layer feed-forward network. Each layer had 512 units (compared to 1024 in (Wu et al., 2016)). The model was trained using 12 replicas running concurrently with asynchronous updates. Dropout of $30 \\%$ was applied as described in (Zaremba et al., 2014). Optimization used a mix of Adam and SGD with gradient clipping. Unlike (Wu et al., 2016), we did not use reinforcement learning to fine-tune our model. We used a beam size of 12 during decoding. For more details, see (Wu et al., 2016). ", + "bbox": [ + 174, + 430, + 825, + 541 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For label smoothing, we performed a grid search over values $[ 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 ]$ and found 0.1 to work best for both uniform and unigram smoothing. For the confidence penalty, we searched over values of [0.5, 2.5, 4.5] and found a value of 2.5 to work best . For machine translation, we found label smoothing slightly outperformed confidence penalty. When applied without dropout, both lead to an improvement of just over 1 BLEU point (dropout leads to an improvement of just over 2 BLEU points). However, when combined with dropout, the effect of both regularizers was diminished. ", + "bbox": [ + 173, + 547, + 825, + 646 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/71d18f0a693e364bc78a3ea27fccfb5df5d5ad28e9fc97b07f84cf2deeb05352.jpg", + "table_caption": [], + "table_footnote": [ + "Table 4: Validation and test BLEU for WMT’14 English-to-German. For the last four model configurations, we report the mean and standard error of the mean (SEM) over 5 random initializations. " + ], + "table_body": "
ModelParametersValidationTest
Buck et al. (2014) - PBMT20.7
Cho et al. (2015) - RNNSearch16.9
Zhou et al. (2016) - Deep-Att20.6
Luong et al. (2015) - P-Attention164M20.9
Wu et al. (2016) - WPM-16K167M24.4
Wu et al. (2016) - WPM-32K278M24.6
WPM-32K (without dropout)94M22.3321.24
WPM-32K (label smoothing)94M23.8522.42
WPM-32K (confidence penalty)94M23.2522.52
WPM-32K (dropout)94M24.1 ±0.123.41 ± 0.04
WPM-32K (dropout + label smoothing)94M24.3 ± 0.123.52 ±0.03
WPM-32K (dropout + unigram smoothing)94M24.3 ± 0.123.57 ± 0.02
WPM-32K (dropout + confidence penalty)94M24.3 ± 0.123.4 ± 0.1
", + "bbox": [ + 189, + 659, + 807, + 873 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.4 SPEECH RECOGNITION ", + "text_level": 1, + "bbox": [ + 174, + 103, + 374, + 117 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.4.1 TIMIT ", + "text_level": 1, + "bbox": [ + 174, + 128, + 279, + 143 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the TIMIT corpus, the training set consists of 3512 utterances, the validation set consists of 184 utterances and the test set consists of 192 utterances. All 61 phonemes were used during training and decoding, and during scoring, these 61 phonemes were reduced to 39 to compute the phoneme error rate (PER). ", + "bbox": [ + 174, + 154, + 825, + 210 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As our base model, we used a sequence-to-sequence model with attention. The encoder consisted of 3 bidirectional LSTM layers, the decoder consisted of a single unidirectional LSTM layer, and the attention network consisted of a single layer feed-forward network. All layers consisted of 256 units. Dropout of $15 \\%$ was applied as described in Zaremba et al. (2014). We trained the model with asynchronous SGD with 5 replicas. We used a batch size of 32, a learning rate of 0.01, and momentum of 0.9. Gradients were clipped at 5.0. For more details, see Norouzi et al. (2016). ", + "bbox": [ + 174, + 217, + 825, + 301 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For label smoothing, we performed a grid search over values [0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8] and found 0.2 to work best. For the confidence penalty, we performed a grid search over values of $[ 0 . 1 2 5 , 0 . 2 5 , 0 . 5 , 1 . 0 , 2 . 0 , 4 . 0 , 8 . 0 ]$ and found a value of 1.0 to work best. Label smoothing led to an absolute improvement over the dropout baseline of $1 . 6 \\%$ , while the confidence penalty led to an absolute improvement of $1 . 2 \\%$ . ", + "bbox": [ + 174, + 308, + 823, + 377 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d9b388288ec158df1718552a054738bdd87782d32ae1af2e5cecbc8d42af707c.jpg", + "table_caption": [ + "Table 5: Validation and test phoneme error rates (PER) for TIMIT. We report the mean and SEM over 5 random initializations. " + ], + "table_footnote": [], + "table_body": "
ModelParametersValidationTest
Mohamed et al. (2012) - DNN-HMM120.7
Norouzi et al. (2016) - RML6.5M18.019.9
Graves et al. (2006) - CTC6.8M-18.4
Graves et al. (2013) - RNN Transducer4.3M117.7
T6th (2014) - CNN113.916.7
Dropout6.5M21.0 ±0.123.2 ± 0.4
Dropout + Label Smoothing6.5M19.3 ± 0.121.6 ± 0.2
Dropout + Confidence Penalty6.5M19.9 ± 0.222.0± 0.4
", + "bbox": [ + 215, + 392, + 782, + 539 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.4.2 WALL STREET JOURNAL ", + "text_level": 1, + "bbox": [ + 176, + 606, + 398, + 621 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the WSJ corpus we used attention-based sequence-to-sequence networks that directly predicted characters. We used the SI284 subset for training, DEV93 for validation, and EVAL92 for testing. We used 240-dimensional vectors consisting of 80-bin filterbank features augmented with their deltas and delta-deltas with per-speaker normalized mean and variances computed with Kaldi Povey et al. (2011). We did not use text-only data or separate language models during decoding. ", + "bbox": [ + 174, + 631, + 823, + 702 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Network architecture details were as follows. The encoder of the network consisted of 4 bidirectional LSTM layers each having 256 units, interleaved with 3 time-subsampling layers, configured to drop every second frame (Bahdanau et al., 2016; Chan et al., 2015). The decoder used a single LSTM layer with 256 units. The attention vectors were computed with a single layer feedforward network having 64 hidden units and the convolutional filters as described in Chorowski et al. (2015). Weights were initialized from a uniform distribution $[ - 0 . 0 7 5 , 0 . 0 7 5 ]$ . All models used weight decay of $1 \\bar { 0 ^ { - 6 } }$ , additive Gaussian weight noise with standard deviation 0.075, applied after 20K steps, and were trained for 650K steps. We used the ADAM optimizer asynchronously over 8 GPUs. We used a learning rate of $1 0 ^ { - 3 }$ , which was reduced to $1 0 ^ { - 4 }$ after $4 0 0 \\mathrm { K }$ and $1 0 ^ { - 5 }$ after 500K steps. ", + "bbox": [ + 174, + 708, + 825, + 833 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We tested three methods of increasing the entropy of outputs: the confidence penalty and two variants of label smoothing: uniform and unigram. All resulted in improved Word Error Rates (WER), however the unigram smoothing resulted in the greatest WER reduction, and we found it to be least sensitive to its hyperparameter (the smoothing value). Furthermore, uniform smoothing and confidence penalty required masking network outputs corresponding to tokens that never appeared as labels, such as the start-of-sequence token. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 6 compares the performance of the regularized networks with several recent results. We observe that the benefits of label smoothing (WER reduction from 14.2 to 11) improve over the recently proposed Latent Sequence Decompositions (LSD) method (Chan et al., 2016) which reduces the WER from 14.7 to 12.9 by extending the space of output tokens to dynamically chosen character n-grams. ", + "bbox": [ + 174, + 103, + 823, + 174 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/24524012974a872e39be1c37c96a2e78588fa309ee5c51bd52cc3c5386ee53d3.jpg", + "table_caption": [ + "Table 6: Validation and test word error rates (WER) for WSJ. For Baseline, Uniform Label Smoothing and Confidence Penalty we report the average over two runs. For the best setting (Unigram Label Smoothing), we report the average over 6 runs together with the standard deviation. " + ], + "table_footnote": [], + "table_body": "
ModelParametersValidationTest
Graves & Jaitly (2014) - CTC26.5M27.3
Bahdanau et al. (2016) - seq2seq Chan et al. (2016) - Baseline5.7M18.6
Chan et al. (2016) - LSD5.1M14.7
5.9M112.9
Baseline6.6M17.914.2
Uniform Label Smoothing6.6M14.711.3
Unigram Label Smoothing6.6M14.0 ± 0.2511.0± 0.35
Confidence Penalty6.6M17.212.7
", + "bbox": [ + 227, + 185, + 771, + 329 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 409, + 318, + 425 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Motivated by recent successes of output regularizers (Szegedy et al., 2015b; Xie et al., 2016), we conduct a systematic evaluation of two output regularizers: the confidence penalty and label smoothing. We show that this form of regularization, which has been shown to improve exploration in reinforcement learning, also acts as a strong regularizer in supervised learning. We find that both the confidence penalty and label smoothing improve a wide range of state-of-the-art models, without the need to modify hyper-parameters. ", + "bbox": [ + 174, + 441, + 825, + 525 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 541, + 326, + 554 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We would like to thank Sergey Ioffe, Alex Alemi and Navdeep Jaitly for helpful discussions. We would also like to thank Prajit Ramachandran, Barret Zoph, Mohammad Norouzi, and Yonghui Wu for technical help with the various models used in our experiments. We thank the anonymous reviewers for insightful comments. ", + "bbox": [ + 176, + 564, + 825, + 619 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 642, + 285, + 656 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Martın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. 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", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Julian Georg Zilly, Rupesh Kumar Srivastava, Jan Koutn´ık, and Jurgen Schmidhuber. Recurrent ¨ highway networks. arXiv preprint arXiv:1607.03474, 2016. ", + "bbox": [ + 171, + 141, + 823, + 170 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "6 GRADIENT NORMS ", + "text_level": 1, + "bbox": [ + 174, + 136, + 362, + 151 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/65a42cb6dcec02738c6b4d251b50c0e3e158940e82aa51754f8d301d489346b9.jpg", + "image_caption": [ + "Figure 2: Norm of the gradient as training proceeds on the MNIST dataset. We plot the norm of the gradient while training with confidence penalty, dropout, label smoothing, and without regularization. We use early stopping on the validation set, which explains the difference in training steps between methods. Both confidence penalty and label smoothing result in smaller gradient norm. " + ], + "image_footnote": [], + "bbox": [ + 349, + 186, + 632, + 359 + ], + "page_idx": 11 + } +] \ No newline at end of file diff --git a/parse/train/HkCjNI5ex/HkCjNI5ex_middle.json b/parse/train/HkCjNI5ex/HkCjNI5ex_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..689688d79c864af4e9bccca12526511b72df1f1c --- /dev/null +++ b/parse/train/HkCjNI5ex/HkCjNI5ex_middle.json @@ -0,0 +1,25028 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 97 + ], + "score": 1.0, + "content": "REGULARIZING NEURAL NETWORKS BY PENALIZING", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 385, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 385, + 118 + ], + "score": 1.0, + "content": "CONFIDENT OUTPUT DISTRIBUTIONS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 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We find that both", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 373, + 470, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 373, + 470, + 386 + ], + "score": 1.0, + "content": "label smoothing and the confidence penalty improve state-of-the-art models across", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 383, + 468, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 383, + 468, + 398 + ], + "score": 1.0, + "content": "benchmarks without modifying existing hyperparameters, suggesting the wide ap-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 395, + 272, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 395, + 272, + 408 + ], + "score": 1.0, + "content": "plicability of these regularizers.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 433, + 206, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 208, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 208, + 449 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 504, + 473 + ], + "score": 1.0, + "content": "Large neural networks with millions of parameters achieve strong performance on image classifica-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "tion (Szegedy et al., 2015a), machine translation (Wu et al., 2016), language modeling (Jozefowicz", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "et al., 2016), and speech recognition (Graves et al., 2013). However, despite using large datasets,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "neural networks are still prone to overfitting. Numerous techniques have been proposed to prevent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "overfitting, including early stopping, L1/L2 regularization (weight decay), dropout (Srivastava et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "2014), and batch normalization (Ioffe & Szegedy, 2015). These techniques, along with most other", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "forms of regularization, act on the hidden activations or weights of a neural network. Alternatively,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 538, + 487, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 487, + 550 + ], + "score": 1.0, + "content": "regularizing the output distribution of large, deep neural networks has largely been unexplored.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "To motivate output regularizers, we can view the knowledge of a model as the conditional distribu-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "tion it produces over outputs given an input (Hinton et al., 2015) as opposed to the learned values", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "of its parameters. Given this functional view of knowledge, the probabilities assigned to class labels", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "that are incorrect (according to the training data) are part of the knowledge of the network. For", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 441, + 610 + ], + "score": 1.0, + "content": "example, when shown an image of a BMW, a network that assigns a probability of", + "type": "text" + }, + { + "bbox": [ + 441, + 597, + 462, + 608 + ], + "score": 0.9, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "to “Audi”", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 123, + 622 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 609, + 145, + 619 + ], + "score": 0.9, + "content": "1 \\bar { 0 } ^ { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 608, + 363, + 622 + ], + "score": 1.0, + "content": "to “carrot” is clearly better than a network that assigns", + "type": "text" + }, + { + "bbox": [ + 364, + 609, + 385, + 619 + ], + "score": 0.91, + "content": "1 0 ^ { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 608, + 444, + 622 + ], + "score": 1.0, + "content": "to “Audi” and", + "type": "text" + }, + { + "bbox": [ + 444, + 609, + 465, + 619 + ], + "score": 0.9, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "to carrot,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "all else being equal. 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Furthermore, we connect a maximum", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 319, + 470, + 330 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 470, + 330 + ], + "score": 1.0, + "content": "entropy based confidence penalty to label smoothing through the direction of the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 329, + 470, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 329, + 470, + 342 + ], + "score": 1.0, + "content": "KL divergence. We exhaustively evaluate the proposed confidence penalty and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 341, + 470, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 341, + 470, + 353 + ], + "score": 1.0, + "content": "label smoothing on 6 common benchmarks: image classification (MNIST and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 352, + 469, + 363 + ], + "spans": [ + { + "bbox": [ + 141, + 352, + 469, + 363 + ], + "score": 1.0, + "content": "Cifar-10), language modeling (Penn Treebank), machine translation (WMT’14", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 362, + 470, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 470, + 375 + ], + "score": 1.0, + "content": "English-to-German), and speech recognition (TIMIT and WSJ). We find that both", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 373, + 470, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 373, + 470, + 386 + ], + "score": 1.0, + "content": "label smoothing and the confidence penalty improve state-of-the-art models across", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 383, + 468, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 383, + 468, + 398 + ], + "score": 1.0, + "content": "benchmarks without modifying existing hyperparameters, suggesting the wide ap-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 395, + 272, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 395, + 272, + 408 + ], + "score": 1.0, + "content": "plicability of these regularizers.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23.5, + "bbox_fs": [ + 141, + 273, + 470, + 408 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 433, + 206, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 208, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 208, + 449 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 504, + 473 + ], + "score": 1.0, + "content": "Large neural networks with millions of parameters achieve strong performance on image classifica-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "tion (Szegedy et al., 2015a), machine translation (Wu et al., 2016), language modeling (Jozefowicz", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "et al., 2016), and speech recognition (Graves et al., 2013). However, despite using large datasets,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "neural networks are still prone to overfitting. Numerous techniques have been proposed to prevent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "overfitting, including early stopping, L1/L2 regularization (weight decay), dropout (Srivastava et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "2014), and batch normalization (Ioffe & Szegedy, 2015). These techniques, along with most other", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "forms of regularization, act on the hidden activations or weights of a neural network. Alternatively,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 538, + 487, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 487, + 550 + ], + "score": 1.0, + "content": "regularizing the output distribution of large, deep neural networks has largely been unexplored.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 461, + 505, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "To motivate output regularizers, we can view the knowledge of a model as the conditional distribu-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "tion it produces over outputs given an input (Hinton et al., 2015) as opposed to the learned values", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "of its parameters. Given this functional view of knowledge, the probabilities assigned to class labels", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "that are incorrect (according to the training data) are part of the knowledge of the network. 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Further, by operating on", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "the output distribution that has a natural scale rather than on internal weights, whose significance", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "depends on the values of the other weights, output regularization has the property that it is invariant", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 686, + 336, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 336, + 698 + ], + "score": 1.0, + "content": "to the parameterization of the underlying neural network.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 553, + 506, + 698 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In this paper, we systematically evaluated two output regularizers: a maximum entropy based confi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "dence penalty and label smoothing (uniform and unigram) for large, deep neural networks on 6 com-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "mon benchmarks: image classification (MNIST and Cifar-10), language modeling (Penn Treebank),", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "machine translation (WMT’14 English-to-German), and speech recognition (TIMIT and WSJ). We", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "find that both label smoothing and the confidence penalty improve state-of-the-art models across", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 339, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 339, + 150 + ], + "score": 1.0, + "content": "benchmarks without modifying existing hyperparameters.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 173, + 210, + 186 + ], + "lines": [ + { + "bbox": [ + 104, + 172, + 213, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 172, + 213, + 189 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "score": 1.0, + "content": "The maximum entropy principle (Jaynes, 1957) has a long history with deep connections to many", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "areas of machine learning including unsupervised learning, supervised learning, and reinforcement", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "learning. In supervised learning, we can search for the model with maximum entropy subject to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "constraints on empirical statistics, which naturally gives rise to maximum likelihood in log-linear", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "models (see (Berger et al., 1996) for a review). Deterministic annealing Rose (1998) is a general", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "approach for optimization that is widely applicable, avoids local minima, and can minimize discrete", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 269, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 504, + 282 + ], + "score": 1.0, + "content": "objectives, and it can be derived from the maximum entropy principle. Closely related to our work,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "Miller et al. 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However, their focus is avoiding poor initial-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "ization and local minima, and while they find that deterministic annealing helps, the improvement", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 364, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 364, + 325 + ], + "score": 1.0, + "content": "diminishes quickly as the number of hidden units exceeds eight.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "In reinforcement learning, encouraging the policy to have an output distribution with high entropy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "has been used to improve exploration (Williams & Peng, 1991). This prevents the policy from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 350, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 350, + 505, + 366 + ], + "score": 1.0, + "content": "converging early and leads to improved performance (Mnih et al., 2016). Penalizing low entropy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "has also been used when combining reinforcement learning and supervised learning to train a neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "speech recognition model to learn when to emit tokens (Luo et al., 2016). When learning to emit,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "the entropy of the emission policy was added to the training objective and was annealed throughout", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "training. 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Simply adding label noise has also been shown to be effective", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "at regularizing neural networks (Xie et al., 2016). Instead of smoothing the labels with a uniform", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "distribution, as in label smoothing, we can smooth the labels with a teacher model (Hinton et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "2015) or the model’s own distribution (Reed et al., 2014). Distillation and self-distillation both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 479, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 479, + 546 + ], + "score": 1.0, + "content": "regularize a network by incorporating information about the ratios between incorrect classes.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "Virtual adversarial training (VAT) (Miyato et al., 2015) is another promising smoothing regularizer.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "However, we did not compare to VAT because it has multiple hyperparameters and the approximated", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "gradient of the local distributional smoothness can be computed with no more than three pairs of for-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 597 + ], + "score": 1.0, + "content": "ward and back propagations, which is significantly more computation in grid-searching and training", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 279, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 279, + 606 + ], + "score": 1.0, + "content": "than the other approaches we compared to.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 108, + 630, + 316, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 628, + 318, + 644 + ], + "spans": [ + { + "bbox": [ + 104, + 628, + 318, + 644 + ], + "score": 1.0, + "content": "3 DIRECTLY PENALIZING CONFIDENCE", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "Confident predictions correspond to output distributions that have low entropy. A network is over-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "confident when it places all probability on a single class in the training set, which is often a symptom", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "of overfitting (Szegedy et al., 2015b). The confidence penalty constitutes a regularization term that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 693, + 380, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 380, + 705 + ], + "score": 1.0, + "content": "prevents these peaked distributions, leading to better generalization.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 331, + 723 + ], + "score": 1.0, + "content": "A neural network produces a conditional distribution", + "type": "text" + }, + { + "bbox": [ + 331, + 710, + 365, + 722 + ], + "score": 0.92, + "content": "p _ { \\theta } ( \\pmb { y } | \\pmb { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 708, + 421, + 723 + ], + "score": 1.0, + "content": "over classes", + "type": "text" + }, + { + "bbox": [ + 421, + 712, + 429, + 721 + ], + "score": 0.75, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 708, + 496, + 723 + ], + "score": 1.0, + "content": "given an input", + "type": "text" + }, + { + "bbox": [ + 496, + 712, + 504, + 720 + ], + "score": 0.66, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 440, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 440, + 733 + ], + "score": 1.0, + "content": "through a softmax function. 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We", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "find that both label smoothing and the confidence penalty improve state-of-the-art models across", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 339, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 339, + 150 + ], + "score": 1.0, + "content": "benchmarks without modifying existing hyperparameters.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 150 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 173, + 210, + 186 + ], + "lines": [ + { + "bbox": [ + 104, + 172, + 213, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 172, + 213, + 189 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 217 + ], + "score": 1.0, + "content": "The maximum entropy principle (Jaynes, 1957) has a long history with deep connections to many", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "areas of machine learning including unsupervised learning, supervised learning, and reinforcement", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "learning. In supervised learning, we can search for the model with maximum entropy subject to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "constraints on empirical statistics, which naturally gives rise to maximum likelihood in log-linear", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "models (see (Berger et al., 1996) for a review). Deterministic annealing Rose (1998) is a general", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "approach for optimization that is widely applicable, avoids local minima, and can minimize discrete", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 269, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 504, + 282 + ], + "score": 1.0, + "content": "objectives, and it can be derived from the maximum entropy principle. Closely related to our work,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "Miller et al. (1996) apply deterministic annealing to train multilayer perceptrons, where an entropy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "based regularizer is introduced and slowly annealed. However, their focus is avoiding poor initial-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "ization and local minima, and while they find that deterministic annealing helps, the improvement", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 364, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 364, + 325 + ], + "score": 1.0, + "content": "diminishes quickly as the number of hidden units exceeds eight.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 202, + 506, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 343 + ], + "score": 1.0, + "content": "In reinforcement learning, encouraging the policy to have an output distribution with high entropy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "has been used to improve exploration (Williams & Peng, 1991). This prevents the policy from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 350, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 350, + 505, + 366 + ], + "score": 1.0, + "content": "converging early and leads to improved performance (Mnih et al., 2016). Penalizing low entropy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "has also been used when combining reinforcement learning and supervised learning to train a neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "speech recognition model to learn when to emit tokens (Luo et al., 2016). When learning to emit,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "the entropy of the emission policy was added to the training objective and was annealed throughout", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "training. Indeed, in recent work on reward augmented maximum likelihood (Norouzi et al., 2016),", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "this entropy augmented reinforcement learning objective played a direct role in linking maximum", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 304, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 304, + 430 + ], + "score": 1.0, + "content": "likelihood and reinforcement learning objectives.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 329, + 506, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Penalizing the entropy of a network’s output distribution has not been evaluated for large deep neural", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "networks in supervised learning, but a closely related idea, label smoothing regularization, has been", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "shown to improve generalization (Szegedy et al., 2015b). Label smoothing regularization estimates", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "the marginalized effect of label-dropout during training, reducing overfitting by preventing a network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "from assigning full probability to each training example and maintaining a reasonable ratio between", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "the logits of the incorrect classes. Simply adding label noise has also been shown to be effective", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "at regularizing neural networks (Xie et al., 2016). Instead of smoothing the labels with a uniform", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "distribution, as in label smoothing, we can smooth the labels with a teacher model (Hinton et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "2015) or the model’s own distribution (Reed et al., 2014). Distillation and self-distillation both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 479, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 479, + 546 + ], + "score": 1.0, + "content": "regularize a network by incorporating information about the ratios between incorrect classes.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 434, + 505, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "Virtual adversarial training (VAT) (Miyato et al., 2015) is another promising smoothing regularizer.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "However, we did not compare to VAT because it has multiple hyperparameters and the approximated", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "gradient of the local distributional smoothness can be computed with no more than three pairs of for-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 597 + ], + "score": 1.0, + "content": "ward and back propagations, which is significantly more computation in grid-searching and training", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 279, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 279, + 606 + ], + "score": 1.0, + "content": "than the other approaches we compared to.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 549, + 506, + 606 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 630, + 316, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 628, + 318, + 644 + ], + "spans": [ + { + "bbox": [ + 104, + 628, + 318, + 644 + ], + "score": 1.0, + "content": "3 DIRECTLY PENALIZING CONFIDENCE", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "Confident predictions correspond to output distributions that have low entropy. A network is over-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "confident when it places all probability on a single class in the training set, which is often a symptom", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "of overfitting (Szegedy et al., 2015b). 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This interpretation suggests further confidence regularizers that use alternative target distri-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "butions instead of the uniform distribution. 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ModelLayersSizeTest
Wan et al. (2013) - Unregularized28001.40%
Srivastava et al. (2014) - Dropout310241.25%
Wan et al. (2013) - DropConnect28001.20%
Srivastava et al. (2014) - MaxNorm + Dropout281920.95%
Dropout210241.28 ± 0.06%
Label Smoothing210241.23 ± 0.06%
Confidence Penalty210241.17 ± 0.06%
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This interpretation suggests further confidence regularizers that use alternative target distri-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "butions instead of the uniform distribution. We leave the exploration of these regularizers to future", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 218, + 132, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 132, + 230 + ], + "score": 1.0, + "content": "work.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 185, + 506, + 230 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 249, + 200, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 201, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 201, + 263 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "We evaluated the confidence penalty and label smoothing on MNIST and CIFAR-10 for image clas-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "sification, Penn Treebank for language modeling, WMT’14 English-to-German for machine transla-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "tion, and TIMIT and WSJ for speech recognition. All models were implemented using TensorFlow", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 309, + 387, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 387, + 321 + ], + "score": 1.0, + "content": "(Abadi et al., 2016) and trained on NVIDIA Tesla K40 or K80 GPUs.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 275, + 505, + 321 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 336, + 235, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 235, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 235, + 348 + ], + "score": 1.0, + "content": "4.1 IMAGE CLASSIFICATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 357, + 174, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 175, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 175, + 371 + ], + "score": 1.0, + "content": "4.1.1 MNIST", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "As a preliminary experiment, we evaluated the approaches on the standard MNIST digit recognition", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "task. We used the standard split into 60k training images and 10k testing images. We use the last 10k", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "images of the training set as a held-out validation set for hyper-parameter tuning and then retrained", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 411, + 346, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 346, + 423 + ], + "score": 1.0, + "content": "the models on the entire dataset with the best configuration.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 378, + 505, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "We trained fully-connected, ReLu activation neural networks with 1024 units per layer and two", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "hidden layers. Weights were initialized from a normal distribution with standard deviation 0.01.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 104, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "Models were optimized with stochastic gradient descent with a constant learning rate 0.05 (except", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 461, + 317, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 317, + 473 + ], + "score": 1.0, + "content": "for dropout where we set the learning rate to 0.001).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 428, + 506, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 504, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "For label smoothing, we varied the smoothing parameter in the range [0.05, 0.1, 0.2, 0.3, 0.4, 0.5],", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "and found 0.1 to work best for both methods. For the confidence penalty, we varied the weight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "score": 1.0, + "content": "values over [0.1, 0.3, 0.5, 1.0, 2.0, 4.0, 8.0] and found a confidence penalty weight of 1.0 to work", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 511, + 128, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 128, + 523 + ], + "score": 1.0, + "content": "best.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 478, + 506, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 527, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "We also plotted the norm of the gradient as training progressed in Figure 2. We observed that label", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 538, + 504, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 504, + 550 + ], + "score": 1.0, + "content": "smoothing and confidence penalty had smaller gradient norms and converged more quickly than", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 504, + 562 + ], + "score": 1.0, + "content": "models regularized with dropout. If the output distributions is peaked on a misclassified example,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "the model receives a large gradient. This may explain why the regularized models have smaller", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 572, + 172, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 172, + 583 + ], + "score": 1.0, + "content": "gradient norms.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 527, + 505, + 583 + ] + }, + { + "type": "table", + "bbox": [ + 132, + 596, + 478, + 702 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 132, + 596, + 478, + 702 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 596, + 478, + 702 + ], + "spans": [ + { + "bbox": [ + 132, + 596, + 478, + 702 + ], + "score": 0.982, + "html": "
ModelLayersSizeTest
Wan et al. (2013) - Unregularized28001.40%
Srivastava et al. (2014) - Dropout310241.25%
Wan et al. (2013) - DropConnect28001.20%
Srivastava et al. (2014) - MaxNorm + Dropout281920.95%
Dropout210241.28 ± 0.06%
Label Smoothing210241.23 ± 0.06%
Confidence Penalty210241.17 ± 0.06%
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The", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 185, + 130 + ], + "score": 1.0, + "content": "dataset is split into", + "type": "text" + }, + { + "bbox": [ + 186, + 118, + 203, + 128 + ], + "score": 0.39, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 118, + 287, + 130 + ], + "score": 1.0, + "content": "training images and", + "type": "text" + }, + { + "bbox": [ + 287, + 118, + 303, + 128 + ], + "score": 0.51, + "content": "1 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 118, + 434, + 130 + ], + "score": 1.0, + "content": "testing images. We use the last", + "type": "text" + }, + { + "bbox": [ + 434, + 118, + 446, + 128 + ], + "score": 0.31, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "images of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 469, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 469, + 141 + ], + "score": 1.0, + "content": "training set as a held-out validation set for hyper-parameter tuning, as is common practice.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 146, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "For our experiments, we used a densely connected convolutional neural network, which represents", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 157, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 505, + 168 + ], + "score": 1.0, + "content": "the current state-of-the-art on CIFAR-10 (Huang et al., 2016a). We use the small configuration from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "(Huang et al., 2016a), which consists of 40-layers, with a growth rate of 12. All models were trained", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "for 300 epochs, with a batch-size of 50 and a learning rate 0.1. The learning rate was reduced by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "a factor of 10 at 150 and 225 epochs. We present results for training without data-augmentation.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "We found that the confidence penalty did not lead to improved performance when training with data", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 451, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 451, + 224 + ], + "score": 1.0, + "content": "augmentation, however neither did other regularization techniques, including dropout.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "For our final test scores, we trained on the entire training set. For label smoothing, we tried smooth-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "ing parameter values of [0.05, 0.1, 0.2, 0.3, 0.4, 0.5], and found 0.1 to work best. For the confidence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "penalty, we performed a grid search over confidence penalty weight values of [0.1, 0.25, 0.5, 1.0,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 361, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 361, + 273 + ], + "score": 1.0, + "content": "1.5] and found a confidence penalty weight of 0.1 to work best.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "table", + "bbox": [ + 109, + 294, + 502, + 422 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 294, + 502, + 422 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 294, + 502, + 422 + ], + "spans": [ + { + "bbox": [ + 109, + 294, + 502, + 422 + ], + "score": 0.98, + "html": "
ModelLayersParametersTest
He et al. (2015) - Residual CNN1101.7M13.63%
Huang et al. (2016b) - Stochastic Depth Residual CNN1101.7M11.66%
Larsson et al. (2016) - Fractal CNN2138.6M10.18%
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Densely Connected CNN (Dropout)401.0M7.04%
Densely Connected CNN (Dropout + Label Smoothing)401.0M6.89%
Densely Connected CNN (Dropout + Confidence Penalty)401.0M6.77%
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ModelLayersParametersTest
He et al. (2015) - Residual CNN1101.7M13.63%
Huang et al. (2016b) - Stochastic Depth Residual CNN1101.7M11.66%
Larsson et al. (2016) - Fractal CNN2138.6M10.18%
Larsson et al. (2016) - Fractal CNN (Dropout)2138.6M7.33%
Huang et al. (2O16a) - Densely Connected CNN401.0M7.00%
Huang et al. (2O16a) - Densely Connected CNN1007.0M5.77%
Densely Connected CNN (Dropout)401.0M7.04%
Densely Connected CNN (Dropout + Label Smoothing)401.0M6.89%
Densely Connected CNN (Dropout + Confidence Penalty)401.0M6.77%
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Variational dropout (Gal, 2015) applies a fixed dropout mask (stochas-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "tic for each sample) at each time-step, instead of resampling at each time-step as in traditional", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "dropout. Note, that we do not include the variational dropout results that use Monte Carlo (MC)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "model averaging, which achieves lower perplexity on the test set but requires 1000 model evalua-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "tions, which are then averaged. Recurrent highway networks (Zilly et al., 2016) currently represent", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 721, + 271, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 271, + 732 + ], + "score": 1.0, + "content": "the state-of-the-art performance on PTB.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 655, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 133, + 80, + 478, + 219 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 133, + 80, + 478, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 80, + 478, + 219 + ], + "spans": [ + { + "bbox": [ + 133, + 80, + 478, + 219 + ], + "score": 0.982, + "html": "
ModelParametersValidationTest
Zaremba et al. (2014) - Regularized LSTM Gal (2015) - Variational LSTM66M82.278.4
Press& Wolf(2016) - TiedVariationalLSTM66M77.975.2
51M79.675.0
Merity et al. (2016) - Pointer Sentinel LSTM21M72.470.9
Zilly et al. (2016) - Variational RHN32M71.268.5
Zilly et al. (2016) - Tied Variational RHN24M68.166.0
Regularized LSTM (label noise)66M79.777.7
Regularized LSTM (label smoothing)66M78.976.6
Regularized LSTM (unigram smoothing)66M79.176.3
Regularized LSTM (confidence penalty)66M77.874.7
", + "type": "table", + "image_path": "f770da827ec692ae26f593b4ac543bbc8df419d37ab0e3f3a2579dbccbc6456c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 133, + 80, + 478, + 126.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 133, + 126.33333333333334, + 478, + 172.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 133, + 172.66666666666669, + 478, + 219.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 165, + 227, + 444, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 226, + 444, + 239 + ], + "spans": [ + { + "bbox": [ + 166, + 226, + 444, + 239 + ], + "score": 1.0, + "content": "Table 3: Validation and test perplexity for word-level Penn Treebank.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 107, + 259, + 238, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 238, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 238, + 271 + ], + "score": 1.0, + "content": "4.3 MACHINE TRANSLATION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "For machine translation, we evaluated the confidence penalty on the WMT’14 English-to-German", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "translation task using Google’s production-level translation system Wu et al. (2016). The training set", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "consists of 5M sentence pairs, and we used newstest2012 and newtests2013 for validation and new-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "stest2014 for testing. We report tokenized BLEU scores as computed by the multi-bleu.perl", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 324, + 358, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 358, + 338 + ], + "score": 1.0, + "content": "script from the Moses translation machine translation package.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "Our model was an 8-layer sequence-to-sequence model with attention (Bahdanau et al., 2014). The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "first encoder was a bidirectional LSTM, the remaining encoder and decoder layers were unidirec-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "tional LSTMs, and the attention network was a single layer feed-forward network. Each layer had", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "512 units (compared to 1024 in (Wu et al., 2016)). The model was trained using 12 replicas running", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 326, + 398 + ], + "score": 1.0, + "content": "concurrently with asynchronous updates. Dropout of", + "type": "text" + }, + { + "bbox": [ + 326, + 385, + 346, + 396 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "was applied as described in (Zaremba", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "et al., 2014). Optimization used a mix of Adam and SGD with gradient clipping. Unlike (Wu et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "2016), we did not use reinforcement learning to fine-tune our model. We used a beam size of 12", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 419, + 334, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 334, + 430 + ], + "score": 1.0, + "content": "during decoding. For more details, see (Wu et al., 2016).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 352, + 448 + ], + "score": 1.0, + "content": "For label smoothing, we performed a grid search over values", + "type": "text" + }, + { + "bbox": [ + 352, + 435, + 461, + 447 + ], + "score": 0.66, + "content": "[ 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "and found", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "0.1 to work best for both uniform and unigram smoothing. For the confidence penalty, we searched", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "over values of [0.5, 2.5, 4.5] and found a value of 2.5 to work best . For machine translation, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "found label smoothing slightly outperformed confidence penalty. When applied without dropout,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "both lead to an improvement of just over 1 BLEU point (dropout leads to an improvement of just", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "over 2 BLEU points). However, when combined with dropout, the effect of both regularizers was", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 501, + 156, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 156, + 512 + ], + "score": 1.0, + "content": "diminished.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "table", + "bbox": [ + 116, + 522, + 494, + 692 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 116, + 522, + 494, + 692 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 116, + 522, + 494, + 692 + ], + "spans": [ + { + "bbox": [ + 116, + 522, + 494, + 692 + ], + "score": 0.984, + "html": "
ModelParametersValidationTest
Buck et al. (2014) - PBMT20.7
Cho et al. (2015) - RNNSearch16.9
Zhou et al. (2016) - Deep-Att20.6
Luong et al. (2015) - P-Attention164M20.9
Wu et al. (2016) - WPM-16K167M24.4
Wu et al. (2016) - WPM-32K278M24.6
WPM-32K (without dropout)94M22.3321.24
WPM-32K (label smoothing)94M23.8522.42
WPM-32K (confidence penalty)94M23.2522.52
WPM-32K (dropout)94M24.1 ±0.123.41 ± 0.04
WPM-32K (dropout + label smoothing)94M24.3 ± 0.123.52 ±0.03
WPM-32K (dropout + unigram smoothing)94M24.3 ± 0.123.57 ± 0.02
WPM-32K (dropout + confidence penalty)94M24.3 ± 0.123.4 ± 0.1
", + "type": "table", + "image_path": "71d18f0a693e364bc78a3ea27fccfb5df5d5ad28e9fc97b07f84cf2deeb05352.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 116, + 522, + 494, + 578.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 116, + 578.6666666666666, + 494, + 635.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 116, + 635.3333333333333, + 494, + 691.9999999999999 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 107, + 700, + 504, + 723 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "Table 4: Validation and test BLEU for WMT’14 English-to-German. For the last four model con-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 712, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 505, + 724 + ], + "score": 1.0, + "content": "figurations, we report the mean and standard error of the mean (SEM) over 5 random initializations.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "index": 27.25 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 133, + 80, + 478, + 219 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 133, + 80, + 478, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 80, + 478, + 219 + ], + "spans": [ + { + "bbox": [ + 133, + 80, + 478, + 219 + ], + "score": 0.982, + "html": "
ModelParametersValidationTest
Zaremba et al. (2014) - Regularized LSTM Gal (2015) - Variational LSTM66M82.278.4
Press& Wolf(2016) - TiedVariationalLSTM66M77.975.2
51M79.675.0
Merity et al. (2016) - Pointer Sentinel LSTM21M72.470.9
Zilly et al. (2016) - Variational RHN32M71.268.5
Zilly et al. (2016) - Tied Variational RHN24M68.166.0
Regularized LSTM (label noise)66M79.777.7
Regularized LSTM (label smoothing)66M78.976.6
Regularized LSTM (unigram smoothing)66M79.176.3
Regularized LSTM (confidence penalty)66M77.874.7
", + "type": "table", + "image_path": "f770da827ec692ae26f593b4ac543bbc8df419d37ab0e3f3a2579dbccbc6456c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 133, + 80, + 478, + 126.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 133, + 126.33333333333334, + 478, + 172.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 133, + 172.66666666666669, + 478, + 219.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 165, + 227, + 444, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 226, + 444, + 239 + ], + "spans": [ + { + "bbox": [ + 166, + 226, + 444, + 239 + ], + "score": 1.0, + "content": "Table 3: Validation and test perplexity for word-level Penn Treebank.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 107, + 259, + 238, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 238, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 238, + 271 + ], + "score": 1.0, + "content": "4.3 MACHINE TRANSLATION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "For machine translation, we evaluated the confidence penalty on the WMT’14 English-to-German", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "translation task using Google’s production-level translation system Wu et al. (2016). The training set", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "consists of 5M sentence pairs, and we used newstest2012 and newtests2013 for validation and new-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "stest2014 for testing. We report tokenized BLEU scores as computed by the multi-bleu.perl", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 324, + 358, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 358, + 338 + ], + "score": 1.0, + "content": "script from the Moses translation machine translation package.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 281, + 505, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 354 + ], + "score": 1.0, + "content": "Our model was an 8-layer sequence-to-sequence model with attention (Bahdanau et al., 2014). The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "first encoder was a bidirectional LSTM, the remaining encoder and decoder layers were unidirec-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "tional LSTMs, and the attention network was a single layer feed-forward network. Each layer had", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "512 units (compared to 1024 in (Wu et al., 2016)). The model was trained using 12 replicas running", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 326, + 398 + ], + "score": 1.0, + "content": "concurrently with asynchronous updates. Dropout of", + "type": "text" + }, + { + "bbox": [ + 326, + 385, + 346, + 396 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "was applied as described in (Zaremba", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "et al., 2014). Optimization used a mix of Adam and SGD with gradient clipping. Unlike (Wu et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "2016), we did not use reinforcement learning to fine-tune our model. We used a beam size of 12", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 419, + 334, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 334, + 430 + ], + "score": 1.0, + "content": "during decoding. For more details, see (Wu et al., 2016).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 340, + 505, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 352, + 448 + ], + "score": 1.0, + "content": "For label smoothing, we performed a grid search over values", + "type": "text" + }, + { + "bbox": [ + 352, + 435, + 461, + 447 + ], + "score": 0.66, + "content": "[ 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "and found", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "0.1 to work best for both uniform and unigram smoothing. For the confidence penalty, we searched", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "over values of [0.5, 2.5, 4.5] and found a value of 2.5 to work best . For machine translation, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "found label smoothing slightly outperformed confidence penalty. When applied without dropout,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "both lead to an improvement of just over 1 BLEU point (dropout leads to an improvement of just", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "over 2 BLEU points). However, when combined with dropout, the effect of both regularizers was", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 501, + 156, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 156, + 512 + ], + "score": 1.0, + "content": "diminished.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 434, + 506, + 512 + ] + }, + { + "type": "table", + "bbox": [ + 116, + 522, + 494, + 692 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 116, + 522, + 494, + 692 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 116, + 522, + 494, + 692 + ], + "spans": [ + { + "bbox": [ + 116, + 522, + 494, + 692 + ], + "score": 0.984, + "html": "
ModelParametersValidationTest
Buck et al. (2014) - PBMT20.7
Cho et al. (2015) - RNNSearch16.9
Zhou et al. (2016) - Deep-Att20.6
Luong et al. (2015) - P-Attention164M20.9
Wu et al. (2016) - WPM-16K167M24.4
Wu et al. (2016) - WPM-32K278M24.6
WPM-32K (without dropout)94M22.3321.24
WPM-32K (label smoothing)94M23.8522.42
WPM-32K (confidence penalty)94M23.2522.52
WPM-32K (dropout)94M24.1 ±0.123.41 ± 0.04
WPM-32K (dropout + label smoothing)94M24.3 ± 0.123.52 ±0.03
WPM-32K (dropout + unigram smoothing)94M24.3 ± 0.123.57 ± 0.02
WPM-32K (dropout + confidence penalty)94M24.3 ± 0.123.4 ± 0.1
", + "type": "table", + "image_path": "71d18f0a693e364bc78a3ea27fccfb5df5d5ad28e9fc97b07f84cf2deeb05352.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 116, + 522, + 494, + 578.6666666666666 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 116, + 578.6666666666666, + 494, + 635.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 116, + 635.3333333333333, + 494, + 691.9999999999999 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 107, + 700, + 504, + 723 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "Table 4: Validation and test BLEU for WMT’14 English-to-German. For the last four model con-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 712, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 505, + 724 + ], + "score": 1.0, + "content": "figurations, we report the mean and standard error of the mean (SEM) over 5 random initializations.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "index": 27.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 229, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 230, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 230, + 95 + ], + "score": 1.0, + "content": "4.4 SPEECH RECOGNITION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 102, + 171, + 114 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 172, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 172, + 116 + ], + "score": 1.0, + "content": "4.4.1 TIMIT", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 505, + 135 + ], + "score": 1.0, + "content": "In the TIMIT corpus, the training set consists of 3512 utterances, the validation set consists of 184", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "score": 1.0, + "content": "utterances and the test set consists of 192 utterances. All 61 phonemes were used during training", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "score": 1.0, + "content": "and decoding, and during scoring, these 61 phonemes were reduced to 39 to compute the phoneme", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 155, + 176, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 176, + 168 + ], + "score": 1.0, + "content": "error rate (PER).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "As our base model, we used a sequence-to-sequence model with attention. The encoder consisted", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 184, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 195 + ], + "score": 1.0, + "content": "of 3 bidirectional LSTM layers, the decoder consisted of a single unidirectional LSTM layer, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 206 + ], + "score": 1.0, + "content": "the attention network consisted of a single layer feed-forward network. All layers consisted of 256", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 182, + 218 + ], + "score": 1.0, + "content": "units. Dropout of", + "type": "text" + }, + { + "bbox": [ + 182, + 205, + 201, + 216 + ], + "score": 0.86, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "was applied as described in Zaremba et al. (2014). We trained the model", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "with asynchronous SGD with 5 replicas. We used a batch size of 32, a learning rate of 0.01, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 228, + 479, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 479, + 239 + ], + "score": 1.0, + "content": "momentum of 0.9. Gradients were clipped at 5.0. For more details, see Norouzi et al. (2016).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 504, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "For label smoothing, we performed a grid search over values [0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "0.8] and found 0.2 to work best. For the confidence penalty, we performed a grid search over values", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 118, + 279 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 266, + 255, + 278 + ], + "score": 0.49, + "content": "[ 0 . 1 2 5 , 0 . 2 5 , 0 . 5 , 1 . 0 , 2 . 0 , 4 . 0 , 8 . 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "and found a value of 1.0 to work best. Label smoothing led to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 325, + 291 + ], + "score": 1.0, + "content": "an absolute improvement over the dropout baseline of", + "type": "text" + }, + { + "bbox": [ + 325, + 277, + 347, + 288 + ], + "score": 0.86, + "content": "1 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 277, + 506, + 291 + ], + "score": 1.0, + "content": ", while the confidence penalty led to an", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 234, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 208, + 301 + ], + "score": 1.0, + "content": "absolute improvement of", + "type": "text" + }, + { + "bbox": [ + 209, + 288, + 230, + 299 + ], + "score": 0.87, + "content": "1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 288, + 234, + 301 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "table", + "bbox": [ + 132, + 311, + 479, + 427 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 132, + 311, + 479, + 427 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 311, + 479, + 427 + ], + "spans": [ + { + "bbox": [ + 132, + 311, + 479, + 427 + ], + "score": 0.982, + "html": "
ModelParametersValidationTest
Mohamed et al. (2012) - DNN-HMM120.7
Norouzi et al. (2016) - RML6.5M18.019.9
Graves et al. (2006) - CTC6.8M-18.4
Graves et al. (2013) - RNN Transducer4.3M117.7
T6th (2014) - CNN113.916.7
Dropout6.5M21.0 ±0.123.2 ± 0.4
Dropout + Label Smoothing6.5M19.3 ± 0.121.6 ± 0.2
Dropout + Confidence Penalty6.5M19.9 ± 0.222.0± 0.4
", + "type": "table", + "image_path": "d9b388288ec158df1718552a054738bdd87782d32ae1af2e5cecbc8d42af707c.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 132, + 311, + 479, + 349.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 132, + 349.6666666666667, + 479, + 388.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 132, + 388.33333333333337, + 479, + 427.00000000000006 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 435, + 504, + 457 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "Table 5: Validation and test phoneme error rates (PER) for TIMIT. We report the mean and SEM", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 446, + 226, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 226, + 458 + ], + "score": 1.0, + "content": "over 5 random initializations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + } + ], + "index": 19.25 + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 244, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 246, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 246, + 493 + ], + "score": 1.0, + "content": "4.4.2 WALL STREET JOURNAL", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 504, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "For the WSJ corpus we used attention-based sequence-to-sequence networks that directly predicted", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "characters. We used the SI284 subset for training, DEV93 for validation, and EVAL92 for test-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "ing. We used 240-dimensional vectors consisting of 80-bin filterbank features augmented with their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 547 + ], + "score": 1.0, + "content": "deltas and delta-deltas with per-speaker normalized mean and variances computed with Kaldi Povey", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 543, + 465, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 465, + 559 + ], + "score": 1.0, + "content": "et al. (2011). We did not use text-only data or separate language models during decoding.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 572 + ], + "score": 1.0, + "content": "Network architecture details were as follows. The encoder of the network consisted of 4 bidirectional", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "LSTM layers each having 256 units, interleaved with 3 time-subsampling layers, configured to drop", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "every second frame (Bahdanau et al., 2016; Chan et al., 2015). The decoder used a single LSTM", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "layer with 256 units. The attention vectors were computed with a single layer feedforward network", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "having 64 hidden units and the convolutional filters as described in Chorowski et al. (2015). Weights", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 280, + 629 + ], + "score": 1.0, + "content": "were initialized from a uniform distribution", + "type": "text" + }, + { + "bbox": [ + 280, + 616, + 344, + 628 + ], + "score": 0.78, + "content": "[ - 0 . 0 7 5 , 0 . 0 7 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 615, + 479, + 629 + ], + "score": 1.0, + "content": ". All models used weight decay of", + "type": "text" + }, + { + "bbox": [ + 480, + 615, + 501, + 627 + ], + "score": 0.89, + "content": "1 \\bar { 0 ^ { - 6 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 615, + 506, + 629 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "additive Gaussian weight noise with standard deviation 0.075, applied after 20K steps, and were", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "trained for 650K steps. We used the ADAM optimizer asynchronously over 8 GPUs. We used a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 646, + 463, + 663 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 170, + 663 + ], + "score": 1.0, + "content": "learning rate of", + "type": "text" + }, + { + "bbox": [ + 170, + 648, + 191, + 659 + ], + "score": 0.89, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 646, + 284, + 663 + ], + "score": 1.0, + "content": ", which was reduced to", + "type": "text" + }, + { + "bbox": [ + 285, + 649, + 306, + 659 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 646, + 328, + 663 + ], + "score": 1.0, + "content": "after", + "type": "text" + }, + { + "bbox": [ + 328, + 649, + 352, + 659 + ], + "score": 0.41, + "content": "4 0 0 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 646, + 370, + 663 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 648, + 392, + 659 + ], + "score": 0.9, + "content": "1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 646, + 463, + 663 + ], + "score": 1.0, + "content": "after 500K steps.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We tested three methods of increasing the entropy of outputs: the confidence penalty and two vari-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "ants of label smoothing: uniform and unigram. All resulted in improved Word Error Rates (WER),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "however the unigram smoothing resulted in the greatest WER reduction, and we found it to be least", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "sensitive to its hyperparameter (the smoothing value). 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ModelParametersValidationTest
Mohamed et al. (2012) - DNN-HMM120.7
Norouzi et al. (2016) - RML6.5M18.019.9
Graves et al. (2006) - CTC6.8M-18.4
Graves et al. (2013) - RNN Transducer4.3M117.7
T6th (2014) - CNN113.916.7
Dropout6.5M21.0 ±0.123.2 ± 0.4
Dropout + Label Smoothing6.5M19.3 ± 0.121.6 ± 0.2
Dropout + Confidence Penalty6.5M19.9 ± 0.222.0± 0.4
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We did not use text-only data or separate language models during decoding.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 500, + 506, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 572 + ], + "score": 1.0, + "content": "Network architecture details were as follows. 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We used a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 646, + 463, + 663 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 170, + 663 + ], + "score": 1.0, + "content": "learning rate of", + "type": "text" + }, + { + "bbox": [ + 170, + 648, + 191, + 659 + ], + "score": 0.89, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 646, + 284, + 663 + ], + "score": 1.0, + "content": ", which was reduced to", + "type": "text" + }, + { + "bbox": [ + 285, + 649, + 306, + 659 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 646, + 328, + 663 + ], + "score": 1.0, + "content": "after", + "type": "text" + }, + { + "bbox": [ + 328, + 649, + 352, + 659 + ], + "score": 0.41, + "content": "4 0 0 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 646, + 370, + 663 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 648, + 392, + 659 + ], + "score": 0.9, + "content": "1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 646, + 463, + 663 + ], + "score": 1.0, + "content": "after 500K steps.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 562, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We tested three methods of increasing the entropy of outputs: the confidence penalty and two vari-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "ants of label smoothing: uniform and unigram. 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ModelParametersValidationTest
Graves & Jaitly (2014) - CTC26.5M27.3
Bahdanau et al. (2016) - seq2seq Chan et al. (2016) - Baseline5.7M18.6
Chan et al. (2016) - LSD5.1M14.7
5.9M112.9
Baseline6.6M17.914.2
Uniform Label Smoothing6.6M14.711.3
Unigram Label Smoothing6.6M14.0 ± 0.2511.0± 0.35
Confidence Penalty6.6M17.212.7
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ModelParametersValidationTest
Graves & Jaitly (2014) - CTC26.5M27.3
Bahdanau et al. (2016) - seq2seq Chan et al. (2016) - Baseline5.7M18.6
Chan et al. (2016) - LSD5.1M14.7
5.9M112.9
Baseline6.6M17.914.2
Uniform Label Smoothing6.6M14.711.3
Unigram Label Smoothing6.6M14.0 ± 0.2511.0± 0.35
Confidence Penalty6.6M17.212.7
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We find that both the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "confidence penalty and label smoothing improve a wide range of state-of-the-art models, without", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 405, + 258, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 258, + 417 + ], + "score": 1.0, + "content": "the need to modify hyper-parameters.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 349, + 506, + 417 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 429, + 200, + 439 + ], + "lines": [ + { + "bbox": [ + 107, + 430, + 200, + 440 + ], + "spans": [ + { + "bbox": [ + 107, + 430, + 200, + 440 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 447, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "We would like to thank Sergey Ioffe, Alex Alemi and Navdeep Jaitly for helpful discussions. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "would also like to thank Prajit Ramachandran, Barret Zoph, Mohammad Norouzi, and Yonghui", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "Wu for technical help with the various models used in our experiments. 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ModelLayersSizeTest
Wan et al. (2013) - Unregularized28001.40%
Srivastava et al. (2014) - Dropout310241.25%
Wan et al. (2013) - DropConnect28001.20%
Srivastava et al. (2014) - MaxNorm + Dropout281920.95%
Dropout210241.28 ± 0.06%
Label Smoothing210241.23 ± 0.06%
Confidence Penalty210241.17 ± 0.06%
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ModelLayersParametersTest
He et al. (2015) - Residual CNN1101.7M13.63%
Huang et al. (2016b) - Stochastic Depth Residual CNN1101.7M11.66%
Larsson et al. (2016) - Fractal CNN2138.6M10.18%
Larsson et al. (2016) - Fractal CNN (Dropout)2138.6M7.33%
Huang et al. (2O16a) - Densely Connected CNN401.0M7.00%
Huang et al. (2O16a) - Densely Connected CNN1007.0M5.77%
Densely Connected CNN (Dropout)401.0M7.04%
Densely Connected CNN (Dropout + Label Smoothing)401.0M6.89%
Densely Connected CNN (Dropout + Confidence Penalty)401.0M6.77%
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ModelParametersValidationTest
Buck et al. (2014) - PBMT20.7
Cho et al. (2015) - RNNSearch16.9
Zhou et al. (2016) - Deep-Att20.6
Luong et al. (2015) - P-Attention164M20.9
Wu et al. (2016) - WPM-16K167M24.4
Wu et al. (2016) - WPM-32K278M24.6
WPM-32K (without dropout)94M22.3321.24
WPM-32K (label smoothing)94M23.8522.42
WPM-32K (confidence penalty)94M23.2522.52
WPM-32K (dropout)94M24.1 ±0.123.41 ± 0.04
WPM-32K (dropout + label smoothing)94M24.3 ± 0.123.52 ±0.03
WPM-32K (dropout + unigram smoothing)94M24.3 ± 0.123.57 ± 0.02
WPM-32K (dropout + confidence penalty)94M24.3 ± 0.123.4 ± 0.1
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ModelParametersValidationTest
Zaremba et al. (2014) - Regularized LSTM Gal (2015) - Variational LSTM66M82.278.4
Press& Wolf(2016) - TiedVariationalLSTM66M77.975.2
51M79.675.0
Merity et al. (2016) - Pointer Sentinel LSTM21M72.470.9
Zilly et al. (2016) - Variational RHN32M71.268.5
Zilly et al. (2016) - Tied Variational RHN24M68.166.0
Regularized LSTM (label noise)66M79.777.7
Regularized LSTM (label smoothing)66M78.976.6
Regularized LSTM (unigram smoothing)66M79.176.3
Regularized LSTM (confidence penalty)66M77.874.7
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ModelParametersValidationTest
Mohamed et al. (2012) - DNN-HMM120.7
Norouzi et al. (2016) - RML6.5M18.019.9
Graves et al. (2006) - CTC6.8M-18.4
Graves et al. (2013) - RNN Transducer4.3M117.7
T6th (2014) - CNN113.916.7
Dropout6.5M21.0 ±0.123.2 ± 0.4
Dropout + Label Smoothing6.5M19.3 ± 0.121.6 ± 0.2
Dropout + Confidence Penalty6.5M19.9 ± 0.222.0± 0.4
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ModelParametersValidationTest
Graves & Jaitly (2014) - CTC26.5M27.3
Bahdanau et al. (2016) - seq2seq Chan et al. (2016) - Baseline5.7M18.6
Chan et al. (2016) - LSD5.1M14.7
5.9M112.9
Baseline6.6M17.914.2
Uniform Label Smoothing6.6M14.711.3
Unigram Label Smoothing6.6M14.0 ± 0.2511.0± 0.35
Confidence Penalty6.6M17.212.7
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@@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e401c8c8346f2984be5d61042de3d9f21fdee86071838b594eb683280507746 +size 187017 diff --git a/parse/train/Hkg4TI9xl/Hkg4TI9xl_span.pdf b/parse/train/Hkg4TI9xl/Hkg4TI9xl_span.pdf new file mode 100644 index 0000000000000000000000000000000000000000..63f1b12469e2440238c0c7fcfd196d6b335f4d40 --- /dev/null +++ b/parse/train/Hkg4TI9xl/Hkg4TI9xl_span.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7722fa153b2f206bdee269cb227bb96dc1a9acfdaed882cf2ba0ccb88581d33e +size 287317 diff --git a/parse/train/HkgeGeBYDB/HkgeGeBYDB.md b/parse/train/HkgeGeBYDB/HkgeGeBYDB.md new file mode 100644 index 0000000000000000000000000000000000000000..f0f9e132cb80819ca7edc70fa2fc559b9f20bdcb --- /dev/null +++ b/parse/train/HkgeGeBYDB/HkgeGeBYDB.md @@ -0,0 +1,327 @@ +# RAPP: NOVELTY DETECTION WITH RECONSTRUCTION ALONG PROJECTION PATHWAY + +Ki Hyun Kim, Sangwoo Shim, Yongsub Lim, Jongseob Jeon, +Jeongwoo Choi, Byungchan Kim, Andre S. Yoon +MakinaRocks +{khkim, sangwoo, yongsub, jongseob.jeon}@makinarocks.ai +{jeongwoo, kbc8894, andre}@makinarocks.ai + +# ABSTRACT + +We propose RAPP, a new methodology for novelty detection by utilizing hidden space activation values obtained from a deep autoencoder. Precisely, RAPP compares input and its autoencoder reconstruction not only in the input space but also in the hidden spaces. We show that if we feed a reconstructed input to the same autoencoder again, its activated values in a hidden space are equivalent to the corresponding reconstruction in that hidden space given the original input. We devise two metrics aggregating those hidden activated values to quantify the novelty of the input. Through extensive experiments using diverse datasets, we validate that RAPP improves novelty detection performances of autoencoder-based approaches. Besides, we show that RAPP outperforms recent novelty detection methods evaluated on popular benchmarks. + +# 1 INTRODUCTION + +How can we characterize novelty when only normality information is given? Novelty detection is the mechanism to decide whether a data sample is an outlier with respect to the training data. This mechanism is especially useful in situations where a proportion of detection targets is inherently small. Examples are fraudulent transaction detection (Pawar et al., 2014; Porwal & Mukund, 2018), intrusion detection (Lee, 2017; Aoudi et al., 2018), video surveillance (Ravanbakhsh et al., 2017; Xu et al., 2015b), medical diagnosis (Schlegl et al., 2017; Baur et al., 2018) and equipment failure detection (Kuzin & Borovicka, 2016; Zhao et al., 2017; Beghi et al., 2014). Recently, deep autoencoders and their variants have shown outstanding performances in finding compact representations from complex data, and the reconstruction error has been chosen as a popular metric for detecting novelty (An & Cho, 2015; Vasilev et al., 2018). However, this approach has a limitation of measuring reconstruction quality only in an input space, which does not fully utilize hierarchical representations in hidden spaces identified by the deep autoencoder. + +In this paper, we propose RAPP, a new method of detecting novelty samples exploiting hidden activation values in addition to the input values and their autoencoder reconstruction values. While ordinary reconstruction-based methods carry out novelty detection by comparing differences between input data before the input layer and reconstructed data at the output layer, RAPP extends these comparisons to hidden spaces. We first collect a set of hidden activation values by feeding the original input to the autoencoder. Subsequently, we feed the autoencoder reconstructed input to the autoencoder to calculate another set of activation values in the hidden layers. This procedure does not need additional training of the autoencoder. In turn, we quantify the novelty of the input by aggregating these two sets of hidden activation values. To this end, we devise two metrics. The first metric measures the total amount of reconstruction errors in input and hidden spaces. The second metric normalizes the reconstruction errors before summing up. Note that RAPP falls back to the ordinary reconstruction-based method if we only aggregate input values before the input layer and the reconstructed values at the output layer. + +Also, we explain the motivations that facilitated the development of RAPP. We show that activation values in a hidden space obtained by feeding a reconstructed input to the autoencoder are equivalent to the corresponding reconstruction in that hidden space for the original input. We refer the latter quantity as a hidden reconstruction of the input. Note that this is a natural extension of the reconstruction to the hidden space. Unfortunately, we cannot directly compute the hidden reconstruction as in the computation of the ordinary reconstruction because the autoencoder does not impose any correspondence between encoding-decoding pairs of hidden layers during the training. Nevertheless, we show that it can be computed by feeding a reconstructed input to the autoencoder again. Consequently, RAPP incorporates hidden reconstruction errors as well as the ordinary reconstruction error in detecting novelty. + +With extensive experiments, we demonstrate using diverse datasets that our method effectively improves autoencoder-based novelty detection methods. In addition, we show by evaluating on popular benchmark datasets that RAPP outperforms competing methods recently developed. + +Our contributions are summarized as follows. + +• We propose a new novelty detection method by utilizing hidden activation values of an input and its autoencoder reconstruction, and provide aggregation functions for them to quantify novelty of the input. We provide motivation that RAPP extends the reconstruction concept in the input space into the hidden spaces. Precisely, we show that hidden activation values of a reconstructed input are equivalent to the corresponding hidden reconstruction of the original input. We demonstrate that RAPP improves autoencoder-based novelty detection methods in diverse datasets. Moreover, we validate that RAPP outperforms recent novelty detection methods on popular benchmark datasets. + +# 2 RELATED WORK + +Various novelty detection methods with deep neural networks rely on the reconstruction error (Sakurada & Yairi, 2014; Hoffmann, 2007; An & Cho, 2015), because discriminative learning schemes are not suitable for highly class-imbalanced data which is common in practice. Unsupervised and semi-supervised learning approaches handle such imbalance by focusing on the characterization of normality and detecting samples out of the normality. + +Variational Autoencoders (VAE) (Kingma & Welling, 2014) were reported to outperform vanilla autoencoders for novelty detection based on reconstruction error (An & Cho, 2015). To carry out the novelty detection outlined in this approach, an autoencoder needs to be trained only with normal data. The autoencoder encodes the training data, which comprises of only normal data in this case, into a lower-dimensional space and decodes them to the input space. To test novelty, an input value is fed to the autoencoder to produce a reconstructed value and calculate the distance between the input and reconstructed values. This distance is the reconstruction error. A higher reconstruction error means that the input value cannot be encoded onto the lower-dimensional space that represents normal data. Therefore, the input value can be marked as a novelty if its reconstruction error exceeds a certain threshold. + +Instead of autoencoders, Generative Adversarial Networks (GAN) have been also suggested to model a distribution of normal data (Sabokrou et al., 2018; Schlegl et al., 2017). Despite the same purpose of discovering a simpler, lower-dimensional representation, the training criterion for GAN is focusing on the quality of data generation rather than the reconstruction quality of training data. Recently, several pieces of research have combined autoencoders and adversarial learning to meet both criteria in dimension reduction and data generation (Haloui et al., 2018; Pidhorskyi et al., 2018; Zenati et al., 2018). One limitation of these methods based on the ordinary reconstruction error is that they do not exploit all the information available along the projection pathway of deep autoencoders. We will explain how to leverage this information for novelty detection in the next section. + +From the viewpoint of the diversity and ratio of the normal data in novelty detection, there are two cases available. The first case is when a small fraction of classes are normal. This case has been studied in a one-class classification context, and usually evaluated by organizing training data into a collection of samples belonging to a small number of normal classes (Ruff et al., 2018; Perera & Patel, 2018; Sabokrou et al., 2018; Golan & El-Yaniv, 2018). The second case is when a majority of classes are assigned as normal (An & Cho, 2015; Schlegl et al., 2017; Haloui et al., 2018; Zenati et al., 2018). In this case, normal data is more diverse, and the training data is consist of samples of a relatively large number of normal classes: e.g., nine digits of MNIST. One setup does not dominate the other, but depending on applications, either can be more suitable than the other. Different methods may perform differently in both cases. In this paper, we evaluate RAPP and other competing methods with experiments in both setups. + +# 3 PROPOSED METHOD: RAPP + +In this section, we describe the proposed novelty detection method RAPP based on an autoencoder. The main idea is to compare hidden activations of an input and its hidden reconstructions along the projection pathway of the autoencoder. To be precise, we project the input and its autoencoder reconstruction onto the hidden spaces to obtain pairs of activation values, and aggregate them to quantify the novelty of the input. For the aggregation, we present two metrics to measure the total amount of difference within each pair. + +# 3.1 RECONSTRUCTION BASED NOVELTY DETECTION + +An autoencoder $A$ is a neural network consisting of an encoder $g$ and a decoder $f$ , responsible for dimension reduction and its inverse mapping to the original input space, respectively: i.e. $A = f \circ g$ . For the purpose, training the autoencoder aims to minimize difference between its input $x$ and output $A ( x )$ . The space that the encoder $g$ constitutes is called the latent space, and provides more concise representation for data than the input space. + +Due to this unsupervised representation learning property, the autoencoder has been widely used for novelty detection. Specifically, training an autoencoder on normal data samples, novelty of a test sample $x$ is measured by the following reconstruction error $\epsilon$ : + +$$ +\epsilon = \| x - A ( x ) \| _ { 2 } . +$$ + +The test sample $x$ is more likely to be novel as the error $\epsilon ( x )$ becomes larger, because it means that $x$ is farther from the manifold that the autoencoder describes. + +Although this approach has shown promising results in novelty detection, the reconstruction error alone does not fully exploit information provided by a trained autoencoder especially when its architecture is deep. In other words, hierarchical information identified by the deep architecture is being ignored. This is rather unfortunate because hierarchical representation learning is one of the most successfully proven capabilities of deep neural networks. + +To fully leverage that capability, below we will describe the way to exploit hidden spaces to capture the difference between normal and novel samples in more detail. + +# 3.2 RECONSTRUCTION ERROR IN HIDDEN SPACES + +Let $A = f \circ g$ be a trained autoencoder where $g$ and $f$ are an encoder and a decoder, and $\ell$ be the number of hidden layers of $g$ . Namely, $g = g _ { \ell } \circ \cdots \circ g _ { 1 }$ . We define partial computation of $g$ as follows: + +$$ +g _ { : i } = g _ { i } \circ \cdots \circ g _ { 1 } , +$$ + +Let $x$ be an input vector, and $\hat { x }$ be its reconstruction by $A$ : i.e., ${ \hat { x } } = A ( x )$ . In addition to comparing $x$ and $\hat { x }$ in the input space, as the ordinary approach does, we examine them in hidden spaces along a projection pathway of $A$ . More precisely, feeding $x$ and $\hat { x }$ into $A$ , we obtain pairs $( h _ { i } , \hat { h } _ { i } )$ of their hidden representations where + +$$ +\begin{array} { l } { { h _ { i } ( x ) = g _ { : i } ( x ) , } } \\ { { \hat { h } _ { i } ( x ) = g _ { : i } ( \hat { x } ) = g _ { : i } ( A ( x ) ) . } } \end{array} +$$ + +Figure 1a illustrates the procedure of computing $h _ { i }$ and $\hat { h } _ { i }$ . As a result, novelty of the sample $x$ is quantified by aggregating $H ( x ) = \{ ( h _ { i } ( x ) , \hat { h } _ { i } ( x ) ) : 1 \leq i \leq \ell \}$ . + +The overall procedure of RAPP is summarized in Algorithm 1. To clearly state the required variables to construct $H$ , we write the algorithm with the for loop in Lines 3–5, but in practice, all of them + +![](images/2abd94966a342b376723bb1fe17fa897827f189e1454d050cbb0f7ee403f0ce2.jpg) +Figure 1: (a) Computation of $h _ { i } ( x )$ and $\hat { h } _ { i } ( x )$ . The reconstruction $\hat { x }$ is fed to the same autoencoder that produced itself. (b) Motivation of RAPP. The quantity that RAPP computes, the hidden activation of the reconstruction input, is equivalent to the hidden reconstruction of the input. If $\tilde { f } = f$ , computing $\hat { h } _ { 2 } ^ { \prime } ( x ) = \hat { h } _ { 2 } ( x )$ does not require explicitly evaluating $\tilde { f } _ { i }$ but only $g _ { i }$ and $f = { \tilde { f } }$ . + +Algorithm 1: RAPP to compute a novelty score. + +Input : Sample $x$ , trained autoencoder $A = f \circ g$ , the number of layers $\ell$ , and aggregation $s$ . Output: Novelty score $S$ . 1 ${ \hat { x } } = A ( x )$ . 2 $H = \varnothing$ . 3 foreach $i$ in 1 to $\ell$ do 4 $H = H \cup \{ ( g _ { : i } ( x ) , g _ { : i } ( \hat { x } ) ) \}$ . 5 end 6 $S = s ( H )$ . + +can be computed by feed-forwarding one time each of $x$ and $\hat { x }$ to $g$ . Note that RAPP is indeed a generalization of the ordinary reconstruction method with defining $g _ { 0 }$ as the identity function and $s _ { o r d }$ as follows. + +$$ +s _ { o r d } ( H ( x ) ) = \| h _ { 0 } ( x ) - \hat { h } _ { 0 } ( x ) \| _ { 2 } ^ { 2 } , +$$ + +where $h _ { 0 } ( x ) = g _ { 0 } ( x ) = x$ and $\hat { h } _ { 0 } ( x ) = g _ { 0 } ( \hat { x } ) = \hat { x } .$ + +In this paper, we provide two metrics $s _ { S A P }$ and $s N A P$ which more extensively utilize $H$ than $s _ { o r d }$ . Those are especially suited when no prior knowledge exists for the selection of layers to derive a novelty metric, which commonly happens when modeling with deep neural networks. Note that, however, more elaborate metrics can be designed if we have knowledge on or can characterize the spaces. + +# 3.2.1 SIMPLE AGGREGATION ALONG PATHWAY (SAP) + +This is the most straightforward metric that one can define on $H$ . For a data sample $x$ , SAP is defined by summing the square of Euclidean distances for all pairs in $H$ : + +$$ +s _ { S A P } ( x ) = \sum _ { i = 0 } ^ { \ell } \| h _ { i } ( x ) - \hat { h } _ { i } ( x ) \| _ { 2 } ^ { 2 } = \| \mathbf { h } ( x ) - \hat { \mathbf { h } } ( x ) \| _ { 2 } ^ { 2 } , +$$ + +where $\mathbf { h } ( x )$ and $\hat { \mathbf { h } } ( x )$ are the concatenations of $[ h _ { 0 } ( x ) , \cdot \cdot \cdot , h _ { \ell } ( x ) ]$ and $[ \hat { h } _ { 0 } ( x ) ; \cdot \cdot \cdot ; \hat { h } _ { \ell } ( x ) ]$ , respectively. + +# 3.2.2 NORMALIZED AGGREGATION ALONG PATHWAY (NAP) + +Although SAP is intuitive, it does not consider properties of hidden spaces; distance distributions of pairs in $H$ may be different depending on the individual hidden spaces. For instance, the magnitude of distances can depend on layers, or there may exist correlated neurons even across layers which are unintentionally emphasized in SAP. To capture clearer patterns, we propose to normalize the distances via two steps: orthogonalization and scaling. + +Let ${ \bf d } ( x ) = { \bf h } ( x ) - \hat { { \bf h } } ( x )$ ; given a training set $X$ , let $\mathbf { D }$ be a matrix whose $i$ -th row corresponds to $\mathbf { d } ( x _ { i } )$ for $x _ { i } \in X$ , and $\bar { \bf D }$ be the column-wise centered matrix of $\mathbf { D }$ . For the normalization, we compute $\bar { \mathbf { D } } = U \Sigma V ^ { \top }$ , SVD of $\bar { \bf D }$ , to obtain its singular values $\Sigma$ and right singular vectors $V$ . For a given data sample $x$ , we define $s N A P$ as follows: + +$$ +s _ { N A P } ( \boldsymbol { x } ) = \| \left( \mathbf { d } ( \boldsymbol { x } ) - \mu _ { \boldsymbol { X } } \right) ^ { \top } \boldsymbol { V } \boldsymbol { \Sigma } ^ { - 1 } \| _ { 2 } ^ { 2 } , +$$ + +where $\mu _ { X }$ is the column-wise mean of $D$ , and $\mathbf { d } ( x )$ is expressed as a column vector. Note that $s N A P$ is equal to the Mahalanobis distance with the covariance matrix $V \Sigma \Sigma V ^ { \top }$ . Although SVD computation time is quadratic in the number of columns of the target matrix, we observe that its impact is relatively small in practical setups. See Appendix A for more details. + +# 4 MOTIVATION OF RAPP + +One natural question in using the ordinary reconstruction method is as follows: why do we investigate only the input space? Or, why do we not use information in hidden spaces? While the reconstruction error in the input space is extensively employed, any similar concept does not exist in hidden spaces. One reason is that the corresponding encoding and decoding layers are not guaranteed to express the same space: e.g. permuted dimensions. This is because the autoencoder objective does not have any term involving activations from intermediate hidden layers. As a result, $f _ { \ell : i + 1 } ( g ( x ) )$ cannot be considered a reconstruction of $g _ { : i } ( x )$ , except for $i = 0$ with which they become the ordinary reconstruction of and input to an autoencoder, respectively. + +Nevertheless, in this section, we will show that there is an indirect way to compute the hidden reconstruction. Precisely, we will show that $\hat { h } _ { i } ( x ) = g _ { : i } ( A ( x ) )$ is indeed equivalent to a reconstruction of $g _ { : i } ( x )$ . The overall mechanism is depicted in Figure 1b. + +# 4.1 COMPUTATION OF HIDDEN RECONSTRUCTION + +Let $A = f \circ g$ be a trained autoencoder, and $M _ { 0 } = \{ A ( x ) : x \in \mathbb { R } ^ { n } \}$ be the low dimensional manifold that $A$ describes (Pidhorskyi et al., 2018): i.e., + +$$ +\forall x \in M _ { 0 } , x = A ( x ) . +$$ + +Defining $M _ { i } = \{ g _ { : i } ( x ) : x \in M _ { 0 } \}$ , which is the low dimensional image of $M _ { 0 }$ defined by $g _ { : i } , g$ and $f$ restricted on $M _ { 0 }$ and $M _ { \ell }$ , respectively, are inverse functions of each other. + +Quantifying Hidden Reconstruction We first assume that there exists a decoder $\tilde { f } = \tilde { f } _ { 1 } \circ \cdots \circ \tilde { f } _ { \ell }$ such that + +$$ +\begin{array} { r l } & { \forall x \in M _ { \ell } , \ \tilde { f } ( x ) = f ( x ) , } \\ & { \forall a \in M _ { i } , \ a = ( g _ { i } \circ \tilde { f } _ { i } ) ( a ) . } \end{array} +$$ + +The second condition makes $\tilde { f } _ { \ell : i + 1 }$ a proper decoder corresponding to $g _ { i + 1 } .$ , and thus, $\tilde { f }$ enables to define the $i$ -th hidden reconstruction $\hat { h } _ { i } ^ { \prime } ( x )$ as follows: + +$$ +\hat { h } _ { i } ^ { \prime } ( x ) = ( \tilde { f } _ { \ell : i + 1 } \circ g _ { i + 1 : } ) ( h _ { i } ( x ) ) . +$$ + +Finally, we conclude that $\hat { h } _ { i } ^ { \prime } ( x )$ is equal to $\hat { h } _ { i } ( x )$ for $x \in M _ { 0 }$ as follows. + +$$ +\begin{array} { r l } & { \hat { h } _ { i } ^ { \prime } ( x ) = ( \tilde { f } _ { \ell : i + 1 } \circ g _ { i + 1 : } ) ( h _ { i } ( x ) ) = ( \tilde { f } _ { \ell : i + 1 } \circ g ) ( x ) } \\ & { \qquad = ( g _ { : i } \circ \tilde { f } \circ g ) ( x ) } \\ & { \qquad = ( g _ { : i } \circ A ) ( x ) = h _ { i } ( \hat { x } ) = \hat { h } _ { i } ( x ) . } \end{array} +$$ + +where we do not need $\tilde { f } _ { i }$ for computing $\hat { h } _ { i } ^ { \prime } ( x )$ , but only $g _ { i }$ and $f$ . Note that for $x \in M _ { 0 }$ already on the manifold, its $i$ -th hidden reconstruction $\hat { h } _ { i } ^ { \prime } ( x )$ becomes equal to its corresponding hidden input $h _ { i } ( x ) = \hat { h } _ { i } ( x )$ for every $1 \leq i \leq \ell$ : i.e. $h _ { i } ( x ) = \hat { h } _ { i } ^ { \prime } ( x )$ as $x = A ( x )$ . For $x \notin M _ { 0 }$ , its hidden reconstruction $\hat { h } _ { i } ^ { \prime } ( x )$ will differ from the input $h _ { i } ( x )$ . + +Table 1: Description of datasets used in our evaluation. + +
Name# Samples#Features# ClassDomainNovelty Target
MI-F25,286582CNC millingMachine not completed
MI-V23,12558CNC millingWorkpiece out-of-spec
EOPT90.51520Storage systemSystem failures
NASA4,68733AstronomyHazardous asteroids
RARM20.2216２２２２７RoboticsMalfunctions
STL1,94127SteelSurface defects
OTTO61,878939E-commerceTypes of products
SNSR58,5094811Electric CurrentsDefective conditions
MNIST70,00078410Hand written digitsDigits
F-MNIST70,00078410Fashion articlesArticles
+ +Existence of $\tilde { f }$ Since $x = A ( x )$ for $x \in M _ { 0 }$ , $g _ { i }$ and $f _ { i }$ are one-to-one functions from $M _ { i - 1 }$ and $M _ { i }$ , respectively. Let us define $\tilde { f } _ { i } = g _ { i } ^ { - 1 }$ for $M _ { i }$ ; then it also holds $\tilde { f } = g ^ { - 1 }$ for $M _ { \ell }$ . This implies $x = ( { \tilde { f } } \circ g ) ( x )$ for $x \in M _ { 0 }$ , and consequently, $\tilde { f } = f$ on $M _ { \ell }$ . This definition of $\tilde { f } _ { i }$ satisfies the two conditions above, and as discussed, we are able to compute hidden reconstructions given an input $x$ , through computing the $i$ -th hidden activation of the reconstructed input: i.e. $\hat { h } _ { i } ^ { \prime } ( x ) = ( g _ { : i } \circ A ) ( x ) =$ $\hat { h } _ { i } ( x )$ . + +Existence of $\tilde { f }$ with Neural Networks Given $g _ { i }$ , if the symmetric architecture for $\tilde { f } _ { i }$ is used, we may not be able to learn $\tilde { f _ { i } } = g _ { i } ^ { - 1 }$ . Neural networks are, however, highly flexible frameworks in which we can deal with models of arbitrary function forms by adjusting network architecture. This property enables us to design a layer capable of representing $\bar { \tilde { f } _ { i } }$ . For instance, even if $\tilde { f } _ { i }$ is too complicated to be represented with a single fully connected layer, we can still approximate $\tilde { f } _ { i }$ by stacking multiple layers. Hence, given $g _ { i }$ , $\tilde { f } _ { i }$ can be represented by neural networks. + +# 5 EVALUATION + +In this section, we evaluate RAPP in comparison to existing methods. To this end, we tested the methods on several benchmarks and diverse datasets collected from Kaggle and the UCI repository which are suitable for evaluating novelty detection methods. + +# 5.1 DATASETS AND PROBLEM SETUPS + +The datasets from Kaggle and the UCI repository are chosen from problem sets of anomaly detection and multi-class classification, summarized in Table 1. We note that MI-F and MI-V share the same feature matrix, but are considered to be different datasets because their labels normal and abnormal are assigned by different columns: i.e. machine completed and pass visual inspection, respectively. We use these datasets to compare RAPP with standard autoencoder-based methods described in Section 5.2. + +To compare RAPP with novelty detection methods in recent literatures, we also use popular benchmark datasets for evaluating deep learning techniques: MNIST (LeCun & Cortes, 2010) and FMNIST (Xiao et al., 2017). For theses datasets, we do not take pre-split training and test sets, but instead merge them for post-processing. + +Novelty detection detects novel patterns by focusing on deviations from model-learned normal patterns. Thus, training sets contain only normal samples and test sets contain both normal and anomaly samples in our evaluation setups. Precisely, if a dataset contains an anomaly label, we assign all samples with that label to the test set for detection. If a dataset does not have any anomaly labels, we consider the following two setups. + +• Multimodal Normality: A single class is chosen to be the novelty class and the remaining classes are assigned as the normal class. This setup is repeated to produce sub-datasets with all possible novelty assignments. For instance, MNIST results in a set of datasets with 10 different novelty classes. + +• Unimodal Normality: In contrast to the multimodal normality setup, we take one class for normality, and the others for novelty. For instance, MNIST results in a set of datasets with 10 different normal classes. + +We applied these two setups to STL, OTTO, SNSR, MNIST, and F-MNIST datasets. + +# 5.2 COMPARISON METHOD + +We compare RAPP and the other methods using Area Under Receiver Operating Characteristic (AUROC). Note that we do not employ thresholding-based metrics such as $F 1$ score because access to abnormal samples is only allowed in testing time. Hence, we focus on the separability of models for novelty with AUROC. + +For the datasets in Table 1, we compare the effectiveness of the reconstruction error, SAP and NAP for three models: Autoencoder (AE), Variational Autoencoder (VAE), Adversarial Autoencoder (AAE) (Makhzani et al., 2016). For the benchmark datasets, recent approaches including OCNN (Chalapathy et al., 2018), GPND (Pidhorskyi et al., 2018), DSVDD (Ruff et al., 2018) and GT (Golan & El-Yaniv, 2018) are available. To obtain the performances of the existing approaches, we downloaded their codes and applied against our problem setups. + +For MNIST and F-MNIST, we create test sets of novelty ratios $3 5 \%$ for the multimodal setup and $50 \%$ for the unimodal setup, where novelty samples in the test sets are randomly selected from given novelty classes. For the other datasets, we take all samples in given novelty classes to create test sets. Note that the expectation value of AUROC is invariant to the novelty ratio. + +# 5.3 IMPLEMENTATION DETAILS + +We use symmetric architecture with fully-connected layers for the three base models, AE, VAE, and AAE. Each encoder and decoder has 10 layers with different bottleneck size. For the Kaggle and UCI datasets, we carry out PCA for each dataset first. The minimum number of principal components that explain at least $90 \%$ of the variance is selected as the bottleneck size of the autoencoders. We set bottleneck size to 20 for benchmark datasets. Leaky-ReLU (Xu et al., 2015a) activation and batch normalization (Ioffe & Szegedy, 2015) layers are appended to all layers except the last layer. + +We train AE, VAE and AAE with Adam optimizer (Kingma & Ba, 2015), and select the model with the lowest validation loss as the best model. For training stability of VAE, 10 Monte Carlo samples were averaged in the reparamterization trick (Kingma & Welling, 2014) to obtain reconstruction from the decoder. In the calculation of SAP and NAP, we excluded reconstructions in the input space for MNIST and F-MNIST. + +# 5.4 RESULTS + +Each AUROC score is obtained by averaging AUROC scores from multiple trials to reduce the random errors in training neural networks: 5 trials for MNIST and F-MNIST, and 20 trials for the other datasets. More results are provided in Appendix: standard deviations in Appendix B, comparison to baselines other than autoencoder variants C, and the effect of varying hidden layers involved in RAPP computation in Appendix D. + +# 5.4.1 COMPARISON WITH BASELINES + +Table 2 summarizes the result of our performance evaluation; the best score for each model is in bold, and the best score for each dataset with an underline. Since STL, OTTO, SNSR, MNIST, and F-MNIST do not have anomaly labels, their scores are averaged over all possible anomaly class assignments. For instance, the AUROC value for OTTO in the unimodal normality setup is the average of 9 AUROC values with different novelty class assignments. + +In Table 2, RAPP shows the highest AUROC scores for most of the cases. If we examine the performance for each dataset, RAPP achieves the best for 10 cases out of 15 (see the underlines). + +Table 2: AUROC of RAPP and the baselines. + +
DatasetAEVAEAAE
ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.7230.7030.7110.7000.6750.7110.7260.7110.724
OTTO0.6170.6160.6650.6300.6310.6490.6170.6180.665
SNSR0.6130.6110.6140.6080.6200.6580.6120.6080.609
MNIST0.8250.8810.8990.9000.9650.9650.8470.9110.929
F-MNIST0.7120.7250.7340.7250.7280.7550.7210.7100.727
Unimodal Normality
MI-F0.6070.6700.7050.5910.5720.6780.6320.6920.704
MI-V0.8970.8980.9070.8450.8330.9030.8950.8910.904
EOPT0.6100.6070.6250.6750.6340.5960.6060.6030.634
NASA0.7190.7020.6920.7380.7140.7190.7120.6950.688
RARM0.6870.6740.6860.5870.5650.6480.6640.6750.675
STL0.8700.8560.8300.8500.8240.7920.8750.8610.830
OTTO0.8290.8290.8320.8310.8350.8330.8280.8280.831
SNSR0.9790.9820.9900.9750.9790.9640.9790.9830.991
MNIST0.9720.9800.9790.9800.9870.9890.9720.9660.977
F-MNIST0.9240.9280.9330.9260.9260.9420.9220.9050.928
+ +Table 3: AUROC on benchmark datasets. + +
DatasetOCNNGPNDDSVDDGTNAPAENAPVAENAPAAE
Multimodal Normality (Novelty] Ratio: 35%)
MNIST F-MNIST0.600 0.6090.501 0.6910.622 0.6100.893 0.7250.899 0.7340.965 0.7550.929 0.727
Unimodal Normality (Novelty ] Ratio: 50%)
MNIST0.9270.9710.9220.9740.9790.9890.977
F-MNIST0.9150.9170.9230.9350.9330.9420.928
+ +# 5.4.2 COMPARISON WITH COMPETITORS + +Table 3 summarizes the comparison of RAPP to recent novelty detection methods. As in Table 2, AUROC values are calculated by averaging results from 10 cases with different anomaly class assignments for both datasets. + +Except for the unimodal F-MNIST setup, NAP outperforms all competing methods regardless of base model choice. Notably, NAP combined with VAE always shows the best performance, which is even higher than that of GT relying on image-specific data transformations for all cases. + +# 6 CONCLUSION + +In this paper, we propose a novelty detection method which utilizes hidden reconstructions along a projection pathway of deep autoencoders. To this end, we extend the concept of reconstruction in the input space to hidden spaces found by an autoencoder and present a tractable way to compute the hidden reconstructions, which requires neither modifying nor retraining the autoencoder. Our experimental results show that the proposed method outperforms other competing methods in terms of AUROC for diverse datasets including popular benchmarks. + +# REFERENCES + +Jinwon An and Sungzoon Cho. Variational autoencoder based anomaly detection using reconstruction probability. SNUDM-TR, Mar 2015. + +Wissam Aoudi, Mikel Iturbe, and Magnus Almgren. Truth will out: Departure-based processlevel detection of stealthy attacks on control systems. In Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security. ACM, 2018. + +Christoph Baur, Benedikt Wiestler, Shadi Albarqouni, and Nassir Navab. Deep autoencoding models for unsupervised anomaly segmentation in brain mr images. MICCAI, 2018. + +Alessandro Beghi, Luca Cecchinato, C Corazzol, Mirco Rampazzo, F Simmini, and Gian Antonio Susto. A one-class svm based tool for machine learning novelty detection in hvac chiller systems. IFAC Proceedings Volumes (IFAC-PapersOnline), 19:1953–1958, 2014. + +Raghavendra Chalapathy, Aditya Krishna Menon, and Sanjay Chawla. Anomaly detection using one-class neural networks. arXiv preprint arXiv:1802.06360, 2018. + +EOPT. https://www.kaggle.com/init-owl/high-storage-system-data-for-energy-optimization. + +F-MNIST. https://github.com/zalandoresearch/fashion-mnist. + +Izhak Golan and Ran El-Yaniv. Deep anomaly detection using geometric transformations. NIPS, 2018. + +Ilyass Haloui, Jayant Sen Gupta, and Vincent Feuillard. Anomaly detection with Wasserstein GAN. arXiv e-prints, art. arXiv:1812.02463, Dec 2018. + +Heiko Hoffmann. Kernel pca for novelty detection. Pattern recognition, 40(3):863–874, 2007. + +Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, volume 37, pp. 448–456, 2015. + +Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR, 2015. + +Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In ICLR, 2014. + +Tomas Kuzin and Tom ´ as Borovicka. Early failure detection for predictive maintenance of sensor ´ parts. In ITAT, 2016. + +Yann LeCun and Corinna Cortes. MNIST handwritten digit database. 2010. URL http://yann. lecun.com/exdb/mnist/. + +D. Lee. Anomaly detection in multivariate non-stationary time series for automatic dbms diagnosis. In ICMLA, 2017. + +Alireza Makhzani, Jonathon Shlens, Navdeep Jaitly, and Ian Goodfellow. Adversarial autoencoders. In International Conference on Learning Representations, 2016. URL http://arxiv.org/ abs/1511.05644. + +MI. https://www.kaggle.com/shasun/tool-wear-detection-in-cnc-mill. + +MNIST. http://yann.lecun.com/exdb/mnist/. + +NASA. https://www.kaggle.com/shrutimehta/nasa-asteroids-classification. + +OTTO. https://www.kaggle.com/c/otto-group-product-classification-challenge. + +Anuradha Vijay Pawar, Prakash N. Kalavadekar, and Ms. Swapnali N. Tambe. A survey on outlier detection techniques for credit card fraud detection. IOSR Journal of Computer Engineering, 16: 44–48, 2014. + +Pramuditha Perera and Vishal M. Patel. Learning deep features for one-class classification. CoRR, abs/1801.05365, 2018. + +Stanislav Pidhorskyi, Ranya Almohsen, and Gianfranco Doretto. Generative probabilistic novelty detection with adversarial autoencoders. In NeurIPS, pp. 6823–6834, 2018. + +Utkarsh Porwal and Smruthi Mukund. Credit card fraud detection in e-commerce: An outlier detection approach. CoRR, abs/1811.02196, 2018. + +RARM. https://github.com/narayave/mh5 anomaly detector. + +M. Ravanbakhsh, M. Nabi, E. Sangineto, L. Marcenaro, C. Regazzoni, and N. Sebe. Abnormal event detection in videos using generative adversarial nets. In 2017 IEEE International Conference on Image Processing (ICIP), 2017. + +Lukas Ruff, Robert Vandermeulen, Nico Goernitz, Lucas Deecke, Shoaib Ahmed Siddiqui, Alexander Binder, Emmanuel Muller, and Marius Kloft. Deep one-class classification. In ¨ ICML, 2018. + +Mohammad Sabokrou, Mohammad Khalooei, Mahmood Fathy, and Ehsan Adeli. Adversarially learned one-class classifier for novelty detection. In CVPR, pp. 3379–3388, 2018. + +Mayu Sakurada and Takehisa Yairi. Anomaly detection using autoencoders with nonlinear dimensionality reduction. In MLSDA, 2014. + +Thomas Schlegl, Philipp Seebock, Sebastian M. Waldstein, Ursula Schmidt-Erfurth, and Georg ¨ Langs. Unsupervised anomaly detection with generative adversarial networks to guide marker discovery. In IPMI, 2017. + +SNSR. https://archive.ics.uci.edu/ml/datasets/dataset+for+sensorless+drive+diagnosis. + +STL. https://www.kaggle.com/uciml/faulty-steel-plates. + +Aleksei Vasilev, Vladimir Golkov, Marc Meissner, Ilona Lipp, Eleonora Sgarlata, Valentina Tomassini, Derek K. Jones, and Daniel Cremers. q-Space Novelty Detection with Variational Autoencoders. arXiv, Jun 2018. + +Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. CoRR, abs/1708.07747, 2017. URL http://arxiv.org/ abs/1708.07747. + +Bing Xu, Naiyan Wang, Tianqi Chen, and Mu Li. Empirical evaluation of rectified activations in convolutional network. arXiv, abs/1505.00853, 2015a. + +Dan Xu, Elisa Ricci, Yan Yan, Jingkuan Song, and Nicu Sebe. Learning deep representations of appearance and motion for anomalous event detection. In BMVC, 2015b. + +Houssam Zenati, Chuan Sheng Foo, Bruno Lecouat, Gaurav Manek, and Vijay Ramaseshan Chandrasekhar. Efficient GAN-based anomaly detection. arXiv, abs/1802.06222, 2018. + +Pushe Zhao, Masaru Kurihara, Junichi Tanaka, Tojiro Noda, Shigeyoshi Chikuma, and Tadashi Suzuki. Advanced correlation-based anomaly detection method for predictive maintenance. In ICPHM, 2017. + +# A SVD COMPUTATION TIME + +We compare running times of training an autoencoder and computing SVD for NAP. We choose two packages for the SVD computation: Pytorch SVD and fbpca provided in https://fbpca. readthedocs.io/en/latest/. + +Since the time complexity of SVD is linear in the number of data samples1, we mainly focus on the performance of SVD with varying the number of columns of the input matrix that SVD is applied. To obtain variable sizes of the columns, we vary the depth and bottleneck size of autoencoders. + +The result is shown below. Notably, Pytorch SVD utilizing GPU is at least $4 7 \mathbf { x }$ faster than training neural networks. Even, fbpca running only on CPU achieves at least $2 . 4 \mathbf { x }$ speedup. + +![](images/c3d2c9664c6a9e393392b5f66dac8b7d4c34056cddbceceac73865babee145a3.jpg) +The detailed setups to obtain the matrices for the experiment are given in the table below: + +
MNISTOTTOSNSR
DepthBottleneck SizeDepthBottleneck SizeDepthBottleneck Size
12010020402090
21810018401890
31610016401690
41410014401490
51210012401290
61010010401090
78100840890
86100640690
94100440490
102100240290
11280２２230270
1226020250
1324010230
14220210
+ +# B STANDARD DEVIATIONS OF EXPERIMENTAL RESULTS + +We provide the standard deviations of the result in Table 2. Given a dataset and a setup, multimodal or unimodal, AUROC values are first averaged over multiple cases with different novelty class assignments; then, the standard deviations are calculated for those averaged values over multiple trials. The number of cases and the number of trials depend on datasets, and we refer to Section 5 for more details. + +
AEVAEAAE
Dataset ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.723 (0.005)0.703 (0.007)0.711 (0.006)0.700 (0.012)0.675 (0.018)0.711 (0.012)0.726 (0.009)0.711 (0.011)0.724 (0.016)
OTTO0.617 (0.002)0.616 (0.001)0.665 (0.004)0.630 (0.003)0.631 (0.005)0.649 (0.003)0.617 (0.001)0.618 (0.003)0.665 (0.004)
SNSR0.613 (0.002)0.611 (0.003)0.614 (0.004)0.608 (0.001)0.620 (0.002)0.658 (0.003)0.612 (0.005)0.608 (0.005)0.609 (0.005)
MNIST0.825 (0.001)0.881 (0.007)0.899 (0.008)0.900 (0.004)0.965 (0.005)0.965 (0.003)0.847 (0.005)0.911 (0.006)0.929 (0.005)
F-MNIST0.712 (0.001)0.725 (0.004)0.734 (0.004)0.725 (0.003)0.728 (0.002)0.755 (0.006)0.721 (0.001)0.710 (0.005)0.727 (0.005)
Unimodal Normality
MI-F0.607 (0.023)0.670 (0.041)0.705 (0.018)0.591 (0.013)0.572 (0.019)0.678 (0.047)0.632 (0.028)0.692 (0.042)0.704 (0.024)
MI-V0.897 (0.010)0.898 (0.005)0.907 (0.005)0.845 (0.021)0.833 (0.032)0.903 (0.010)0.895 (0.005)0.891 (0.006)0.904 (0.005)
EOPT0.610 (0.015)0.607 (0.014)0.625 (0.008)0.675 (0.017)0.634 (0.007)0.596 (0.002)0.606 (0.011)0.603 (0.013)0.634 (0.010)
NASA0.719 (0.009)0.702 (0.015)0.692 (0.016)0.738 (0.009)0.714 (0.015)0.719 (0.009)0.712 (0.016)0.695 (0.027)0.688 (0.024)
RARM0.687 (0.038)0.674 (0.031)0.686 (0.030)0.587 (0.021)0.565 (0.029)0.648 (0.022)0.664 (0.015)0.675 (0.023)0.675 (0.040)
STL0.870 (0.007)0.856 (0.007)0.830 (0.007)0.850 (0.006)0.824 (0.003)0.792 (0.034)0.875 (0.003)0.861 (0.004)0.830 (0.008)
OTTO0.829 (0.001)0.829 (0.001)0.832 (0.001)0.831 (0.002)0.835 (0.003)0.833 (0.004)0.828 (0.002)0.828 (0.001)0.831 (0.002)
SNSR0.979 (0.001)0.982 (0.001)0.990 (0.001)0.975 (0.003)0.979 (0.002)0.964 (0.043)0.979 (0.001)0.983 (0.002)0.997 (0.001)
MNIST0.972 (0.001)0.980 (0.001)0.979 (0.000)0.980 (0.002)0.987 (0.002)0.989 (0.001)0.972 (0.001)0.966 (0.004)0.977 (0.001)
F-MNIST0.924 (0.002)0.928 (0.002)0.933 (0.002)0.926 (0.001)0.926 (0.001)0.942 (0.000)0.922 (0.001)0.905 (0.002)0.928 (0.002)
+ +# C COMPARING RAPP TO BASELINES OTHER THAN NEURAL NETWORKS + +The table below shows the result of comparing RAPP and baselines not based on neural networks. Note that the baselines are run with hyperparameters or hyperparameter search spaces provided in Ruff et al. (2018), while NAP is not further tuned. + +
DatasetOCSVMISOFPCAkPCANAPAENAPvAENAPAAE
Multimodal Normality
STL0.6080.6350.7000.6930.7110.7110.724
OTTO0.6280.5480.5720.6440.6650.6490.665
SNSR0.5240.5260.5360.5800.6140.6580.606
MNIST F-MNIST0.566 0.5710.562 0.6500.698 0.6700.701 0.6760.899 0.7340.965 0.7550.929 0.727
Unimodal Normality
MI-F0.7770.8260.5460.8380.7050.6780.704
MI-V0.8400.8390.8740.8880.9070.9030.904
EOPT0.5970.5950.5520.6930.6250.5960.634
NASA0.5730.6640.7650.6990.6920.7190.688
RARM0.7490.7520.5870.7340.7340.6480.675
STL0.8790.8710.8830.8800.8800.7920.830
OTTO0.8210.6990.7990.8230.8230.8330.831
SNSR0.9590.8930.8940.9700.9700.964
0.991
MNIST0.8940.9190.9030.9060.9790.9890.977
F-MNIST0.8840.8540.9500.9430.9330.9420.928
+ +# D NAP PERFORMANCE OVER INCREASING INVOLVED HIDDEN LAYERS + +We investigate the performance of NAP while increasing the number of hidden layers involved in the NAP computation. Specifically, we consider two ways for the increment: 1) adding hidden layers one by one from the input layer (forward addition), and 2) adding hidden layers one by one from the bottleneck layer (backward addition). Experimental results on three datasets are shown below. For most cases, more hidden layers tend to result in higher performance. The values are obtained from one trial without averaging results from multiple trials. + +![](images/64d9367ef1275abc2f0f52dc5a0903a026d55a84ee4e2adb24f3acdba0d3bb08.jpg) \ No newline at end of file diff --git a/parse/train/HkgeGeBYDB/HkgeGeBYDB_content_list.json b/parse/train/HkgeGeBYDB/HkgeGeBYDB_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..89082dcd65c949a1ca7cbfa4ebfe59e7a4b06c7d --- /dev/null +++ b/parse/train/HkgeGeBYDB/HkgeGeBYDB_content_list.json @@ -0,0 +1,1743 @@ +[ + { + "type": "text", + "text": "RAPP: NOVELTY DETECTION WITH RECONSTRUCTION ALONG PROJECTION PATHWAY ", + "text_level": 1, + "bbox": [ + 176, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ki Hyun Kim, Sangwoo Shim, Yongsub Lim, Jongseob Jeon, \nJeongwoo Choi, Byungchan Kim, Andre S. Yoon \nMakinaRocks \n{khkim, sangwoo, yongsub, jongseob.jeon}@makinarocks.ai \n{jeongwoo, kbc8894, andre}@makinarocks.ai ", + "bbox": [ + 181, + 169, + 718, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose RAPP, a new methodology for novelty detection by utilizing hidden space activation values obtained from a deep autoencoder. Precisely, RAPP compares input and its autoencoder reconstruction not only in the input space but also in the hidden spaces. We show that if we feed a reconstructed input to the same autoencoder again, its activated values in a hidden space are equivalent to the corresponding reconstruction in that hidden space given the original input. We devise two metrics aggregating those hidden activated values to quantify the novelty of the input. Through extensive experiments using diverse datasets, we validate that RAPP improves novelty detection performances of autoencoder-based approaches. Besides, we show that RAPP outperforms recent novelty detection methods evaluated on popular benchmarks. ", + "bbox": [ + 233, + 308, + 764, + 460 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 489, + 336, + 505 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "How can we characterize novelty when only normality information is given? Novelty detection is the mechanism to decide whether a data sample is an outlier with respect to the training data. This mechanism is especially useful in situations where a proportion of detection targets is inherently small. Examples are fraudulent transaction detection (Pawar et al., 2014; Porwal & Mukund, 2018), intrusion detection (Lee, 2017; Aoudi et al., 2018), video surveillance (Ravanbakhsh et al., 2017; Xu et al., 2015b), medical diagnosis (Schlegl et al., 2017; Baur et al., 2018) and equipment failure detection (Kuzin & Borovicka, 2016; Zhao et al., 2017; Beghi et al., 2014). Recently, deep autoencoders and their variants have shown outstanding performances in finding compact representations from complex data, and the reconstruction error has been chosen as a popular metric for detecting novelty (An & Cho, 2015; Vasilev et al., 2018). However, this approach has a limitation of measuring reconstruction quality only in an input space, which does not fully utilize hierarchical representations in hidden spaces identified by the deep autoencoder. ", + "bbox": [ + 174, + 520, + 825, + 686 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we propose RAPP, a new method of detecting novelty samples exploiting hidden activation values in addition to the input values and their autoencoder reconstruction values. While ordinary reconstruction-based methods carry out novelty detection by comparing differences between input data before the input layer and reconstructed data at the output layer, RAPP extends these comparisons to hidden spaces. We first collect a set of hidden activation values by feeding the original input to the autoencoder. Subsequently, we feed the autoencoder reconstructed input to the autoencoder to calculate another set of activation values in the hidden layers. This procedure does not need additional training of the autoencoder. In turn, we quantify the novelty of the input by aggregating these two sets of hidden activation values. To this end, we devise two metrics. The first metric measures the total amount of reconstruction errors in input and hidden spaces. The second metric normalizes the reconstruction errors before summing up. Note that RAPP falls back to the ordinary reconstruction-based method if we only aggregate input values before the input layer and the reconstructed values at the output layer. ", + "bbox": [ + 174, + 694, + 825, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Also, we explain the motivations that facilitated the development of RAPP. We show that activation values in a hidden space obtained by feeding a reconstructed input to the autoencoder are equivalent to the corresponding reconstruction in that hidden space for the original input. We refer the latter quantity as a hidden reconstruction of the input. Note that this is a natural extension of the reconstruction to the hidden space. Unfortunately, we cannot directly compute the hidden reconstruction as in the computation of the ordinary reconstruction because the autoencoder does not impose any correspondence between encoding-decoding pairs of hidden layers during the training. Nevertheless, we show that it can be computed by feeding a reconstructed input to the autoencoder again. Consequently, RAPP incorporates hidden reconstruction errors as well as the ordinary reconstruction error in detecting novelty. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 202 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "With extensive experiments, we demonstrate using diverse datasets that our method effectively improves autoencoder-based novelty detection methods. In addition, we show by evaluating on popular benchmark datasets that RAPP outperforms competing methods recently developed. ", + "bbox": [ + 176, + 208, + 825, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions are summarized as follows. ", + "bbox": [ + 174, + 257, + 472, + 271 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose a new novelty detection method by utilizing hidden activation values of an input and its autoencoder reconstruction, and provide aggregation functions for them to quantify novelty of the input. We provide motivation that RAPP extends the reconstruction concept in the input space into the hidden spaces. Precisely, we show that hidden activation values of a reconstructed input are equivalent to the corresponding hidden reconstruction of the original input. We demonstrate that RAPP improves autoencoder-based novelty detection methods in diverse datasets. Moreover, we validate that RAPP outperforms recent novelty detection methods on popular benchmark datasets. ", + "bbox": [ + 215, + 277, + 825, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 438, + 344, + 455 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Various novelty detection methods with deep neural networks rely on the reconstruction error (Sakurada & Yairi, 2014; Hoffmann, 2007; An & Cho, 2015), because discriminative learning schemes are not suitable for highly class-imbalanced data which is common in practice. Unsupervised and semi-supervised learning approaches handle such imbalance by focusing on the characterization of normality and detecting samples out of the normality. ", + "bbox": [ + 174, + 470, + 825, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Variational Autoencoders (VAE) (Kingma & Welling, 2014) were reported to outperform vanilla autoencoders for novelty detection based on reconstruction error (An & Cho, 2015). To carry out the novelty detection outlined in this approach, an autoencoder needs to be trained only with normal data. The autoencoder encodes the training data, which comprises of only normal data in this case, into a lower-dimensional space and decodes them to the input space. To test novelty, an input value is fed to the autoencoder to produce a reconstructed value and calculate the distance between the input and reconstructed values. This distance is the reconstruction error. A higher reconstruction error means that the input value cannot be encoded onto the lower-dimensional space that represents normal data. Therefore, the input value can be marked as a novelty if its reconstruction error exceeds a certain threshold. ", + "bbox": [ + 174, + 547, + 825, + 686 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Instead of autoencoders, Generative Adversarial Networks (GAN) have been also suggested to model a distribution of normal data (Sabokrou et al., 2018; Schlegl et al., 2017). Despite the same purpose of discovering a simpler, lower-dimensional representation, the training criterion for GAN is focusing on the quality of data generation rather than the reconstruction quality of training data. Recently, several pieces of research have combined autoencoders and adversarial learning to meet both criteria in dimension reduction and data generation (Haloui et al., 2018; Pidhorskyi et al., 2018; Zenati et al., 2018). One limitation of these methods based on the ordinary reconstruction error is that they do not exploit all the information available along the projection pathway of deep autoencoders. We will explain how to leverage this information for novelty detection in the next section. ", + "bbox": [ + 173, + 694, + 825, + 819 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "From the viewpoint of the diversity and ratio of the normal data in novelty detection, there are two cases available. The first case is when a small fraction of classes are normal. This case has been studied in a one-class classification context, and usually evaluated by organizing training data into a collection of samples belonging to a small number of normal classes (Ruff et al., 2018; Perera & Patel, 2018; Sabokrou et al., 2018; Golan & El-Yaniv, 2018). The second case is when a majority of classes are assigned as normal (An & Cho, 2015; Schlegl et al., 2017; Haloui et al., 2018; Zenati et al., 2018). In this case, normal data is more diverse, and the training data is consist of samples of a relatively large number of normal classes: e.g., nine digits of MNIST. One setup does not dominate the other, but depending on applications, either can be more suitable than the other. Different methods may perform differently in both cases. In this paper, we evaluate RAPP and other competing methods with experiments in both setups. ", + "bbox": [ + 173, + 825, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PROPOSED METHOD: RAPP ", + "text_level": 1, + "bbox": [ + 176, + 179, + 437, + 195 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we describe the proposed novelty detection method RAPP based on an autoencoder. The main idea is to compare hidden activations of an input and its hidden reconstructions along the projection pathway of the autoencoder. To be precise, we project the input and its autoencoder reconstruction onto the hidden spaces to obtain pairs of activation values, and aggregate them to quantify the novelty of the input. For the aggregation, we present two metrics to measure the total amount of difference within each pair. ", + "bbox": [ + 174, + 210, + 825, + 295 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 RECONSTRUCTION BASED NOVELTY DETECTION", + "text_level": 1, + "bbox": [ + 174, + 310, + 553, + 325 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "An autoencoder $A$ is a neural network consisting of an encoder $g$ and a decoder $f$ , responsible for dimension reduction and its inverse mapping to the original input space, respectively: i.e. $A = f \\circ g$ . For the purpose, training the autoencoder aims to minimize difference between its input $x$ and output $A ( x )$ . The space that the encoder $g$ constitutes is called the latent space, and provides more concise representation for data than the input space. ", + "bbox": [ + 174, + 337, + 825, + 407 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Due to this unsupervised representation learning property, the autoencoder has been widely used for novelty detection. Specifically, training an autoencoder on normal data samples, novelty of a test sample $x$ is measured by the following reconstruction error $\\epsilon$ : ", + "bbox": [ + 174, + 414, + 823, + 457 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9d16e167e2c466a3ffb5d0b3ce51ce3f27162a7795b1ce18652980a4c2826ce8.jpg", + "text": "$$\n\\epsilon = \\| x - A ( x ) \\| _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 437, + 462, + 560, + 478 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The test sample $x$ is more likely to be novel as the error $\\epsilon ( x )$ becomes larger, because it means that $x$ is farther from the manifold that the autoencoder describes. ", + "bbox": [ + 173, + 483, + 823, + 512 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Although this approach has shown promising results in novelty detection, the reconstruction error alone does not fully exploit information provided by a trained autoencoder especially when its architecture is deep. In other words, hierarchical information identified by the deep architecture is being ignored. This is rather unfortunate because hierarchical representation learning is one of the most successfully proven capabilities of deep neural networks. ", + "bbox": [ + 174, + 518, + 825, + 589 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To fully leverage that capability, below we will describe the way to exploit hidden spaces to capture the difference between normal and novel samples in more detail. ", + "bbox": [ + 174, + 594, + 823, + 625 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 RECONSTRUCTION ERROR IN HIDDEN SPACES", + "text_level": 1, + "bbox": [ + 174, + 640, + 535, + 655 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $A = f \\circ g$ be a trained autoencoder where $g$ and $f$ are an encoder and a decoder, and $\\ell$ be the number of hidden layers of $g$ . Namely, $g = g _ { \\ell } \\circ \\cdots \\circ g _ { 1 }$ . We define partial computation of $g$ as follows: ", + "bbox": [ + 174, + 666, + 825, + 708 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b53668f9fecf79c8fc1176ab593249ecf2e0943c923a0ac01c3b866017198840.jpg", + "text": "$$\ng _ { : i } = g _ { i } \\circ \\cdots \\circ g _ { 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 436, + 710, + 560, + 724 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $x$ be an input vector, and $\\hat { x }$ be its reconstruction by $A$ : i.e., ${ \\hat { x } } = A ( x )$ . In addition to comparing $x$ and $\\hat { x }$ in the input space, as the ordinary approach does, we examine them in hidden spaces along a projection pathway of $A$ . More precisely, feeding $x$ and $\\hat { x }$ into $A$ , we obtain pairs $( h _ { i } , \\hat { h } _ { i } )$ of their hidden representations where ", + "bbox": [ + 173, + 746, + 825, + 805 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dbd24823ecb8e3a8fc96e6417a46818428c2dc3c886d8a290910591f86ebec6e.jpg", + "text": "$$\n\\begin{array} { l } { { h _ { i } ( x ) = g _ { : i } ( x ) , } } \\\\ { { \\hat { h } _ { i } ( x ) = g _ { : i } ( \\hat { x } ) = g _ { : i } ( A ( x ) ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 401, + 809, + 594, + 851 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1a illustrates the procedure of computing $h _ { i }$ and $\\hat { h } _ { i }$ . As a result, novelty of the sample $x$ is quantified by aggregating $H ( x ) = \\{ ( h _ { i } ( x ) , \\hat { h } _ { i } ( x ) ) : 1 \\leq i \\leq \\ell \\}$ . ", + "bbox": [ + 173, + 857, + 820, + 890 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The overall procedure of RAPP is summarized in Algorithm 1. To clearly state the required variables to construct $H$ , we write the algorithm with the for loop in Lines 3–5, but in practice, all of them ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/2abd94966a342b376723bb1fe17fa897827f189e1454d050cbb0f7ee403f0ce2.jpg", + "image_caption": [ + "Figure 1: (a) Computation of $h _ { i } ( x )$ and $\\hat { h } _ { i } ( x )$ . The reconstruction $\\hat { x }$ is fed to the same autoencoder that produced itself. (b) Motivation of RAPP. The quantity that RAPP computes, the hidden activation of the reconstruction input, is equivalent to the hidden reconstruction of the input. If $\\tilde { f } = f$ , computing $\\hat { h } _ { 2 } ^ { \\prime } ( x ) = \\hat { h } _ { 2 } ( x )$ does not require explicitly evaluating $\\tilde { f } _ { i }$ but only $g _ { i }$ and $f = { \\tilde { f } }$ . " + ], + "image_footnote": [], + "bbox": [ + 184, + 116, + 812, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1: RAPP to compute a novelty score. ", + "bbox": [ + 174, + 375, + 495, + 390 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Input : Sample $x$ , trained autoencoder $A = f \\circ g$ , the number of layers $\\ell$ , and aggregation $s$ . Output: Novelty score $S$ . 1 ${ \\hat { x } } = A ( x )$ . 2 $H = \\varnothing$ . 3 foreach $i$ in 1 to $\\ell$ do 4 $H = H \\cup \\{ ( g _ { : i } ( x ) , g _ { : i } ( \\hat { x } ) ) \\}$ . 5 end 6 $S = s ( H )$ . ", + "bbox": [ + 161, + 392, + 803, + 508 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "can be computed by feed-forwarding one time each of $x$ and $\\hat { x }$ to $g$ . Note that RAPP is indeed a generalization of the ordinary reconstruction method with defining $g _ { 0 }$ as the identity function and $s _ { o r d }$ as follows. ", + "bbox": [ + 173, + 537, + 825, + 579 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/97a2ba07366c0943f21eb0ad12b5c6ea2ec03c617fd4e785acb973c39d034b64.jpg", + "text": "$$\ns _ { o r d } ( H ( x ) ) = \\| h _ { 0 } ( x ) - \\hat { h } _ { 0 } ( x ) \\| _ { 2 } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 383, + 578, + 611, + 598 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $h _ { 0 } ( x ) = g _ { 0 } ( x ) = x$ and $\\hat { h } _ { 0 } ( x ) = g _ { 0 } ( \\hat { x } ) = \\hat { x } .$ ", + "bbox": [ + 173, + 601, + 511, + 618 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this paper, we provide two metrics $s _ { S A P }$ and $s N A P$ which more extensively utilize $H$ than $s _ { o r d }$ . Those are especially suited when no prior knowledge exists for the selection of layers to derive a novelty metric, which commonly happens when modeling with deep neural networks. Note that, however, more elaborate metrics can be designed if we have knowledge on or can characterize the spaces. ", + "bbox": [ + 173, + 625, + 825, + 694 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2.1 SIMPLE AGGREGATION ALONG PATHWAY (SAP) ", + "text_level": 1, + "bbox": [ + 174, + 709, + 563, + 724 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This is the most straightforward metric that one can define on $H$ . For a data sample $x$ , SAP is defined by summing the square of Euclidean distances for all pairs in $H$ : ", + "bbox": [ + 173, + 733, + 823, + 762 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a5a0f577ece8356cc39ffca8a90b40ce6a1d4b7a8a7b38ecf719c4bc3408baf1.jpg", + "text": "$$\ns _ { S A P } ( x ) = \\sum _ { i = 0 } ^ { \\ell } \\| h _ { i } ( x ) - \\hat { h } _ { i } ( x ) \\| _ { 2 } ^ { 2 } = \\| \\mathbf { h } ( x ) - \\hat { \\mathbf { h } } ( x ) \\| _ { 2 } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 318, + 777, + 679, + 821 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbf { h } ( x )$ and $\\hat { \\mathbf { h } } ( x )$ are the concatenations of $[ h _ { 0 } ( x ) , \\cdot \\cdot \\cdot , h _ { \\ell } ( x ) ]$ and $[ \\hat { h } _ { 0 } ( x ) ; \\cdot \\cdot \\cdot ; \\hat { h } _ { \\ell } ( x ) ]$ , respectively. ", + "bbox": [ + 173, + 825, + 823, + 856 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2.2 NORMALIZED AGGREGATION ALONG PATHWAY (NAP) ", + "text_level": 1, + "bbox": [ + 174, + 871, + 609, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Although SAP is intuitive, it does not consider properties of hidden spaces; distance distributions of pairs in $H$ may be different depending on the individual hidden spaces. For instance, the magnitude of distances can depend on layers, or there may exist correlated neurons even across layers which are unintentionally emphasized in SAP. To capture clearer patterns, we propose to normalize the distances via two steps: orthogonalization and scaling. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 147 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let ${ \\bf d } ( x ) = { \\bf h } ( x ) - \\hat { { \\bf h } } ( x )$ ; given a training set $X$ , let $\\mathbf { D }$ be a matrix whose $i$ -th row corresponds to $\\mathbf { d } ( x _ { i } )$ for $x _ { i } \\in X$ , and $\\bar { \\bf D }$ be the column-wise centered matrix of $\\mathbf { D }$ . For the normalization, we compute $\\bar { \\mathbf { D } } = U \\Sigma V ^ { \\top }$ , SVD of $\\bar { \\bf D }$ , to obtain its singular values $\\Sigma$ and right singular vectors $V$ . For a given data sample $x$ , we define $s N A P$ as follows: ", + "bbox": [ + 173, + 154, + 825, + 210 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/180e28c23a5067d5f01b3ee7ca57e5e8dfd1a531e102a550bb83e1c011295111.jpg", + "text": "$$\ns _ { N A P } ( \\boldsymbol { x } ) = \\| \\left( \\mathbf { d } ( \\boldsymbol { x } ) - \\mu _ { \\boldsymbol { X } } \\right) ^ { \\top } \\boldsymbol { V } \\boldsymbol { \\Sigma } ^ { - 1 } \\| _ { 2 } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 212, + 630, + 232 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mu _ { X }$ is the column-wise mean of $D$ , and $\\mathbf { d } ( x )$ is expressed as a column vector. Note that $s N A P$ is equal to the Mahalanobis distance with the covariance matrix $V \\Sigma \\Sigma V ^ { \\top }$ . Although SVD computation time is quadratic in the number of columns of the target matrix, we observe that its impact is relatively small in practical setups. See Appendix A for more details. ", + "bbox": [ + 174, + 239, + 825, + 296 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 MOTIVATION OF RAPP ", + "text_level": 1, + "bbox": [ + 176, + 315, + 397, + 332 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "One natural question in using the ordinary reconstruction method is as follows: why do we investigate only the input space? Or, why do we not use information in hidden spaces? While the reconstruction error in the input space is extensively employed, any similar concept does not exist in hidden spaces. One reason is that the corresponding encoding and decoding layers are not guaranteed to express the same space: e.g. permuted dimensions. This is because the autoencoder objective does not have any term involving activations from intermediate hidden layers. As a result, $f _ { \\ell : i + 1 } ( g ( x ) )$ cannot be considered a reconstruction of $g _ { : i } ( x )$ , except for $i = 0$ with which they become the ordinary reconstruction of and input to an autoencoder, respectively. ", + "bbox": [ + 173, + 344, + 825, + 458 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Nevertheless, in this section, we will show that there is an indirect way to compute the hidden reconstruction. Precisely, we will show that $\\hat { h } _ { i } ( x ) = g _ { : i } ( A ( x ) )$ is indeed equivalent to a reconstruction of $g _ { : i } ( x )$ . The overall mechanism is depicted in Figure 1b. ", + "bbox": [ + 174, + 463, + 825, + 510 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 COMPUTATION OF HIDDEN RECONSTRUCTION", + "text_level": 1, + "bbox": [ + 174, + 523, + 537, + 539 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let $A = f \\circ g$ be a trained autoencoder, and $M _ { 0 } = \\{ A ( x ) : x \\in \\mathbb { R } ^ { n } \\}$ be the low dimensional manifold that $A$ describes (Pidhorskyi et al., 2018): i.e., ", + "bbox": [ + 174, + 549, + 823, + 579 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/dd45baeeb06bcb3d0e56a2a978ac5bc79bf2441a4c307e4ba2dbf55c588f3c43.jpg", + "text": "$$\n\\forall x \\in M _ { 0 } , x = A ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 423, + 580, + 571, + 597 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Defining $M _ { i } = \\{ g _ { : i } ( x ) : x \\in M _ { 0 } \\}$ , which is the low dimensional image of $M _ { 0 }$ defined by $g _ { : i } , g$ and $f$ restricted on $M _ { 0 }$ and $M _ { \\ell }$ , respectively, are inverse functions of each other. ", + "bbox": [ + 174, + 598, + 823, + 627 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Quantifying Hidden Reconstruction We first assume that there exists a decoder $\\tilde { f } = \\tilde { f } _ { 1 } \\circ \\cdots \\circ \\tilde { f } _ { \\ell }$ such that ", + "bbox": [ + 174, + 640, + 823, + 670 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8cb2792c949fae05c516ac93a2af0889d251c0538037b6047088fedb35d56e51.jpg", + "text": "$$\n\\begin{array} { r l } & { \\forall x \\in M _ { \\ell } , \\ \\tilde { f } ( x ) = f ( x ) , } \\\\ & { \\forall a \\in M _ { i } , \\ a = ( g _ { i } \\circ \\tilde { f } _ { i } ) ( a ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 403, + 669, + 593, + 712 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The second condition makes $\\tilde { f } _ { \\ell : i + 1 }$ a proper decoder corresponding to $g _ { i + 1 } .$ , and thus, $\\tilde { f }$ enables to define the $i$ -th hidden reconstruction $\\hat { h } _ { i } ^ { \\prime } ( x )$ as follows: ", + "bbox": [ + 174, + 713, + 825, + 744 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a7cce2f0e67cc44ff31f9431dbabb5492eebcfd346ffbb68345b73e67dc118d3.jpg", + "text": "$$\n\\hat { h } _ { i } ^ { \\prime } ( x ) = ( \\tilde { f } _ { \\ell : i + 1 } \\circ g _ { i + 1 : } ) ( h _ { i } ( x ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 747, + 607, + 767 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, we conclude that $\\hat { h } _ { i } ^ { \\prime } ( x )$ is equal to $\\hat { h } _ { i } ( x )$ for $x \\in M _ { 0 }$ as follows. ", + "bbox": [ + 174, + 775, + 647, + 792 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a4dfec90b4092152420b11f1b28ee956d641c65474a5acf204419873543da388.jpg", + "text": "$$\n\\begin{array} { r l } & { \\hat { h } _ { i } ^ { \\prime } ( x ) = ( \\tilde { f } _ { \\ell : i + 1 } \\circ g _ { i + 1 : } ) ( h _ { i } ( x ) ) = ( \\tilde { f } _ { \\ell : i + 1 } \\circ g ) ( x ) } \\\\ & { \\qquad = ( g _ { : i } \\circ \\tilde { f } \\circ g ) ( x ) } \\\\ & { \\qquad = ( g _ { : i } \\circ A ) ( x ) = h _ { i } ( \\hat { x } ) = \\hat { h } _ { i } ( x ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 794, + 666, + 856 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where we do not need $\\tilde { f } _ { i }$ for computing $\\hat { h } _ { i } ^ { \\prime } ( x )$ , but only $g _ { i }$ and $f$ . Note that for $x \\in M _ { 0 }$ already on the manifold, its $i$ -th hidden reconstruction $\\hat { h } _ { i } ^ { \\prime } ( x )$ becomes equal to its corresponding hidden input $h _ { i } ( x ) = \\hat { h } _ { i } ( x )$ for every $1 \\leq i \\leq \\ell$ : i.e. $h _ { i } ( x ) = \\hat { h } _ { i } ^ { \\prime } ( x )$ as $x = A ( x )$ . For $x \\notin M _ { 0 }$ , its hidden reconstruction $\\hat { h } _ { i } ^ { \\prime } ( x )$ will differ from the input $h _ { i } ( x )$ . ", + "bbox": [ + 174, + 858, + 825, + 925 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/4c858221b4144c685d885da461aa4956dd2594ac3c5f700063d35dc3f194d365.jpg", + "table_caption": [ + "Table 1: Description of datasets used in our evaluation. " + ], + "table_footnote": [], + "table_body": "
Name# Samples#Features# ClassDomainNovelty Target
MI-F25,286582CNC millingMachine not completed
MI-V23,12558CNC millingWorkpiece out-of-spec
EOPT90.51520Storage systemSystem failures
NASA4,68733AstronomyHazardous asteroids
RARM20.2216２２２２７RoboticsMalfunctions
STL1,94127SteelSurface defects
OTTO61,878939E-commerceTypes of products
SNSR58,5094811Electric CurrentsDefective conditions
MNIST70,00078410Hand written digitsDigits
F-MNIST70,00078410Fashion articlesArticles
", + "bbox": [ + 196, + 119, + 802, + 281 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Existence of $\\tilde { f }$ Since $x = A ( x )$ for $x \\in M _ { 0 }$ , $g _ { i }$ and $f _ { i }$ are one-to-one functions from $M _ { i - 1 }$ and $M _ { i }$ , respectively. Let us define $\\tilde { f } _ { i } = g _ { i } ^ { - 1 }$ for $M _ { i }$ ; then it also holds $\\tilde { f } = g ^ { - 1 }$ for $M _ { \\ell }$ . This implies $x = ( { \\tilde { f } } \\circ g ) ( x )$ for $x \\in M _ { 0 }$ , and consequently, $\\tilde { f } = f$ on $M _ { \\ell }$ . This definition of $\\tilde { f } _ { i }$ satisfies the two conditions above, and as discussed, we are able to compute hidden reconstructions given an input $x$ , through computing the $i$ -th hidden activation of the reconstructed input: i.e. $\\hat { h } _ { i } ^ { \\prime } ( x ) = ( g _ { : i } \\circ A ) ( x ) =$ $\\hat { h } _ { i } ( x )$ . ", + "bbox": [ + 173, + 308, + 825, + 404 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Existence of $\\tilde { f }$ with Neural Networks Given $g _ { i }$ , if the symmetric architecture for $\\tilde { f } _ { i }$ is used, we may not be able to learn $\\tilde { f _ { i } } = g _ { i } ^ { - 1 }$ . Neural networks are, however, highly flexible frameworks in which we can deal with models of arbitrary function forms by adjusting network architecture. This property enables us to design a layer capable of representing $\\bar { \\tilde { f } _ { i } }$ . For instance, even if $\\tilde { f } _ { i }$ is too complicated to be represented with a single fully connected layer, we can still approximate $\\tilde { f } _ { i }$ by stacking multiple layers. Hence, given $g _ { i }$ , $\\tilde { f } _ { i }$ can be represented by neural networks. ", + "bbox": [ + 174, + 421, + 825, + 515 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 536, + 315, + 553 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we evaluate RAPP in comparison to existing methods. To this end, we tested the methods on several benchmarks and diverse datasets collected from Kaggle and the UCI repository which are suitable for evaluating novelty detection methods. ", + "bbox": [ + 174, + 569, + 825, + 612 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 DATASETS AND PROBLEM SETUPS ", + "text_level": 1, + "bbox": [ + 176, + 632, + 450, + 646 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The datasets from Kaggle and the UCI repository are chosen from problem sets of anomaly detection and multi-class classification, summarized in Table 1. We note that MI-F and MI-V share the same feature matrix, but are considered to be different datasets because their labels normal and abnormal are assigned by different columns: i.e. machine completed and pass visual inspection, respectively. We use these datasets to compare RAPP with standard autoencoder-based methods described in Section 5.2. ", + "bbox": [ + 174, + 659, + 825, + 742 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To compare RAPP with novelty detection methods in recent literatures, we also use popular benchmark datasets for evaluating deep learning techniques: MNIST (LeCun & Cortes, 2010) and FMNIST (Xiao et al., 2017). For theses datasets, we do not take pre-split training and test sets, but instead merge them for post-processing. ", + "bbox": [ + 174, + 750, + 825, + 805 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Novelty detection detects novel patterns by focusing on deviations from model-learned normal patterns. Thus, training sets contain only normal samples and test sets contain both normal and anomaly samples in our evaluation setups. Precisely, if a dataset contains an anomaly label, we assign all samples with that label to the test set for detection. If a dataset does not have any anomaly labels, we consider the following two setups. ", + "bbox": [ + 176, + 813, + 823, + 882 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• Multimodal Normality: A single class is chosen to be the novelty class and the remaining classes are assigned as the normal class. This setup is repeated to produce sub-datasets with all possible novelty assignments. For instance, MNIST results in a set of datasets with 10 different novelty classes. ", + "bbox": [ + 215, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 228, + 103, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• Unimodal Normality: In contrast to the multimodal normality setup, we take one class for normality, and the others for novelty. For instance, MNIST results in a set of datasets with 10 different normal classes. ", + "bbox": [ + 217, + 137, + 823, + 179 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We applied these two setups to STL, OTTO, SNSR, MNIST, and F-MNIST datasets. ", + "bbox": [ + 176, + 190, + 727, + 205 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 COMPARISON METHOD ", + "text_level": 1, + "bbox": [ + 176, + 223, + 377, + 238 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We compare RAPP and the other methods using Area Under Receiver Operating Characteristic (AUROC). Note that we do not employ thresholding-based metrics such as $F 1$ score because access to abnormal samples is only allowed in testing time. Hence, we focus on the separability of models for novelty with AUROC. ", + "bbox": [ + 174, + 250, + 825, + 305 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the datasets in Table 1, we compare the effectiveness of the reconstruction error, SAP and NAP for three models: Autoencoder (AE), Variational Autoencoder (VAE), Adversarial Autoencoder (AAE) (Makhzani et al., 2016). For the benchmark datasets, recent approaches including OCNN (Chalapathy et al., 2018), GPND (Pidhorskyi et al., 2018), DSVDD (Ruff et al., 2018) and GT (Golan & El-Yaniv, 2018) are available. To obtain the performances of the existing approaches, we downloaded their codes and applied against our problem setups. ", + "bbox": [ + 174, + 313, + 825, + 396 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For MNIST and F-MNIST, we create test sets of novelty ratios $3 5 \\%$ for the multimodal setup and $50 \\%$ for the unimodal setup, where novelty samples in the test sets are randomly selected from given novelty classes. For the other datasets, we take all samples in given novelty classes to create test sets. Note that the expectation value of AUROC is invariant to the novelty ratio. ", + "bbox": [ + 174, + 404, + 825, + 459 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 477, + 405, + 491 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use symmetric architecture with fully-connected layers for the three base models, AE, VAE, and AAE. Each encoder and decoder has 10 layers with different bottleneck size. For the Kaggle and UCI datasets, we carry out PCA for each dataset first. The minimum number of principal components that explain at least $90 \\%$ of the variance is selected as the bottleneck size of the autoencoders. We set bottleneck size to 20 for benchmark datasets. Leaky-ReLU (Xu et al., 2015a) activation and batch normalization (Ioffe & Szegedy, 2015) layers are appended to all layers except the last layer. ", + "bbox": [ + 174, + 503, + 825, + 587 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We train AE, VAE and AAE with Adam optimizer (Kingma & Ba, 2015), and select the model with the lowest validation loss as the best model. For training stability of VAE, 10 Monte Carlo samples were averaged in the reparamterization trick (Kingma & Welling, 2014) to obtain reconstruction from the decoder. In the calculation of SAP and NAP, we excluded reconstructions in the input space for MNIST and F-MNIST. ", + "bbox": [ + 176, + 593, + 825, + 664 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.4 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 681, + 277, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Each AUROC score is obtained by averaging AUROC scores from multiple trials to reduce the random errors in training neural networks: 5 trials for MNIST and F-MNIST, and 20 trials for the other datasets. More results are provided in Appendix: standard deviations in Appendix B, comparison to baselines other than autoencoder variants C, and the effect of varying hidden layers involved in RAPP computation in Appendix D. ", + "bbox": [ + 174, + 708, + 825, + 779 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.4.1 COMPARISON WITH BASELINES", + "text_level": 1, + "bbox": [ + 176, + 794, + 446, + 808 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2 summarizes the result of our performance evaluation; the best score for each model is in bold, and the best score for each dataset with an underline. Since STL, OTTO, SNSR, MNIST, and F-MNIST do not have anomaly labels, their scores are averaged over all possible anomaly class assignments. For instance, the AUROC value for OTTO in the unimodal normality setup is the average of 9 AUROC values with different novelty class assignments. ", + "bbox": [ + 174, + 819, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Table 2, RAPP shows the highest AUROC scores for most of the cases. If we examine the performance for each dataset, RAPP achieves the best for 10 cases out of 15 (see the underlines). ", + "bbox": [ + 173, + 895, + 821, + 922 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/c0ef9c23da7ad8b1bf2294528236e6f142d5d87ec7cecf7327a115a9a0da49c2.jpg", + "table_caption": [ + "Table 2: AUROC of RAPP and the baselines. " + ], + "table_footnote": [], + "table_body": "
DatasetAEVAEAAE
ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.7230.7030.7110.7000.6750.7110.7260.7110.724
OTTO0.6170.6160.6650.6300.6310.6490.6170.6180.665
SNSR0.6130.6110.6140.6080.6200.6580.6120.6080.609
MNIST0.8250.8810.8990.9000.9650.9650.8470.9110.929
F-MNIST0.7120.7250.7340.7250.7280.7550.7210.7100.727
Unimodal Normality
MI-F0.6070.6700.7050.5910.5720.6780.6320.6920.704
MI-V0.8970.8980.9070.8450.8330.9030.8950.8910.904
EOPT0.6100.6070.6250.6750.6340.5960.6060.6030.634
NASA0.7190.7020.6920.7380.7140.7190.7120.6950.688
RARM0.6870.6740.6860.5870.5650.6480.6640.6750.675
STL0.8700.8560.8300.8500.8240.7920.8750.8610.830
OTTO0.8290.8290.8320.8310.8350.8330.8280.8280.831
SNSR0.9790.9820.9900.9750.9790.9640.9790.9830.991
MNIST0.9720.9800.9790.9800.9870.9890.9720.9660.977
F-MNIST0.9240.9280.9330.9260.9260.9420.9220.9050.928
", + "bbox": [ + 173, + 119, + 825, + 436 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/975664314fc5c7b8ae521ff6743dd750d7b74e85f9665444518fa6e5712853ee.jpg", + "table_caption": [ + "Table 3: AUROC on benchmark datasets. " + ], + "table_footnote": [], + "table_body": "
DatasetOCNNGPNDDSVDDGTNAPAENAPVAENAPAAE
Multimodal Normality (Novelty] Ratio: 35%)
MNIST F-MNIST0.600 0.6090.501 0.6910.622 0.6100.893 0.7250.899 0.7340.965 0.7550.929 0.727
Unimodal Normality (Novelty ] Ratio: 50%)
MNIST0.9270.9710.9220.9740.9790.9890.977
F-MNIST0.9150.9170.9230.9350.9330.9420.928
", + "bbox": [ + 215, + 479, + 782, + 613 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.4.2 COMPARISON WITH COMPETITORS ", + "text_level": 1, + "bbox": [ + 176, + 648, + 470, + 662 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3 summarizes the comparison of RAPP to recent novelty detection methods. As in Table 2, AUROC values are calculated by averaging results from 10 cases with different anomaly class assignments for both datasets. ", + "bbox": [ + 176, + 678, + 820, + 719 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Except for the unimodal F-MNIST setup, NAP outperforms all competing methods regardless of base model choice. Notably, NAP combined with VAE always shows the best performance, which is even higher than that of GT relying on image-specific data transformations for all cases. ", + "bbox": [ + 173, + 726, + 825, + 768 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 801, + 318, + 816 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper, we propose a novelty detection method which utilizes hidden reconstructions along a projection pathway of deep autoencoders. To this end, we extend the concept of reconstruction in the input space to hidden spaces found by an autoencoder and present a tractable way to compute the hidden reconstructions, which requires neither modifying nor retraining the autoencoder. Our experimental results show that the proposed method outperforms other competing methods in terms of AUROC for diverse datasets including popular benchmarks. 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", + "bbox": [ + 173, + 518, + 825, + 560 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Bing Xu, Naiyan Wang, Tianqi Chen, and Mu Li. Empirical evaluation of rectified activations in convolutional network. arXiv, abs/1505.00853, 2015a. ", + "bbox": [ + 173, + 570, + 823, + 599 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Dan Xu, Elisa Ricci, Yan Yan, Jingkuan Song, and Nicu Sebe. Learning deep representations of appearance and motion for anomalous event detection. In BMVC, 2015b. ", + "bbox": [ + 173, + 607, + 823, + 637 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Houssam Zenati, Chuan Sheng Foo, Bruno Lecouat, Gaurav Manek, and Vijay Ramaseshan Chandrasekhar. Efficient GAN-based anomaly detection. arXiv, abs/1802.06222, 2018. ", + "bbox": [ + 173, + 645, + 820, + 674 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Pushe Zhao, Masaru Kurihara, Junichi Tanaka, Tojiro Noda, Shigeyoshi Chikuma, and Tadashi Suzuki. Advanced correlation-based anomaly detection method for predictive maintenance. In ICPHM, 2017. ", + "bbox": [ + 174, + 683, + 825, + 727 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A SVD COMPUTATION TIME ", + "text_level": 1, + "bbox": [ + 176, + 102, + 431, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We compare running times of training an autoencoder and computing SVD for NAP. We choose two packages for the SVD computation: Pytorch SVD and fbpca provided in https://fbpca. readthedocs.io/en/latest/. ", + "bbox": [ + 174, + 133, + 825, + 175 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Since the time complexity of SVD is linear in the number of data samples1, we mainly focus on the performance of SVD with varying the number of columns of the input matrix that SVD is applied. To obtain variable sizes of the columns, we vary the depth and bottleneck size of autoencoders. ", + "bbox": [ + 174, + 183, + 825, + 224 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The result is shown below. Notably, Pytorch SVD utilizing GPU is at least $4 7 \\mathbf { x }$ faster than training neural networks. Even, fbpca running only on CPU achieves at least $2 . 4 \\mathbf { x }$ speedup. ", + "bbox": [ + 174, + 232, + 821, + 261 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/c3d2c9664c6a9e393392b5f66dac8b7d4c34056cddbceceac73865babee145a3.jpg", + "image_caption": [ + "The detailed setups to obtain the matrices for the experiment are given in the table below: " + ], + "image_footnote": [], + "bbox": [ + 174, + 268, + 821, + 404 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/8ae7bb8f46008832d295c4e14ea36cce9fa1ecb2895baef1c92b5b68ed252747.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MNISTOTTOSNSR
DepthBottleneck SizeDepthBottleneck SizeDepthBottleneck Size
12010020402090
21810018401890
31610016401690
41410014401490
51210012401290
61010010401090
78100840890
86100640690
94100440490
102100240290
11280２２230270
1226020250
1324010230
14220210
", + "bbox": [ + 183, + 441, + 816, + 689 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B STANDARD DEVIATIONS OF EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 656, + 118 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/557fc47fb2ab04303cad84b2e11500b40e864616a19c896da3bf5280bc872b42.jpg", + "table_caption": [ + "We provide the standard deviations of the result in Table 2. Given a dataset and a setup, multimodal or unimodal, AUROC values are first averaged over multiple cases with different novelty class assignments; then, the standard deviations are calculated for those averaged values over multiple trials. The number of cases and the number of trials depend on datasets, and we refer to Section 5 for more details. " + ], + "table_footnote": [], + "table_body": "
AEVAEAAE
Dataset ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.723 (0.005)0.703 (0.007)0.711 (0.006)0.700 (0.012)0.675 (0.018)0.711 (0.012)0.726 (0.009)0.711 (0.011)0.724 (0.016)
OTTO0.617 (0.002)0.616 (0.001)0.665 (0.004)0.630 (0.003)0.631 (0.005)0.649 (0.003)0.617 (0.001)0.618 (0.003)0.665 (0.004)
SNSR0.613 (0.002)0.611 (0.003)0.614 (0.004)0.608 (0.001)0.620 (0.002)0.658 (0.003)0.612 (0.005)0.608 (0.005)0.609 (0.005)
MNIST0.825 (0.001)0.881 (0.007)0.899 (0.008)0.900 (0.004)0.965 (0.005)0.965 (0.003)0.847 (0.005)0.911 (0.006)0.929 (0.005)
F-MNIST0.712 (0.001)0.725 (0.004)0.734 (0.004)0.725 (0.003)0.728 (0.002)0.755 (0.006)0.721 (0.001)0.710 (0.005)0.727 (0.005)
Unimodal Normality
MI-F0.607 (0.023)0.670 (0.041)0.705 (0.018)0.591 (0.013)0.572 (0.019)0.678 (0.047)0.632 (0.028)0.692 (0.042)0.704 (0.024)
MI-V0.897 (0.010)0.898 (0.005)0.907 (0.005)0.845 (0.021)0.833 (0.032)0.903 (0.010)0.895 (0.005)0.891 (0.006)0.904 (0.005)
EOPT0.610 (0.015)0.607 (0.014)0.625 (0.008)0.675 (0.017)0.634 (0.007)0.596 (0.002)0.606 (0.011)0.603 (0.013)0.634 (0.010)
NASA0.719 (0.009)0.702 (0.015)0.692 (0.016)0.738 (0.009)0.714 (0.015)0.719 (0.009)0.712 (0.016)0.695 (0.027)0.688 (0.024)
RARM0.687 (0.038)0.674 (0.031)0.686 (0.030)0.587 (0.021)0.565 (0.029)0.648 (0.022)0.664 (0.015)0.675 (0.023)0.675 (0.040)
STL0.870 (0.007)0.856 (0.007)0.830 (0.007)0.850 (0.006)0.824 (0.003)0.792 (0.034)0.875 (0.003)0.861 (0.004)0.830 (0.008)
OTTO0.829 (0.001)0.829 (0.001)0.832 (0.001)0.831 (0.002)0.835 (0.003)0.833 (0.004)0.828 (0.002)0.828 (0.001)0.831 (0.002)
SNSR0.979 (0.001)0.982 (0.001)0.990 (0.001)0.975 (0.003)0.979 (0.002)0.964 (0.043)0.979 (0.001)0.983 (0.002)0.997 (0.001)
MNIST0.972 (0.001)0.980 (0.001)0.979 (0.000)0.980 (0.002)0.987 (0.002)0.989 (0.001)0.972 (0.001)0.966 (0.004)0.977 (0.001)
F-MNIST0.924 (0.002)0.928 (0.002)0.933 (0.002)0.926 (0.001)0.926 (0.001)0.942 (0.000)0.922 (0.001)0.905 (0.002)0.928 (0.002)
", + "bbox": [ + 171, + 205, + 825, + 737 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C COMPARING RAPP TO BASELINES OTHER THAN NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 171, + 102, + 787, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The table below shows the result of comparing RAPP and baselines not based on neural networks. Note that the baselines are run with hyperparameters or hyperparameter search spaces provided in Ruff et al. (2018), while NAP is not further tuned. ", + "bbox": [ + 174, + 133, + 825, + 176 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/7e38f4a07d332330fa7adcf28aaad730181b0145f7b875e22b48c3dce626eeee.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DatasetOCSVMISOFPCAkPCANAPAENAPvAENAPAAE
Multimodal Normality
STL0.6080.6350.7000.6930.7110.7110.724
OTTO0.6280.5480.5720.6440.6650.6490.665
SNSR0.5240.5260.5360.5800.6140.6580.606
MNIST F-MNIST0.566 0.5710.562 0.6500.698 0.6700.701 0.6760.899 0.7340.965 0.7550.929 0.727
Unimodal Normality
MI-F0.7770.8260.5460.8380.7050.6780.704
MI-V0.8400.8390.8740.8880.9070.9030.904
EOPT0.5970.5950.5520.6930.6250.5960.634
NASA0.5730.6640.7650.6990.6920.7190.688
RARM0.7490.7520.5870.7340.7340.6480.675
STL0.8790.8710.8830.8800.8800.7920.830
OTTO0.8210.6990.7990.8230.8230.8330.831
SNSR0.9590.8930.8940.9700.9700.964
0.991
MNIST0.8940.9190.9030.9060.9790.9890.977
F-MNIST0.8840.8540.9500.9430.9330.9420.928
", + "bbox": [ + 220, + 188, + 776, + 484 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D NAP PERFORMANCE OVER INCREASING INVOLVED HIDDEN LAYERS ", + "text_level": 1, + "bbox": [ + 171, + 102, + 784, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We investigate the performance of NAP while increasing the number of hidden layers involved in the NAP computation. Specifically, we consider two ways for the increment: 1) adding hidden layers one by one from the input layer (forward addition), and 2) adding hidden layers one by one from the bottleneck layer (backward addition). Experimental results on three datasets are shown below. For most cases, more hidden layers tend to result in higher performance. The values are obtained from one trial without averaging results from multiple trials. ", + "bbox": [ + 173, + 133, + 826, + 217 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/64d9367ef1275abc2f0f52dc5a0903a026d55a84ee4e2adb24f3acdba0d3bb08.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 176, + 226, + 826, + 786 + ], + "page_idx": 13 + } +] \ No newline at end of file diff --git a/parse/train/HkgeGeBYDB/HkgeGeBYDB_middle.json b/parse/train/HkgeGeBYDB/HkgeGeBYDB_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..cc02e07038702f7e56d41b3b9e465e1151f3751e --- /dev/null +++ b/parse/train/HkgeGeBYDB/HkgeGeBYDB_middle.json @@ -0,0 +1,32431 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 502, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 504, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 504, + 97 + ], + "score": 1.0, + "content": "RAPP: NOVELTY DETECTION WITH RECONSTRUC-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 373, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 373, + 117 + ], + "score": 1.0, + "content": "TION ALONG PROJECTION PATHWAY", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 111, + 134, + 440, + 190 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 379, + 148 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 379, + 148 + ], + "score": 1.0, + "content": "Ki Hyun Kim, Sangwoo Shim, Yongsub Lim, Jongseob Jeon,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 326, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 326, + 158 + ], + "score": 1.0, + "content": "Jeongwoo Choi, Byungchan Kim, Andre S. Yoon", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 111, + 156, + 171, + 168 + ], + "spans": [ + { + "bbox": [ + 111, + 156, + 171, + 168 + ], + "score": 1.0, + "content": "MakinaRocks", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 112, + 167, + 442, + 181 + ], + "spans": [ + { + "bbox": [ + 112, + 167, + 442, + 181 + ], + "score": 1.0, + "content": "{khkim, sangwoo, yongsub, jongseob.jeon}@makinarocks.ai", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 112, + 179, + 358, + 191 + ], + "spans": [ + { + "bbox": [ + 112, + 179, + 358, + 191 + ], + "score": 1.0, + "content": "{jeongwoo, kbc8894, andre}@makinarocks.ai", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 278, + 219, + 333, + 231 + ], + "lines": [ + { + "bbox": [ + 276, + 219, + 335, + 232 + ], + "spans": [ + { + "bbox": [ + 276, + 219, + 335, + 232 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 143, + 244, + 468, + 365 + ], + "lines": [ + { + "bbox": [ + 142, + 244, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 244, + 469, + 257 + ], + "score": 1.0, + "content": "We propose RAPP, a new methodology for novelty detection by utilizing hid-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "score": 1.0, + "content": "den space activation values obtained from a deep autoencoder. Precisely, RAPP", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "compares input and its autoencoder reconstruction not only in the input space but", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "score": 1.0, + "content": "also in the hidden spaces. We show that if we feed a reconstructed input to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "score": 1.0, + "content": "same autoencoder again, its activated values in a hidden space are equivalent to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 311 + ], + "score": 1.0, + "content": "the corresponding reconstruction in that hidden space given the original input. We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "score": 1.0, + "content": "devise two metrics aggregating those hidden activated values to quantify the nov-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "score": 1.0, + "content": "elty of the input. Through extensive experiments using diverse datasets, we val-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "score": 1.0, + "content": "idate that RAPP improves novelty detection performances of autoencoder-based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "score": 1.0, + "content": "approaches. Besides, we show that RAPP outperforms recent novelty detection", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 317, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 317, + 366 + ], + "score": 1.0, + "content": "methods evaluated on popular benchmarks.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 206, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 208, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 208, + 403 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "How can we characterize novelty when only normality information is given? Novelty detection is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "score": 1.0, + "content": "the mechanism to decide whether a data sample is an outlier with respect to the training data. This", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 504, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 504, + 447 + ], + "score": 1.0, + "content": "mechanism is especially useful in situations where a proportion of detection targets is inherently", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "score": 1.0, + "content": "small. Examples are fraudulent transaction detection (Pawar et al., 2014; Porwal & Mukund, 2018),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "intrusion detection (Lee, 2017; Aoudi et al., 2018), video surveillance (Ravanbakhsh et al., 2017;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Xu et al., 2015b), medical diagnosis (Schlegl et al., 2017; Baur et al., 2018) and equipment failure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "detection (Kuzin & Borovicka, 2016; Zhao et al., 2017; Beghi et al., 2014). Recently, deep autoen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "coders and their variants have shown outstanding performances in finding compact representations", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "from complex data, and the reconstruction error has been chosen as a popular metric for detect-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "ing novelty (An & Cho, 2015; Vasilev et al., 2018). However, this approach has a limitation of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "measuring reconstruction quality only in an input space, which does not fully utilize hierarchical", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 379, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 379, + 546 + ], + "score": 1.0, + "content": "representations in hidden spaces identified by the deep autoencoder.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "In this paper, we propose RAPP, a new method of detecting novelty samples exploiting hidden ac-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "tivation values in addition to the input values and their autoencoder reconstruction values. While", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "ordinary reconstruction-based methods carry out novelty detection by comparing differences be-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "tween input data before the input layer and reconstructed data at the output layer, RAPP extends", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "these comparisons to hidden spaces. We first collect a set of hidden activation values by feeding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "the original input to the autoencoder. Subsequently, we feed the autoencoder reconstructed input to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "the autoencoder to calculate another set of activation values in the hidden layers. This procedure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "score": 1.0, + "content": "does not need additional training of the autoencoder. In turn, we quantify the novelty of the input by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "aggregating these two sets of hidden activation values. To this end, we devise two metrics. The first", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "metric measures the total amount of reconstruction errors in input and hidden spaces. The second", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "metric normalizes the reconstruction errors before summing up. Note that RAPP falls back to the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 685 + ], + "score": 1.0, + "content": "ordinary reconstruction-based method if we only aggregate input values before the input layer and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 681, + 281, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 281, + 695 + ], + "score": 1.0, + "content": "the reconstructed values at the output layer.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Also, we explain the motivations that facilitated the development of RAPP. We show that activation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "values in a hidden space obtained by feeding a reconstructed input to the autoencoder are equivalent", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "to the corresponding reconstruction in that hidden space for the original input. We refer the latter", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 502, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 504, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 504, + 97 + ], + "score": 1.0, + "content": "RAPP: NOVELTY DETECTION WITH RECONSTRUC-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 373, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 373, + 117 + ], + "score": 1.0, + "content": "TION ALONG PROJECTION PATHWAY", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "list", + "bbox": [ + 111, + 134, + 440, + 190 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 379, + 148 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 379, + 148 + ], + "score": 1.0, + "content": "Ki Hyun Kim, Sangwoo Shim, Yongsub Lim, Jongseob Jeon,", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 146, + 326, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 326, + 158 + ], + "score": 1.0, + "content": "Jeongwoo Choi, Byungchan Kim, Andre S. Yoon", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 156, + 171, + 168 + ], + "spans": [ + { + "bbox": [ + 111, + 156, + 171, + 168 + ], + "score": 1.0, + "content": "MakinaRocks", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 112, + 167, + 442, + 181 + ], + "spans": [ + { + "bbox": [ + 112, + 167, + 442, + 181 + ], + "score": 1.0, + "content": "{khkim, sangwoo, yongsub, jongseob.jeon}@makinarocks.ai", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 112, + 179, + 358, + 191 + ], + "spans": [ + { + "bbox": [ + 112, + 179, + 358, + 191 + ], + "score": 1.0, + "content": "{jeongwoo, kbc8894, andre}@makinarocks.ai", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + } + ], + "index": 4, + "bbox_fs": [ + 111, + 134, + 442, + 191 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 219, + 333, + 231 + ], + "lines": [ + { + "bbox": [ + 276, + 219, + 335, + 232 + ], + "spans": [ + { + "bbox": [ + 276, + 219, + 335, + 232 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 143, + 244, + 468, + 365 + ], + "lines": [ + { + "bbox": [ + 142, + 244, + 469, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 244, + 469, + 257 + ], + "score": 1.0, + "content": "We propose RAPP, a new methodology for novelty detection by utilizing hid-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 469, + 268 + ], + "score": 1.0, + "content": "den space activation values obtained from a deep autoencoder. Precisely, RAPP", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "compares input and its autoencoder reconstruction not only in the input space but", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 290 + ], + "score": 1.0, + "content": "also in the hidden spaces. We show that if we feed a reconstructed input to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 470, + 300 + ], + "score": 1.0, + "content": "same autoencoder again, its activated values in a hidden space are equivalent to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 311 + ], + "score": 1.0, + "content": "the corresponding reconstruction in that hidden space given the original input. We", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "score": 1.0, + "content": "devise two metrics aggregating those hidden activated values to quantify the nov-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "score": 1.0, + "content": "elty of the input. Through extensive experiments using diverse datasets, we val-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "score": 1.0, + "content": "idate that RAPP improves novelty detection performances of autoencoder-based", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "score": 1.0, + "content": "approaches. Besides, we show that RAPP outperforms recent novelty detection", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 317, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 317, + 366 + ], + "score": 1.0, + "content": "methods evaluated on popular benchmarks.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 244, + 470, + 366 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 206, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 208, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 208, + 403 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "How can we characterize novelty when only normality information is given? Novelty detection is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "score": 1.0, + "content": "the mechanism to decide whether a data sample is an outlier with respect to the training data. This", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 504, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 504, + 447 + ], + "score": 1.0, + "content": "mechanism is especially useful in situations where a proportion of detection targets is inherently", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 504, + 458 + ], + "score": 1.0, + "content": "small. Examples are fraudulent transaction detection (Pawar et al., 2014; Porwal & Mukund, 2018),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "intrusion detection (Lee, 2017; Aoudi et al., 2018), video surveillance (Ravanbakhsh et al., 2017;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Xu et al., 2015b), medical diagnosis (Schlegl et al., 2017; Baur et al., 2018) and equipment failure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "detection (Kuzin & Borovicka, 2016; Zhao et al., 2017; Beghi et al., 2014). Recently, deep autoen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "coders and their variants have shown outstanding performances in finding compact representations", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "from complex data, and the reconstruction error has been chosen as a popular metric for detect-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "ing novelty (An & Cho, 2015; Vasilev et al., 2018). However, this approach has a limitation of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "measuring reconstruction quality only in an input space, which does not fully utilize hierarchical", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 379, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 379, + 546 + ], + "score": 1.0, + "content": "representations in hidden spaces identified by the deep autoencoder.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 413, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "In this paper, we propose RAPP, a new method of detecting novelty samples exploiting hidden ac-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "tivation values in addition to the input values and their autoencoder reconstruction values. While", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "ordinary reconstruction-based methods carry out novelty detection by comparing differences be-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "tween input data before the input layer and reconstructed data at the output layer, RAPP extends", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "these comparisons to hidden spaces. We first collect a set of hidden activation values by feeding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "the original input to the autoencoder. Subsequently, we feed the autoencoder reconstructed input to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "the autoencoder to calculate another set of activation values in the hidden layers. This procedure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "score": 1.0, + "content": "does not need additional training of the autoencoder. In turn, we quantify the novelty of the input by", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "aggregating these two sets of hidden activation values. To this end, we devise two metrics. The first", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "metric measures the total amount of reconstruction errors in input and hidden spaces. The second", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "metric normalizes the reconstruction errors before summing up. Note that RAPP falls back to the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 685 + ], + "score": 1.0, + "content": "ordinary reconstruction-based method if we only aggregate input values before the input layer and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 681, + 281, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 281, + 695 + ], + "score": 1.0, + "content": "the reconstructed values at the output layer.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 549, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Also, we explain the motivations that facilitated the development of RAPP. We show that activation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "values in a hidden space obtained by feeding a reconstructed input to the autoencoder are equivalent", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "to the corresponding reconstruction in that hidden space for the original input. We refer the latter", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "quantity as a hidden reconstruction of the input. Note that this is a natural extension of the recon-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "struction to the hidden space. Unfortunately, we cannot directly compute the hidden reconstruction", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "as in the computation of the ordinary reconstruction because the autoencoder does not impose any", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "correspondence between encoding-decoding pairs of hidden layers during the training. Neverthe-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 504, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 504, + 139 + ], + "score": 1.0, + "content": "less, we show that it can be computed by feeding a reconstructed input to the autoencoder again.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Consequently, RAPP incorporates hidden reconstruction errors as well as the ordinary reconstruc-", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 229, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 229, + 161 + ], + "score": 1.0, + "content": "tion error in detecting novelty.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 46, + "bbox_fs": [ + 106, + 699, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "quantity as a hidden reconstruction of the input. Note that this is a natural extension of the recon-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "struction to the hidden space. Unfortunately, we cannot directly compute the hidden reconstruction", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "as in the computation of the ordinary reconstruction because the autoencoder does not impose any", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "correspondence between encoding-decoding pairs of hidden layers during the training. Neverthe-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 504, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 504, + 139 + ], + "score": 1.0, + "content": "less, we show that it can be computed by feeding a reconstructed input to the autoencoder again.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "Consequently, RAPP incorporates hidden reconstruction errors as well as the ordinary reconstruc-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 229, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 229, + 161 + ], + "score": 1.0, + "content": "tion error in detecting novelty.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 165, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "score": 1.0, + "content": "With extensive experiments, we demonstrate using diverse datasets that our method effectively im-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "proves autoencoder-based novelty detection methods. In addition, we show by evaluating on popular", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 444, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 444, + 200 + ], + "score": 1.0, + "content": "benchmark datasets that RAPP outperforms competing methods recently developed.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 289, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 291, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 291, + 217 + ], + "score": 1.0, + "content": "Our contributions are summarized as follows.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 132, + 220, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 132, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 132, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "• We propose a new novelty detection method by utilizing hidden activation values of an", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 142, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "input and its autoencoder reconstruction, and provide aggregation functions for them to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 243, + 260, + 254 + ], + "spans": [ + { + "bbox": [ + 142, + 243, + 260, + 254 + ], + "score": 1.0, + "content": "quantify novelty of the input.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 140, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 140, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "We provide motivation that RAPP extends the reconstruction concept in the input space", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 270, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 142, + 270, + 506, + 281 + ], + "score": 1.0, + "content": "into the hidden spaces. 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Moreover, we validate that RAPP outperforms recent novelty detection", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 319, + 307, + 331 + ], + "spans": [ + { + "bbox": [ + 142, + 319, + 307, + 331 + ], + "score": 1.0, + "content": "methods on popular benchmark datasets.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 347, + 211, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 213, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 213, + 362 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 373, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "Various novelty detection methods with deep neural networks rely on the reconstruction error (Saku-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "rada & Yairi, 2014; Hoffmann, 2007; An & Cho, 2015), because discriminative learning schemes", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "are not suitable for highly class-imbalanced data which is common in practice. Unsupervised and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "semi-supervised learning approaches handle such imbalance by focusing on the characterization of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 321, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 321, + 432 + ], + "score": 1.0, + "content": "normality and detecting samples out of the normality.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Variational Autoencoders (VAE) (Kingma & Welling, 2014) were reported to outperform vanilla", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "autoencoders for novelty detection based on reconstruction error (An & Cho, 2015). To carry out", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "the novelty detection outlined in this approach, an autoencoder needs to be trained only with normal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "data. The autoencoder encodes the training data, which comprises of only normal data in this case,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "into a lower-dimensional space and decodes them to the input space. To test novelty, an input value", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "is fed to the autoencoder to produce a reconstructed value and calculate the distance between the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "input and reconstructed values. This distance is the reconstruction error. A higher reconstruction", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "error means that the input value cannot be encoded onto the lower-dimensional space that represents", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "normal data. 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Despite the same", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "purpose of discovering a simpler, lower-dimensional representation, the training criterion for GAN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "is focusing on the quality of data generation rather than the reconstruction quality of training data.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Recently, several pieces of research have combined autoencoders and adversarial learning to meet", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "both criteria in dimension reduction and data generation (Haloui et al., 2018; Pidhorskyi et al., 2018;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Zenati et al., 2018). One limitation of these methods based on the ordinary reconstruction error is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "that they do not exploit all the information available along the projection pathway of deep autoen-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 639, + 498, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 498, + 650 + ], + "score": 1.0, + "content": "coders. We will explain how to leverage this information for novelty detection in the next section.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "From the viewpoint of the diversity and ratio of the normal data in novelty detection, there are two", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "cases available. The first case is when a small fraction of classes are normal. This case has been", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "studied in a one-class classification context, and usually evaluated by organizing training data into", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "a collection of samples belonging to a small number of normal classes (Ruff et al., 2018; Perera &", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Patel, 2018; Sabokrou et al., 2018; Golan & El-Yaniv, 2018). The second case is when a majority", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "of classes are assigned as normal (An & Cho, 2015; Schlegl et al., 2017; Haloui et al., 2018; Zenati", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "et al., 2018). 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Precisely, we show that hidden activation values of a reconstructed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 281, + 481, + 293 + ], + "spans": [ + { + "bbox": [ + 142, + 281, + 481, + 293 + ], + "score": 1.0, + "content": "input are equivalent to the corresponding hidden reconstruction of the original input.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 140, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 140, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "We demonstrate that RAPP improves autoencoder-based novelty detection methods in di-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 142, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "verse datasets. Moreover, we validate that RAPP outperforms recent novelty detection", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 319, + 307, + 331 + ], + "spans": [ + { + "bbox": [ + 142, + 319, + 307, + 331 + ], + "score": 1.0, + "content": "methods on popular benchmark datasets.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 132, + 219, + 506, + 331 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 347, + 211, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 213, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 213, + 362 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 373, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "Various novelty detection methods with deep neural networks rely on the reconstruction error (Saku-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "rada & Yairi, 2014; Hoffmann, 2007; An & Cho, 2015), because discriminative learning schemes", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "are not suitable for highly class-imbalanced data which is common in practice. Unsupervised and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "semi-supervised learning approaches handle such imbalance by focusing on the characterization of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 321, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 321, + 432 + ], + "score": 1.0, + "content": "normality and detecting samples out of the normality.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 374, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Variational Autoencoders (VAE) (Kingma & Welling, 2014) were reported to outperform vanilla", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "autoencoders for novelty detection based on reconstruction error (An & Cho, 2015). To carry out", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "the novelty detection outlined in this approach, an autoencoder needs to be trained only with normal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "data. The autoencoder encodes the training data, which comprises of only normal data in this case,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "into a lower-dimensional space and decodes them to the input space. To test novelty, an input value", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "is fed to the autoencoder to produce a reconstructed value and calculate the distance between the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "input and reconstructed values. This distance is the reconstruction error. A higher reconstruction", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "error means that the input value cannot be encoded onto the lower-dimensional space that represents", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "normal data. Therefore, the input value can be marked as a novelty if its reconstruction error exceeds", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 534, + 184, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 184, + 543 + ], + "score": 1.0, + "content": "a certain threshold.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 434, + 505, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "Instead of autoencoders, Generative Adversarial Networks (GAN) have been also suggested to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "model a distribution of normal data (Sabokrou et al., 2018; Schlegl et al., 2017). Despite the same", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "purpose of discovering a simpler, lower-dimensional representation, the training criterion for GAN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "is focusing on the quality of data generation rather than the reconstruction quality of training data.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Recently, several pieces of research have combined autoencoders and adversarial learning to meet", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "both criteria in dimension reduction and data generation (Haloui et al., 2018; Pidhorskyi et al., 2018;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Zenati et al., 2018). One limitation of these methods based on the ordinary reconstruction error is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "that they do not exploit all the information available along the projection pathway of deep autoen-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 639, + 498, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 498, + 650 + ], + "score": 1.0, + "content": "coders. We will explain how to leverage this information for novelty detection in the next section.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 549, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "From the viewpoint of the diversity and ratio of the normal data in novelty detection, there are two", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "cases available. The first case is when a small fraction of classes are normal. This case has been", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "studied in a one-class classification context, and usually evaluated by organizing training data into", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "a collection of samples belonging to a small number of normal classes (Ruff et al., 2018; Perera &", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Patel, 2018; Sabokrou et al., 2018; Golan & El-Yaniv, 2018). The second case is when a majority", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "of classes are assigned as normal (An & Cho, 2015; Schlegl et al., 2017; Haloui et al., 2018; Zenati", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "et al., 2018). In this case, normal data is more diverse, and the training data is consist of samples of a", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "relatively large number of normal classes: e.g., nine digits of MNIST. One setup does not dominate", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "the other, but depending on applications, either can be more suitable than the other. Different meth-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "ods may perform differently in both cases. In this paper, we evaluate RAPP and other competing", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 275, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 275, + 129 + ], + "score": 1.0, + "content": "methods with experiments in both setups.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 654, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "relatively large number of normal classes: e.g., nine digits of MNIST. One setup does not dominate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "the other, but depending on applications, either can be more suitable than the other. Different meth-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "ods may perform differently in both cases. In this paper, we evaluate RAPP and other competing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 275, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 275, + 129 + ], + "score": 1.0, + "content": "methods with experiments in both setups.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 142, + 268, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 270, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 270, + 157 + ], + "score": 1.0, + "content": "3 PROPOSED METHOD: RAPP", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 504, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 504, + 179 + ], + "score": 1.0, + "content": "In this section, we describe the proposed novelty detection method RAPP based on an autoencoder.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "The main idea is to compare hidden activations of an input and its hidden reconstructions along", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "the projection pathway of the autoencoder. To be precise, we project the input and its autoencoder", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "reconstruction onto the hidden spaces to obtain pairs of activation values, and aggregate them to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "score": 1.0, + "content": "quantify the novelty of the input. For the aggregation, we present two metrics to measure the total", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 223, + 260, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 260, + 234 + ], + "score": 1.0, + "content": "amount of difference within each pair.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 246, + 339, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 340, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 340, + 259 + ], + "score": 1.0, + "content": "3.1 RECONSTRUCTION BASED NOVELTY DETECTION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 267, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 106, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 173, + 280 + ], + "score": 1.0, + "content": "An autoencoder", + "type": "text" + }, + { + "bbox": [ + 173, + 268, + 182, + 277 + ], + "score": 0.68, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 268, + 364, + 280 + ], + "score": 1.0, + "content": "is a neural network consisting of an encoder", + "type": "text" + }, + { + "bbox": [ + 364, + 270, + 371, + 279 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 268, + 431, + 280 + ], + "score": 1.0, + "content": "and a decoder", + "type": "text" + }, + { + "bbox": [ + 432, + 268, + 438, + 279 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 268, + 505, + 280 + ], + "score": 1.0, + "content": ", responsible for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 277, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 461, + 293 + ], + "score": 1.0, + "content": "dimension reduction and its inverse mapping to the original input space, respectively: i.e.", + "type": "text" + }, + { + "bbox": [ + 461, + 279, + 501, + 290 + ], + "score": 0.92, + "content": "A = f \\circ g", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 277, + 506, + 293 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 288, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 452, + 303 + ], + "score": 1.0, + "content": "For the purpose, training the autoencoder aims to minimize difference between its input", + "type": "text" + }, + { + "bbox": [ + 453, + 291, + 460, + 299 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 288, + 506, + 303 + ], + "score": 1.0, + "content": "and output", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 107, + 301, + 128, + 312 + ], + "score": 0.9, + "content": "A ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 299, + 241, + 313 + ], + "score": 1.0, + "content": ". The space that the encoder", + "type": "text" + }, + { + "bbox": [ + 241, + 302, + 249, + 312 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "constitutes is called the latent space, and provides more concise", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 311, + 282, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 282, + 325 + ], + "score": 1.0, + "content": "representation for data than the input space.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "Due to this unsupervised representation learning property, the autoencoder has been widely used for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "novelty detection. 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To clearly state the required variables", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 157, + 733 + ], + "score": 1.0, + "content": "to construct", + "type": "text" + }, + { + "bbox": [ + 157, + 721, + 167, + 730 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", we write the algorithm with the for loop in Lines 3–5, but in practice, all of them", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 505, + 129 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 142, + 268, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 270, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 270, + 157 + ], + "score": 1.0, + "content": "3 PROPOSED METHOD: RAPP", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 504, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 504, + 179 + ], + "score": 1.0, + "content": "In this section, we describe the proposed novelty detection method RAPP based on an autoencoder.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "The main idea is to compare hidden activations of an input and its hidden reconstructions along", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "the projection pathway of the autoencoder. 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Or, why do we not use information in hidden spaces? While the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "reconstruction error in the input space is extensively employed, any similar concept does not ex-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "ist in hidden spaces. One reason is that the corresponding encoding and decoding layers are not", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "guaranteed to express the same space: e.g. permuted dimensions. This is because the autoencoder", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "objective does not have any term involving activations from intermediate hidden layers. As a re-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 339, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 126, + 354 + ], + "score": 1.0, + "content": "sult,", + "type": "text" + }, + { + "bbox": [ + 126, + 340, + 177, + 352 + ], + "score": 0.91, + "content": "f _ { \\ell : i + 1 } ( g ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 339, + 342, + 354 + ], + "score": 1.0, + "content": "cannot be considered a reconstruction of", + "type": "text" + }, + { + "bbox": [ + 343, + 340, + 367, + 352 + ], + "score": 0.92, + "content": "g _ { : i } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 339, + 414, + 354 + ], + "score": 1.0, + "content": ", except for", + "type": "text" + }, + { + "bbox": [ + 414, + 340, + 437, + 351 + ], + "score": 0.89, + "content": "i = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 339, + 505, + 354 + ], + "score": 1.0, + "content": "with which they", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 352, + 428, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 428, + 364 + ], + "score": 1.0, + "content": "become the ordinary reconstruction of and input to an autoencoder, respectively.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 275, + 506, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 367, + 504, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 504, + 381 + ], + "score": 1.0, + "content": "Nevertheless, in this section, we will show that there is an indirect way to compute the hidden recon-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 379, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 260, + 394 + ], + "score": 1.0, + "content": "struction. Precisely, we will show that", + "type": "text" + }, + { + "bbox": [ + 260, + 379, + 336, + 393 + ], + "score": 0.93, + "content": "\\hat { h } _ { i } ( x ) = g _ { : i } ( A ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 379, + 506, + 394 + ], + "score": 1.0, + "content": "is indeed equivalent to a reconstruction of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 391, + 331, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 131, + 404 + ], + "score": 0.91, + "content": "g _ { : i } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 391, + 331, + 404 + ], + "score": 1.0, + "content": ". The overall mechanism is depicted in Figure 1b.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 367, + 506, + 404 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 415, + 329, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 330, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 330, + 428 + ], + "score": 1.0, + "content": "4.1 COMPUTATION OF HIDDEN RECONSTRUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 123, + 449 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 437, + 171, + 448 + ], + "score": 0.92, + "content": "A = f \\circ g", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 435, + 297, + 449 + ], + "score": 1.0, + "content": "be a trained autoencoder, and", + "type": "text" + }, + { + "bbox": [ + 297, + 436, + 405, + 448 + ], + "score": 0.92, + "content": "M _ { 0 } = \\{ A ( x ) : x \\in \\mathbb { R } ^ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "be the low dimensional", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 445, + 335, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 162, + 461 + ], + "score": 1.0, + "content": "manifold that", + "type": "text" + }, + { + "bbox": [ + 163, + 448, + 171, + 457 + ], + "score": 0.77, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 445, + 335, + 461 + ], + "score": 1.0, + "content": "describes (Pidhorskyi et al., 2018): i.e.,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 435, + 506, + 461 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 460, + 350, + 473 + ], + "lines": [ + { + "bbox": [ + 259, + 460, + 350, + 473 + ], + 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Name# Samples#Features# ClassDomainNovelty Target
MI-F25,286582CNC millingMachine not completed
MI-V23,12558CNC millingWorkpiece out-of-spec
EOPT90.51520Storage systemSystem failures
NASA4,68733AstronomyHazardous asteroids
RARM20.2216２２２２７RoboticsMalfunctions
STL1,94127SteelSurface defects
OTTO61,878939E-commerceTypes of products
SNSR58,5094811Electric CurrentsDefective conditions
MNIST70,00078410Hand written digitsDigits
F-MNIST70,00078410Fashion articlesArticles
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Thus, training sets contain only normal samples and test sets contain both normal and anomaly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "samples in our evaluation setups. Precisely, if a dataset contains an anomaly label, we assign all sam-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "ples with that label to the test set for detection. 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This setup is repeated to produce sub-datasets", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 120, + 95, + 491, + 223 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 194, + 81, + 415, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 195, + 80, + 416, + 93 + ], + "spans": [ + { + "bbox": [ + 195, + 80, + 416, + 93 + ], + "score": 1.0, + "content": "Table 1: Description of datasets used in our evaluation.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 120, + 95, + 491, + 223 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 95, + 491, + 223 + ], + "spans": [ + { + "bbox": [ + 120, + 95, + 491, + 223 + ], + "score": 0.984, + "html": "
Name# Samples#Features# ClassDomainNovelty Target
MI-F25,286582CNC millingMachine not completed
MI-V23,12558CNC millingWorkpiece out-of-spec
EOPT90.51520Storage systemSystem failures
NASA4,68733AstronomyHazardous asteroids
RARM20.2216２２２２７RoboticsMalfunctions
STL1,94127SteelSurface defects
OTTO61,878939E-commerceTypes of products
SNSR58,5094811Electric CurrentsDefective conditions
MNIST70,00078410Hand written digitsDigits
F-MNIST70,00078410Fashion articlesArticles
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To this end, we tested the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 476 + ], + "score": 1.0, + "content": "methods on several benchmarks and diverse datasets collected from Kaggle and the UCI repository", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 473, + 349, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 349, + 486 + ], + "score": 1.0, + "content": "which are suitable for evaluating novelty detection methods.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 452, + 505, + 486 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 501, + 276, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 278, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 278, + 514 + ], + "score": 1.0, + "content": "5.1 DATASETS AND PROBLEM SETUPS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "The datasets from Kaggle and the UCI repository are chosen from problem sets of anomaly detection", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "and multi-class classification, summarized in Table 1. We note that MI-F and MI-V share the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "feature matrix, but are considered to be different datasets because their labels normal and abnormal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "are assigned by different columns: i.e. machine completed and pass visual inspection, respectively.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "We use these datasets to compare RAPP with standard autoencoder-based methods described in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 577, + 156, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 156, + 588 + ], + "score": 1.0, + "content": "Section 5.2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 523, + 506, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "To compare RAPP with novelty detection methods in recent literatures, we also use popular bench-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "mark datasets for evaluating deep learning techniques: MNIST (LeCun & Cortes, 2010) and F-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "MNIST (Xiao et al., 2017). For theses datasets, we do not take pre-split training and test sets, but", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 626, + 268, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 268, + 641 + ], + "score": 1.0, + "content": "instead merge them for post-processing.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 594, + 505, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 644, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "Novelty detection detects novel patterns by focusing on deviations from model-learned normal pat-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "terns. Thus, training sets contain only normal samples and test sets contain both normal and anomaly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "samples in our evaluation setups. Precisely, if a dataset contains an anomaly label, we assign all sam-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "ples with that label to the test set for detection. If a dataset does not have any anomaly labels, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 245, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 245, + 702 + ], + "score": 1.0, + "content": "consider the following two setups.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 643, + 506, + 702 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 131, + 707, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 131, + 707, + 505, + 723 + ], + "score": 1.0, + "content": "• Multimodal Normality: A single class is chosen to be the novelty class and the remaining", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 142, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "classes are assigned as the normal class. This setup is repeated to produce sub-datasets", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 81, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 142, + 81, + 504, + 94 + ], + "score": 1.0, + "content": "with all possible novelty assignments. For instance, MNIST results in a set of datasets with", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 93, + 256, + 107 + ], + "spans": [ + { + "bbox": [ + 142, + 93, + 256, + 107 + ], + "score": 1.0, + "content": "10 different novelty classes.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 36.5, + "bbox_fs": [ + 131, + 707, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 140, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 142, + 81, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 142, + 81, + 504, + 94 + ], + "score": 1.0, + "content": "with all possible novelty assignments. For instance, MNIST results in a set of datasets with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 93, + 256, + 107 + ], + "spans": [ + { + "bbox": [ + 142, + 93, + 256, + 107 + ], + "score": 1.0, + "content": "10 different novelty classes.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 133, + 109, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 132, + 109, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 132, + 109, + 505, + 121 + ], + "score": 1.0, + "content": "• Unimodal Normality: In contrast to the multimodal normality setup, we take one class for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 120, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 141, + 120, + 505, + 131 + ], + "score": 1.0, + "content": "normality, and the others for novelty. For instance, MNIST results in a set of datasets with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 142, + 129, + 255, + 143 + ], + "spans": [ + { + "bbox": [ + 142, + 129, + 255, + 143 + ], + "score": 1.0, + "content": "10 different normal classes.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 151, + 445, + 163 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 447, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 447, + 165 + ], + "score": 1.0, + "content": "We applied these two setups to STL, OTTO, SNSR, MNIST, and F-MNIST datasets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 177, + 231, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 232, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 232, + 191 + ], + "score": 1.0, + "content": "5.2 COMPARISON METHOD", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "We compare RAPP and the other methods using Area Under Receiver Operating Characteristic", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 405, + 222 + ], + "score": 1.0, + "content": "(AUROC). Note that we do not employ thresholding-based metrics such as", + "type": "text" + }, + { + "bbox": [ + 405, + 210, + 419, + 220 + ], + "score": 0.72, + "content": "F 1", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "score because access", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "to abnormal samples is only allowed in testing time. Hence, we focus on the separability of models", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 212, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 212, + 244 + ], + "score": 1.0, + "content": "for novelty with AUROC.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "For the datasets in Table 1, we compare the effectiveness of the reconstruction error, SAP and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "NAP for three models: Autoencoder (AE), Variational Autoencoder (VAE), Adversarial Autoen-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "score": 1.0, + "content": "coder (AAE) (Makhzani et al., 2016). For the benchmark datasets, recent approaches including", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "OCNN (Chalapathy et al., 2018), GPND (Pidhorskyi et al., 2018), DSVDD (Ruff et al., 2018) and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "GT (Golan & El-Yaniv, 2018) are available. To obtain the performances of the existing approaches,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 378, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 378, + 317 + ], + "score": 1.0, + "content": "we downloaded their codes and applied against our problem setups.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 364, + 332 + ], + "score": 1.0, + "content": "For MNIST and F-MNIST, we create test sets of novelty ratios", + "type": "text" + }, + { + "bbox": [ + 364, + 320, + 384, + 331 + ], + "score": 0.86, + "content": "3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "for the multimodal setup and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 126, + 342 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "for the unimodal setup, where novelty samples in the test sets are randomly selected from given", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "score": 1.0, + "content": "novelty classes. For the other datasets, we take all samples in given novelty classes to create test", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 426, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 426, + 365 + ], + "score": 1.0, + "content": "sets. Note that the expectation value of AUROC is invariant to the novelty ratio.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 248, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 250, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 250, + 391 + ], + "score": 1.0, + "content": "5.3 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "We use symmetric architecture with fully-connected layers for the three base models, AE, VAE, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "AAE. Each encoder and decoder has 10 layers with different bottleneck size. For the Kaggle and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "UCI datasets, we carry out PCA for each dataset first. The minimum number of principal compo-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 209, + 444 + ], + "score": 1.0, + "content": "nents that explain at least", + "type": "text" + }, + { + "bbox": [ + 209, + 432, + 229, + 443 + ], + "score": 0.87, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "of the variance is selected as the bottleneck size of the autoencoders.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "We set bottleneck size to 20 for benchmark datasets. Leaky-ReLU (Xu et al., 2015a) activation and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 503, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 503, + 468 + ], + "score": 1.0, + "content": "batch normalization (Ioffe & Szegedy, 2015) layers are appended to all layers except the last layer.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 108, + 470, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "We train AE, VAE and AAE with Adam optimizer (Kingma & Ba, 2015), and select the model with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "the lowest validation loss as the best model. For training stability of VAE, 10 Monte Carlo samples", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "were averaged in the reparamterization trick (Kingma & Welling, 2014) to obtain reconstruction", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "score": 1.0, + "content": "from the decoder. In the calculation of SAP and NAP, we excluded reconstructions in the input", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 515, + 240, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 240, + 527 + ], + "score": 1.0, + "content": "space for MNIST and F-MNIST.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 107, + 540, + 170, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 171, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 171, + 553 + ], + "score": 1.0, + "content": "5.4 RESULTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "Each AUROC score is obtained by averaging AUROC scores from multiple trials to reduce the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "random errors in training neural networks: 5 trials for MNIST and F-MNIST, and 20 trials for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "the other datasets. More results are provided in Appendix: standard deviations in Appendix B,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "comparison to baselines other than autoencoder variants C, and the effect of varying hidden layers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 297, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 297, + 617 + ], + "score": 1.0, + "content": "involved in RAPP computation in Appendix D.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 629, + 273, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 275, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 275, + 641 + ], + "score": 1.0, + "content": "5.4.1 COMPARISON WITH BASELINES", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "Table 2 summarizes the result of our performance evaluation; the best score for each model is in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "bold, and the best score for each dataset with an underline. Since STL, OTTO, SNSR, MNIST,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 669, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 684 + ], + "score": 1.0, + "content": "and F-MNIST do not have anomaly labels, their scores are averaged over all possible anomaly class", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "assignments. For instance, the AUROC value for OTTO in the unimodal normality setup is the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 691, + 386, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 386, + 706 + ], + "score": 1.0, + "content": "average of 9 AUROC values with different novelty class assignments.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In Table 2, RAPP shows the highest AUROC scores for most of the cases. If we examine the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 496, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 496, + 733 + ], + "score": 1.0, + "content": "performance for each dataset, RAPP achieves the best for 10 cases out of 15 (see the underlines).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 140, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 142, + 81, + 504, + 107 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 133, + 109, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 132, + 109, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 132, + 109, + 505, + 121 + ], + "score": 1.0, + "content": "• Unimodal Normality: In contrast to the multimodal normality setup, we take one class for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 120, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 141, + 120, + 505, + 131 + ], + "score": 1.0, + "content": "normality, and the others for novelty. For instance, MNIST results in a set of datasets with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 142, + 129, + 255, + 143 + ], + "spans": [ + { + "bbox": [ + 142, + 129, + 255, + 143 + ], + "score": 1.0, + "content": "10 different normal classes.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 132, + 109, + 505, + 143 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 151, + 445, + 163 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 447, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 447, + 165 + ], + "score": 1.0, + "content": "We applied these two setups to STL, OTTO, SNSR, MNIST, and F-MNIST datasets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 151, + 447, + 165 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 177, + 231, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 232, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 232, + 191 + ], + "score": 1.0, + "content": "5.2 COMPARISON METHOD", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "We compare RAPP and the other methods using Area Under Receiver Operating Characteristic", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 405, + 222 + ], + "score": 1.0, + "content": "(AUROC). Note that we do not employ thresholding-based metrics such as", + "type": "text" + }, + { + "bbox": [ + 405, + 210, + 419, + 220 + ], + "score": 0.72, + "content": "F 1", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "score because access", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "to abnormal samples is only allowed in testing time. Hence, we focus on the separability of models", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 212, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 212, + 244 + ], + "score": 1.0, + "content": "for novelty with AUROC.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 197, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "For the datasets in Table 1, we compare the effectiveness of the reconstruction error, SAP and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "NAP for three models: Autoencoder (AE), Variational Autoencoder (VAE), Adversarial Autoen-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "score": 1.0, + "content": "coder (AAE) (Makhzani et al., 2016). For the benchmark datasets, recent approaches including", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "OCNN (Chalapathy et al., 2018), GPND (Pidhorskyi et al., 2018), DSVDD (Ruff et al., 2018) and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "GT (Golan & El-Yaniv, 2018) are available. To obtain the performances of the existing approaches,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 378, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 378, + 317 + ], + "score": 1.0, + "content": "we downloaded their codes and applied against our problem setups.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 248, + 505, + 317 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 364, + 332 + ], + "score": 1.0, + "content": "For MNIST and F-MNIST, we create test sets of novelty ratios", + "type": "text" + }, + { + "bbox": [ + 364, + 320, + 384, + 331 + ], + "score": 0.86, + "content": "3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "for the multimodal setup and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 126, + 342 + ], + "score": 0.85, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "for the unimodal setup, where novelty samples in the test sets are randomly selected from given", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 354 + ], + "score": 1.0, + "content": "novelty classes. For the other datasets, we take all samples in given novelty classes to create test", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 426, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 426, + 365 + ], + "score": 1.0, + "content": "sets. Note that the expectation value of AUROC is invariant to the novelty ratio.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 320, + 505, + 365 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 248, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 250, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 250, + 391 + ], + "score": 1.0, + "content": "5.3 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "We use symmetric architecture with fully-connected layers for the three base models, AE, VAE, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "AAE. Each encoder and decoder has 10 layers with different bottleneck size. For the Kaggle and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "UCI datasets, we carry out PCA for each dataset first. The minimum number of principal compo-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 209, + 444 + ], + "score": 1.0, + "content": "nents that explain at least", + "type": "text" + }, + { + "bbox": [ + 209, + 432, + 229, + 443 + ], + "score": 0.87, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "of the variance is selected as the bottleneck size of the autoencoders.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "We set bottleneck size to 20 for benchmark datasets. Leaky-ReLU (Xu et al., 2015a) activation and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 452, + 503, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 503, + 468 + ], + "score": 1.0, + "content": "batch normalization (Ioffe & Szegedy, 2015) layers are appended to all layers except the last layer.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 399, + 505, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 470, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "We train AE, VAE and AAE with Adam optimizer (Kingma & Ba, 2015), and select the model with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "the lowest validation loss as the best model. For training stability of VAE, 10 Monte Carlo samples", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "were averaged in the reparamterization trick (Kingma & Welling, 2014) to obtain reconstruction", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 517 + ], + "score": 1.0, + "content": "from the decoder. In the calculation of SAP and NAP, we excluded reconstructions in the input", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 515, + 240, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 240, + 527 + ], + "score": 1.0, + "content": "space for MNIST and F-MNIST.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 470, + 505, + 527 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 540, + 170, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 171, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 171, + 553 + ], + "score": 1.0, + "content": "5.4 RESULTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "Each AUROC score is obtained by averaging AUROC scores from multiple trials to reduce the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "random errors in training neural networks: 5 trials for MNIST and F-MNIST, and 20 trials for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "the other datasets. More results are provided in Appendix: standard deviations in Appendix B,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "comparison to baselines other than autoencoder variants C, and the effect of varying hidden layers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 297, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 297, + 617 + ], + "score": 1.0, + "content": "involved in RAPP computation in Appendix D.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 561, + 506, + 617 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 629, + 273, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 275, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 275, + 641 + ], + "score": 1.0, + "content": "5.4.1 COMPARISON WITH BASELINES", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "Table 2 summarizes the result of our performance evaluation; the best score for each model is in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "bold, and the best score for each dataset with an underline. Since STL, OTTO, SNSR, MNIST,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 669, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 684 + ], + "score": 1.0, + "content": "and F-MNIST do not have anomaly labels, their scores are averaged over all possible anomaly class", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "assignments. For instance, the AUROC value for OTTO in the unimodal normality setup is the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 691, + 386, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 386, + 706 + ], + "score": 1.0, + "content": "average of 9 AUROC values with different novelty class assignments.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 649, + 505, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In Table 2, RAPP shows the highest AUROC scores for most of the cases. If we examine the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 496, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 496, + 733 + ], + "score": 1.0, + "content": "performance for each dataset, RAPP achieves the best for 10 cases out of 15 (see the underlines).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 95, + 505, + 346 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 215, + 80, + 396, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 213, + 79, + 397, + 93 + ], + "spans": [ + { + "bbox": [ + 213, + 79, + 397, + 93 + ], + "score": 1.0, + "content": "Table 2: AUROC of RAPP and the baselines.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 95, + 505, + 346 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 95, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 346 + ], + "score": 0.985, + "html": "
DatasetAEVAEAAE
ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.7230.7030.7110.7000.6750.7110.7260.7110.724
OTTO0.6170.6160.6650.6300.6310.6490.6170.6180.665
SNSR0.6130.6110.6140.6080.6200.6580.6120.6080.609
MNIST0.8250.8810.8990.9000.9650.9650.8470.9110.929
F-MNIST0.7120.7250.7340.7250.7280.7550.7210.7100.727
Unimodal Normality
MI-F0.6070.6700.7050.5910.5720.6780.6320.6920.704
MI-V0.8970.8980.9070.8450.8330.9030.8950.8910.904
EOPT0.6100.6070.6250.6750.6340.5960.6060.6030.634
NASA0.7190.7020.6920.7380.7140.7190.7120.6950.688
RARM0.6870.6740.6860.5870.5650.6480.6640.6750.675
STL0.8700.8560.8300.8500.8240.7920.8750.8610.830
OTTO0.8290.8290.8320.8310.8350.8330.8280.8280.831
SNSR0.9790.9820.9900.9750.9790.9640.9790.9830.991
MNIST0.9720.9800.9790.9800.9870.9890.9720.9660.977
F-MNIST0.9240.9280.9330.9260.9260.9420.9220.9050.928
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DatasetOCNNGPNDDSVDDGTNAPAENAPVAENAPAAE
Multimodal Normality (Novelty] Ratio: 35%)
MNIST F-MNIST0.600 0.6090.501 0.6910.622 0.6100.893 0.7250.899 0.7340.965 0.7550.929 0.727
Unimodal Normality (Novelty ] Ratio: 50%)
MNIST0.9270.9710.9220.9740.9790.9890.977
F-MNIST0.9150.9170.9230.9350.9330.9420.928
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DatasetAEVAEAAE
ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.7230.7030.7110.7000.6750.7110.7260.7110.724
OTTO0.6170.6160.6650.6300.6310.6490.6170.6180.665
SNSR0.6130.6110.6140.6080.6200.6580.6120.6080.609
MNIST0.8250.8810.8990.9000.9650.9650.8470.9110.929
F-MNIST0.7120.7250.7340.7250.7280.7550.7210.7100.727
Unimodal Normality
MI-F0.6070.6700.7050.5910.5720.6780.6320.6920.704
MI-V0.8970.8980.9070.8450.8330.9030.8950.8910.904
EOPT0.6100.6070.6250.6750.6340.5960.6060.6030.634
NASA0.7190.7020.6920.7380.7140.7190.7120.6950.688
RARM0.6870.6740.6860.5870.5650.6480.6640.6750.675
STL0.8700.8560.8300.8500.8240.7920.8750.8610.830
OTTO0.8290.8290.8320.8310.8350.8330.8280.8280.831
SNSR0.9790.9820.9900.9750.9790.9640.9790.9830.991
MNIST0.9720.9800.9790.9800.9870.9890.9720.9660.977
F-MNIST0.9240.9280.9330.9260.9260.9420.9220.9050.928
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DatasetOCNNGPNDDSVDDGTNAPAENAPVAENAPAAE
Multimodal Normality (Novelty] Ratio: 35%)
MNIST F-MNIST0.600 0.6090.501 0.6910.622 0.6100.893 0.7250.899 0.7340.965 0.7550.929 0.727
Unimodal Normality (Novelty ] Ratio: 50%)
MNIST0.9270.9710.9220.9740.9790.9890.977
F-MNIST0.9150.9170.9230.9350.9330.9420.928
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Even, fbpca running only on CPU achieves at least", + "type": "text" + }, + { + "bbox": [ + 380, + 195, + 400, + 205 + ], + "score": 0.64, + "content": "2 . 4 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 194, + 438, + 208 + ], + "score": 1.0, + "content": "speedup.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 182, + 505, + 208 + ] + }, + { + "type": "image", + "bbox": [ + 107, + 213, + 503, + 320 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 213, + 503, + 320 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 213, + 503, + 320 + ], + "spans": [ + { + "bbox": [ + 107, + 213, + 503, + 320 + ], + "score": 0.967, + "type": "image", + "image_path": "c3d2c9664c6a9e393392b5f66dac8b7d4c34056cddbceceac73865babee145a3.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 107, + 213, + 503, + 248.66666666666666 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 107, + 248.66666666666666, + 503, + 284.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 107, + 284.3333333333333, + 503, + 320.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 329, + 465, + 342 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 329, + 466, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 466, + 343 + ], + "score": 1.0, + "content": "The detailed setups to obtain the matrices for the experiment are given in the table below:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + } + ], + "index": 11.0 + }, + { + "type": "table", + "bbox": [ + 112, + 350, + 500, + 546 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 112, + 350, + 500, + 546 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 350, + 500, + 546 + ], + "spans": [ + { + "bbox": [ + 112, + 350, + 500, + 546 + ], + "score": 0.982, + "html": "
MNISTOTTOSNSR
DepthBottleneck SizeDepthBottleneck SizeDepthBottleneck Size
12010020402090
21810018401890
31610016401690
41410014401490
51210012401290
61010010401090
78100840890
86100640690
94100440490
102100240290
11280２２230270
1226020250
1324010230
14220210
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AEVAEAAE
Dataset ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.723 (0.005)0.703 (0.007)0.711 (0.006)0.700 (0.012)0.675 (0.018)0.711 (0.012)0.726 (0.009)0.711 (0.011)0.724 (0.016)
OTTO0.617 (0.002)0.616 (0.001)0.665 (0.004)0.630 (0.003)0.631 (0.005)0.649 (0.003)0.617 (0.001)0.618 (0.003)0.665 (0.004)
SNSR0.613 (0.002)0.611 (0.003)0.614 (0.004)0.608 (0.001)0.620 (0.002)0.658 (0.003)0.612 (0.005)0.608 (0.005)0.609 (0.005)
MNIST0.825 (0.001)0.881 (0.007)0.899 (0.008)0.900 (0.004)0.965 (0.005)0.965 (0.003)0.847 (0.005)0.911 (0.006)0.929 (0.005)
F-MNIST0.712 (0.001)0.725 (0.004)0.734 (0.004)0.725 (0.003)0.728 (0.002)0.755 (0.006)0.721 (0.001)0.710 (0.005)0.727 (0.005)
Unimodal Normality
MI-F0.607 (0.023)0.670 (0.041)0.705 (0.018)0.591 (0.013)0.572 (0.019)0.678 (0.047)0.632 (0.028)0.692 (0.042)0.704 (0.024)
MI-V0.897 (0.010)0.898 (0.005)0.907 (0.005)0.845 (0.021)0.833 (0.032)0.903 (0.010)0.895 (0.005)0.891 (0.006)0.904 (0.005)
EOPT0.610 (0.015)0.607 (0.014)0.625 (0.008)0.675 (0.017)0.634 (0.007)0.596 (0.002)0.606 (0.011)0.603 (0.013)0.634 (0.010)
NASA0.719 (0.009)0.702 (0.015)0.692 (0.016)0.738 (0.009)0.714 (0.015)0.719 (0.009)0.712 (0.016)0.695 (0.027)0.688 (0.024)
RARM0.687 (0.038)0.674 (0.031)0.686 (0.030)0.587 (0.021)0.565 (0.029)0.648 (0.022)0.664 (0.015)0.675 (0.023)0.675 (0.040)
STL0.870 (0.007)0.856 (0.007)0.830 (0.007)0.850 (0.006)0.824 (0.003)0.792 (0.034)0.875 (0.003)0.861 (0.004)0.830 (0.008)
OTTO0.829 (0.001)0.829 (0.001)0.832 (0.001)0.831 (0.002)0.835 (0.003)0.833 (0.004)0.828 (0.002)0.828 (0.001)0.831 (0.002)
SNSR0.979 (0.001)0.982 (0.001)0.990 (0.001)0.975 (0.003)0.979 (0.002)0.964 (0.043)0.979 (0.001)0.983 (0.002)0.997 (0.001)
MNIST0.972 (0.001)0.980 (0.001)0.979 (0.000)0.980 (0.002)0.987 (0.002)0.989 (0.001)0.972 (0.001)0.966 (0.004)0.977 (0.001)
F-MNIST0.924 (0.002)0.928 (0.002)0.933 (0.002)0.926 (0.001)0.926 (0.001)0.942 (0.000)0.922 (0.001)0.905 (0.002)0.928 (0.002)
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AEVAEAAE
Dataset ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.723 (0.005)0.703 (0.007)0.711 (0.006)0.700 (0.012)0.675 (0.018)0.711 (0.012)0.726 (0.009)0.711 (0.011)0.724 (0.016)
OTTO0.617 (0.002)0.616 (0.001)0.665 (0.004)0.630 (0.003)0.631 (0.005)0.649 (0.003)0.617 (0.001)0.618 (0.003)0.665 (0.004)
SNSR0.613 (0.002)0.611 (0.003)0.614 (0.004)0.608 (0.001)0.620 (0.002)0.658 (0.003)0.612 (0.005)0.608 (0.005)0.609 (0.005)
MNIST0.825 (0.001)0.881 (0.007)0.899 (0.008)0.900 (0.004)0.965 (0.005)0.965 (0.003)0.847 (0.005)0.911 (0.006)0.929 (0.005)
F-MNIST0.712 (0.001)0.725 (0.004)0.734 (0.004)0.725 (0.003)0.728 (0.002)0.755 (0.006)0.721 (0.001)0.710 (0.005)0.727 (0.005)
Unimodal Normality
MI-F0.607 (0.023)0.670 (0.041)0.705 (0.018)0.591 (0.013)0.572 (0.019)0.678 (0.047)0.632 (0.028)0.692 (0.042)0.704 (0.024)
MI-V0.897 (0.010)0.898 (0.005)0.907 (0.005)0.845 (0.021)0.833 (0.032)0.903 (0.010)0.895 (0.005)0.891 (0.006)0.904 (0.005)
EOPT0.610 (0.015)0.607 (0.014)0.625 (0.008)0.675 (0.017)0.634 (0.007)0.596 (0.002)0.606 (0.011)0.603 (0.013)0.634 (0.010)
NASA0.719 (0.009)0.702 (0.015)0.692 (0.016)0.738 (0.009)0.714 (0.015)0.719 (0.009)0.712 (0.016)0.695 (0.027)0.688 (0.024)
RARM0.687 (0.038)0.674 (0.031)0.686 (0.030)0.587 (0.021)0.565 (0.029)0.648 (0.022)0.664 (0.015)0.675 (0.023)0.675 (0.040)
STL0.870 (0.007)0.856 (0.007)0.830 (0.007)0.850 (0.006)0.824 (0.003)0.792 (0.034)0.875 (0.003)0.861 (0.004)0.830 (0.008)
OTTO0.829 (0.001)0.829 (0.001)0.832 (0.001)0.831 (0.002)0.835 (0.003)0.833 (0.004)0.828 (0.002)0.828 (0.001)0.831 (0.002)
SNSR0.979 (0.001)0.982 (0.001)0.990 (0.001)0.975 (0.003)0.979 (0.002)0.964 (0.043)0.979 (0.001)0.983 (0.002)0.997 (0.001)
MNIST0.972 (0.001)0.980 (0.001)0.979 (0.000)0.980 (0.002)0.987 (0.002)0.989 (0.001)0.972 (0.001)0.966 (0.004)0.977 (0.001)
F-MNIST0.924 (0.002)0.928 (0.002)0.933 (0.002)0.926 (0.001)0.926 (0.001)0.942 (0.000)0.922 (0.001)0.905 (0.002)0.928 (0.002)
", + "type": "table", + "image_path": "557fc47fb2ab04303cad84b2e11500b40e864616a19c896da3bf5280bc872b42.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 105, + 163, + 505, + 303.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 105, + 303.33333333333337, + 505, + 443.66666666666674 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 105, + 443.66666666666674, + 505, + 584.0000000000001 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 5.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 105, + 81, + 482, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 483, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 483, + 96 + ], + "score": 1.0, + "content": "C COMPARING RAPP TO BASELINES OTHER THAN NEURAL NETWORKS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "The table below shows the result of comparing RAPP and baselines not based on neural networks.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "Note that the baselines are run with hyperparameters or hyperparameter search spaces provided", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 318, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 318, + 140 + ], + "score": 1.0, + "content": "in Ruff et al. (2018), while NAP is not further tuned.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "table", + "bbox": [ + 135, + 149, + 475, + 384 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 135, + 149, + 475, + 384 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 149, + 475, + 384 + ], + "spans": [ + { + "bbox": [ + 135, + 149, + 475, + 384 + ], + "score": 0.984, + "html": "
DatasetOCSVMISOFPCAkPCANAPAENAPvAENAPAAE
Multimodal Normality
STL0.6080.6350.7000.6930.7110.7110.724
OTTO0.6280.5480.5720.6440.6650.6490.665
SNSR0.5240.5260.5360.5800.6140.6580.606
MNIST F-MNIST0.566 0.5710.562 0.6500.698 0.6700.701 0.6760.899 0.7340.965 0.7550.929 0.727
Unimodal Normality
MI-F0.7770.8260.5460.8380.7050.6780.704
MI-V0.8400.8390.8740.8880.9070.9030.904
EOPT0.5970.5950.5520.6930.6250.5960.634
NASA0.5730.6640.7650.6990.6920.7190.688
RARM0.7490.7520.5870.7340.7340.6480.675
STL0.8790.8710.8830.8800.8800.7920.830
OTTO0.8210.6990.7990.8230.8230.8330.831
SNSR0.9590.8930.8940.9700.9700.964
0.991
MNIST0.8940.9190.9030.9060.9790.9890.977
F-MNIST0.8840.8540.9500.9430.9330.9420.928
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DatasetOCSVMISOFPCAkPCANAPAENAPvAENAPAAE
Multimodal Normality
STL0.6080.6350.7000.6930.7110.7110.724
OTTO0.6280.5480.5720.6440.6650.6490.665
SNSR0.5240.5260.5360.5800.6140.6580.606
MNIST F-MNIST0.566 0.5710.562 0.6500.698 0.6700.701 0.6760.899 0.7340.965 0.7550.929 0.727
Unimodal Normality
MI-F0.7770.8260.5460.8380.7050.6780.704
MI-V0.8400.8390.8740.8880.9070.9030.904
EOPT0.5970.5950.5520.6930.6250.5960.634
NASA0.5730.6640.7650.6990.6920.7190.688
RARM0.7490.7520.5870.7340.7340.6480.675
STL0.8790.8710.8830.8800.8800.7920.830
OTTO0.8210.6990.7990.8230.8230.8330.831
SNSR0.9590.8930.8940.9700.9700.964
0.991
MNIST0.8940.9190.9030.9060.9790.9890.977
F-MNIST0.8840.8540.9500.9430.9330.9420.928
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Name# Samples#Features# ClassDomainNovelty Target
MI-F25,286582CNC millingMachine not completed
MI-V23,12558CNC millingWorkpiece out-of-spec
EOPT90.51520Storage systemSystem failures
NASA4,68733AstronomyHazardous asteroids
RARM20.2216２２２２７RoboticsMalfunctions
STL1,94127SteelSurface defects
OTTO61,878939E-commerceTypes of products
SNSR58,5094811Electric CurrentsDefective conditions
MNIST70,00078410Hand written digitsDigits
F-MNIST70,00078410Fashion articlesArticles
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DatasetAEVAEAAE
ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.7230.7030.7110.7000.6750.7110.7260.7110.724
OTTO0.6170.6160.6650.6300.6310.6490.6170.6180.665
SNSR0.6130.6110.6140.6080.6200.6580.6120.6080.609
MNIST0.8250.8810.8990.9000.9650.9650.8470.9110.929
F-MNIST0.7120.7250.7340.7250.7280.7550.7210.7100.727
Unimodal Normality
MI-F0.6070.6700.7050.5910.5720.6780.6320.6920.704
MI-V0.8970.8980.9070.8450.8330.9030.8950.8910.904
EOPT0.6100.6070.6250.6750.6340.5960.6060.6030.634
NASA0.7190.7020.6920.7380.7140.7190.7120.6950.688
RARM0.6870.6740.6860.5870.5650.6480.6640.6750.675
STL0.8700.8560.8300.8500.8240.7920.8750.8610.830
OTTO0.8290.8290.8320.8310.8350.8330.8280.8280.831
SNSR0.9790.9820.9900.9750.9790.9640.9790.9830.991
MNIST0.9720.9800.9790.9800.9870.9890.9720.9660.977
F-MNIST0.9240.9280.9330.9260.9260.9420.9220.9050.928
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DatasetOCNNGPNDDSVDDGTNAPAENAPVAENAPAAE
Multimodal Normality (Novelty] Ratio: 35%)
MNIST F-MNIST0.600 0.6090.501 0.6910.622 0.6100.893 0.7250.899 0.7340.965 0.7550.929 0.727
Unimodal Normality (Novelty ] Ratio: 50%)
MNIST0.9270.9710.9220.9740.9790.9890.977
F-MNIST0.9150.9170.9230.9350.9330.9420.928
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AEVAEAAE
Dataset ReconSAPNAPReconSAPNAPReconSAPNAP
Multimodal Normality
STL0.723 (0.005)0.703 (0.007)0.711 (0.006)0.700 (0.012)0.675 (0.018)0.711 (0.012)0.726 (0.009)0.711 (0.011)0.724 (0.016)
OTTO0.617 (0.002)0.616 (0.001)0.665 (0.004)0.630 (0.003)0.631 (0.005)0.649 (0.003)0.617 (0.001)0.618 (0.003)0.665 (0.004)
SNSR0.613 (0.002)0.611 (0.003)0.614 (0.004)0.608 (0.001)0.620 (0.002)0.658 (0.003)0.612 (0.005)0.608 (0.005)0.609 (0.005)
MNIST0.825 (0.001)0.881 (0.007)0.899 (0.008)0.900 (0.004)0.965 (0.005)0.965 (0.003)0.847 (0.005)0.911 (0.006)0.929 (0.005)
F-MNIST0.712 (0.001)0.725 (0.004)0.734 (0.004)0.725 (0.003)0.728 (0.002)0.755 (0.006)0.721 (0.001)0.710 (0.005)0.727 (0.005)
Unimodal Normality
MI-F0.607 (0.023)0.670 (0.041)0.705 (0.018)0.591 (0.013)0.572 (0.019)0.678 (0.047)0.632 (0.028)0.692 (0.042)0.704 (0.024)
MI-V0.897 (0.010)0.898 (0.005)0.907 (0.005)0.845 (0.021)0.833 (0.032)0.903 (0.010)0.895 (0.005)0.891 (0.006)0.904 (0.005)
EOPT0.610 (0.015)0.607 (0.014)0.625 (0.008)0.675 (0.017)0.634 (0.007)0.596 (0.002)0.606 (0.011)0.603 (0.013)0.634 (0.010)
NASA0.719 (0.009)0.702 (0.015)0.692 (0.016)0.738 (0.009)0.714 (0.015)0.719 (0.009)0.712 (0.016)0.695 (0.027)0.688 (0.024)
RARM0.687 (0.038)0.674 (0.031)0.686 (0.030)0.587 (0.021)0.565 (0.029)0.648 (0.022)0.664 (0.015)0.675 (0.023)0.675 (0.040)
STL0.870 (0.007)0.856 (0.007)0.830 (0.007)0.850 (0.006)0.824 (0.003)0.792 (0.034)0.875 (0.003)0.861 (0.004)0.830 (0.008)
OTTO0.829 (0.001)0.829 (0.001)0.832 (0.001)0.831 (0.002)0.835 (0.003)0.833 (0.004)0.828 (0.002)0.828 (0.001)0.831 (0.002)
SNSR0.979 (0.001)0.982 (0.001)0.990 (0.001)0.975 (0.003)0.979 (0.002)0.964 (0.043)0.979 (0.001)0.983 (0.002)0.997 (0.001)
MNIST0.972 (0.001)0.980 (0.001)0.979 (0.000)0.980 (0.002)0.987 (0.002)0.989 (0.001)0.972 (0.001)0.966 (0.004)0.977 (0.001)
F-MNIST0.924 (0.002)0.928 (0.002)0.933 (0.002)0.926 (0.001)0.926 (0.001)0.942 (0.000)0.922 (0.001)0.905 (0.002)0.928 (0.002)
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DatasetOCSVMISOFPCAkPCANAPAENAPvAENAPAAE
Multimodal Normality
STL0.6080.6350.7000.6930.7110.7110.724
OTTO0.6280.5480.5720.6440.6650.6490.665
SNSR0.5240.5260.5360.5800.6140.6580.606
MNIST F-MNIST0.566 0.5710.562 0.6500.698 0.6700.701 0.6760.899 0.7340.965 0.7550.929 0.727
Unimodal Normality
MI-F0.7770.8260.5460.8380.7050.6780.704
MI-V0.8400.8390.8740.8880.9070.9030.904
EOPT0.5970.5950.5520.6930.6250.5960.634
NASA0.5730.6640.7650.6990.6920.7190.688
RARM0.7490.7520.5870.7340.7340.6480.675
STL0.8790.8710.8830.8800.8800.7920.830
OTTO0.8210.6990.7990.8230.8230.8330.831
SNSR0.9590.8930.8940.9700.9700.964
0.991
MNIST0.8940.9190.9030.9060.9790.9890.977
F-MNIST0.8840.8540.9500.9430.9330.9420.928
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0000000000000000000000000000000000000000..393a311ec87eb0f2e1623aeaf71f10ae4e8c45f4 --- /dev/null +++ b/parse/train/Hkl1iRNFwS/Hkl1iRNFwS_content_list.json @@ -0,0 +1,1403 @@ +[ + { + "type": "text", + "text": "THE EARLY PHASE OF NEURAL NETWORK TRAINING ", + "text_level": 1, + "bbox": [ + 173, + 98, + 818, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jonathan Frankle† MIT CSAIL ", + "bbox": [ + 183, + 145, + 315, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "David J. Schwab CUNY ITS Facebook AI Research ", + "bbox": [ + 383, + 145, + 534, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ari S. Morcos Facebook AI Research ", + "bbox": [ + 607, + 145, + 758, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 223, + 544, + 238 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent studies have shown that many important aspects of neural network learning take place within the very earliest iterations or epochs of training. For example, sparse, trainable sub-networks emerge (Frankle et al., 2019), gradient descent moves into a small subspace (Gur-Ari et al., 2018), and the network undergoes a critical period (Achille et al., 2019). Here we examine the changes that deep neural networks undergo during this early phase of training. We perform extensive measurements of the network state during these early iterations of training and leverage the framework of Frankle et al. (2019) to quantitatively probe the weight distribution and its reliance on various aspects of the dataset. We find that, within this framework, deep networks are not robust to reinitializing with random weights while maintaining signs, and that weight distributions are highly non-independent even after only a few hundred iterations. Despite this behavior, pre-training with blurred inputs or an auxiliary self-supervised task can approximate the changes in supervised networks, suggesting that these changes are not inherently label-dependent, though labels significantly accelerate this process. Together, these results help to elucidate the network changes occurring during this pivotal initial period of learning. ", + "bbox": [ + 233, + 257, + 764, + 492 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 522, + 336, + 537 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Over the past decade, methods for successfully training big, deep neural networks have revolutionized machine learning. Yet surprisingly, the underlying reasons for the success of these approaches remain poorly understood, despite remarkable empirical performance (Santurkar et al., 2018; Zhang et al., 2017). A large body of work has focused on understanding what happens during the later stages of training (Neyshabur et al., 2019; Yaida, 2019; Chaudhuri & Soatto, 2017; Wei & Schwab, 2019), while the initial phase has been less explored. However, a number of distinct observations indicate that significant and consequential changes are occurring during the most early stage of training. These include the presence of critical periods during training (Achille et al., 2019), the dramatic reshaping of the local loss landscape (Sagun et al., 2017; Gur-Ari et al., 2018), and the necessity of rewinding in the context of the lottery ticket hypothesis (Frankle et al., 2019). Here we perform a thorough investigation of the state of the network in this early stage. ", + "bbox": [ + 174, + 555, + 825, + 708 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To provide a unified framework for understanding the changes the network undergoes during the early phase, we employ the methodology of iterative magnitude pruning with rewinding (IMP), as detailed below, throughout the bulk of this work (Frankle & Carbin, 2019; Frankle et al., 2019). The initial lottery ticket hypothesis, which was validated on comparatively small networks, proposed that small, sparse sub-networks found via pruning of converged larger models could be trained to high performance provided they were initialized with the same values used in the training of the unpruned model (Frankle & Carbin, 2019). However, follow-up work found that rewinding the weights to their values at some iteration early in the training of the unpruned model, rather than to their initial values, was necessary to achieve good performance on deeper networks such as ResNets (Frankle et al., 2019). This observation suggests that the changes in the network during this initial phase are vital for the success of the training of small, sparse sub-networks. As a result, this paradigm provides a simple and quantitative scheme for measuring the importance of the weights at various points early in training within an actionable and causal framework. ", + "bbox": [ + 174, + 715, + 825, + 895 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We make the following contributions, all evaluated across three different network architectures: ", + "bbox": [ + 173, + 103, + 799, + 118 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We provide an in-depth overview of various statistics summarizing learning over the early part of training. \n2. We evaluate the impact of perturbing the state of the network in various ways during the early phase of training, finding that: (i) counter to observations in smaller networks (Zhou et al., 2019), deeper networks are not robust to reinitializion with random weights, but maintained signs (ii) the distribution of weights after the early phase of training is already highly non-i.i.d., as permuting them dramatically harms performance, even when signs are maintained (iii) both of the above perturbations can roughly be approximated by simply adding noise to the network weights, though this effect is stronger for (ii) than (i) ", + "bbox": [ + 205, + 131, + 825, + 286 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3. We measure the data-dependence of the early phase of training, finding that pre-training using only $p ( x )$ can approximate the changes that occur in the early phase of training, though pre-training must last for far longer ${ \\sim } 3 2 \\times$ longer) and not be fed misleading labels. ", + "bbox": [ + 217, + 291, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 KNOWN PHENOMENA IN THE EARLY PHASE OF TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 353, + 691, + 369 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Lottery ticket rewinding: The original lottery ticket paper (Frankle & Carbin, 2019) rewound weights to initialization, i.e., $k ~ = ~ 0$ , during IMP. Follow up work on larger models demonstrated that it is necessary to rewind to a later point during training for IMP to succeed, i.e., $k \\ < < \\ T$ , where $T$ is total training iterations (Frankle et al., 2019). Notably, the benefit of rewinding to a later point in training saturates quickly, roughly between 500 and 2000 iterations for ResNet-20 on CIFAR-10 (Figure 1). This timescale is strikingly similar to the changes in the Hessian described below. ", + "bbox": [ + 174, + 385, + 483, + 565 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Hessian eigenspectrum: The shape of the loss landscape around the network state also appears ", + "bbox": [ + 174, + 573, + 483, + 599 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/8c4ca4d2642f445b29e873fc950bbc21f6cd35aa36fbbe51f69c185df46c53ba.jpg", + "image_caption": [ + "Figure 1: Accuracy of IMP when rewinding to various iterations of the early phase for ResNet20 sub-networks as a function of sparsity level. " + ], + "image_footnote": [], + "bbox": [ + 501, + 404, + 784, + 531 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "to change rapidly during the early phase of training (Sagun et al., 2017; Gur-Ari et al., 2018). At initialization, the Hessian of the loss contains a number of large positive and negative eigenvalues. However, very rapidly the curvature is reshaped in a few marked ways: a few large eigenvalues emerge, the bulk eigenvalues are close to zero, and the negative eigenvalues become very small. Moreover, once the Hessian spectrum has reshaped, gradient descent appears to occur largely within the top subspace of the Hessian (Gur-Ari et al., 2018). These results have been largely confirmed in large scale studies (Ghorbani et al., 2019), but note they depend to some extent on architecture and (absence of) batch normalization (Ioffe & Szegedy, 2015). A notable exception to this consistency is the presence of substantial $L _ { 1 }$ energy of negative eigenvalues for models trained on ImageNet. ", + "bbox": [ + 173, + 601, + 825, + 724 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Critical periods in deep learning: Achille et al. (2019) found that perturbing the training process by providing corrupted data early on in training can result in irrevocable damage to the final performance of the network. Note that the timescales over which the authors find a critical period extend well beyond those we study here. However, architecture, learning rate schedule, and regularization all modify the timing of the critical period, and follow-up work found that critical periods were also present for regularization, in particular weight decay and data augmentation (Golatkar et al., 2019). ", + "bbox": [ + 174, + 732, + 825, + 815 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 PRELIMINARIES AND METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 835, + 521, + 852 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Networks: Throughout this paper, we study five standard convolutional neural networks for CIFAR-10. These include the ResNet-20 and ResNet-56 architectures designed for CIFAR-10 (He et al., 2015), the ResNet-18 architecture designed for ImageNet but commonly used on CIFAR-10 (He et al., 2015), the WRN-16-8 wide residual network (Zagoruyko & Komodakis, 2016), and the ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/bbc021b67f4758bf1ffddd590865638abb4a0901c8374f61644c3639d1985c0a.jpg", + "image_caption": [ + "Figure 2: Rough timeline of the early phase of training for ResNet-20 on CIFAR-10. " + ], + "image_footnote": [], + "bbox": [ + 197, + 108, + 771, + 236 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "VGG-13 network (Simonyan & Zisserman (2015) as adapted by Liu et al. (2019)). Throughout the main body of the paper, we show ResNet-20; in Appendix B, we present the same experiments for the other networks. Unless otherwise stated, results were qualitatively similar across all three networks. All experiments in this paper display the mean and standard deviation across five replicates with different random seeds. See Appendix A for further model details. ", + "bbox": [ + 174, + 300, + 825, + 369 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Iterative magnitude pruning with rewinding: In order to test the effect of various hypotheses about the state of sparse networks early in training, we use the Iterative Magnitude Pruning with rewinding (IMP) procedure of Frankle et al. (2019) to extract sub-networks from various points in training that could have learned on their own. The procedure involves training a network to completion, pruning the $20 \\%$ of weights with the lowest magnitudes globally throughout the network, and rewinding the remaining weights to their values from an earlier iteration $k$ during the initial, pre-pruning training run. This process is iterated to produce networks with high sparsity levels. As demonstrated in Frankle et al. (2019), IMP with rewinding leads to sparse sub-networks which can train to high performance even at high sparsity levels $> 9 0 \\%$ . ", + "bbox": [ + 174, + 377, + 825, + 502 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 shows the results of the IMP with rewinding procedure, showing the accuracy of ResNet20 at increasing sparsity when performing this procedure for several rewinding values of $k$ . For $k \\geq 5 0 0$ , sub-networks can match the performance of the original network with $1 6 . 8 \\%$ of weights remaining. For $k > 2 0 0 0$ , essentially no further improvement is observed (not shown). ", + "bbox": [ + 174, + 510, + 825, + 565 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 THE STATE OF THE NETWORK EARLY IN TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 594, + 632, + 611 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Many of the aforementioned papers refer to various points in the “early” part of training. In this section, we descriptively chart the state of ResNet-20 during the earliest phase of training to provide context for this related work and our subsequent experiments. We specifically focus on the first 4,000 iterations (10 epochs). See Figure A3 for the characterization of additional networks. We include a summary of these results for ResNet-20 as a timeline in Figure 2, and include a broader timeline including results from several previous papers for ResNet-18 in Figure A1. ", + "bbox": [ + 174, + 631, + 825, + 715 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As shown in Figure 3, during the earliest ten iterations, the network undergoes substantial change. It experiences large gradients that correspond to a rapid increase in distance from the initialization and a large number of sign changes of the weights. After these initial iterations, gradient magnitudes drop and the rate of change in each of the aforementioned quantities gradually slows through the remainder of the period we observe. Interestingly, gradient magnitudes reach a minimum after the first 200 iterations and subsequently increase to a stable level by iteration 500. Evaluation accuracy, improves rapidly, reaching $55 \\%$ by the end of the first epoch (400 iterations), more than halfway to the final $9 1 . 5 \\%$ . By 2000 iterations, accuracy approaches $80 \\%$ . ", + "bbox": [ + 174, + 722, + 825, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "During the first 4000 iterations of training, we observe three sub-phases. In the first phase, lasting only the initial few iterations, gradient magnitudes are very large and, consequently, the network changes rapidly. In the second phase, lasting about 500 iterations, performance quickly improves, weight magnitudes quickly increase, sign differences from initialization quickly increase, and gradient magnitudes reach a minimum before settling at a stable level. Finally, in the third phase, all of these quantities continue to change in the same direction, but begin to decelerate. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/536f691a12fa237fb72a5290ddf148134e074daf7b9eb9951f39a422f822647b.jpg", + "image_caption": [ + "Figure 3: Basic telemetry about the state of ResNet-20 during the first 4000 iterations (10 epochs). Top row: evaluation accuracy/loss; average weight magnitude; percentage of weights that change sign from initialization; the values of ten randomly-selected weights. Bottom row: gradient magnitude; L2 distance of weights from their initial values and final values at the end of training; cosine similarity of weights from their initial values and final values at the end of training. " + ], + "image_footnote": [], + "bbox": [ + 200, + 102, + 797, + 285 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/9b938cd5121d512198fefd1d83a99771476d10b7fe225ff94604b45fa851893d.jpg", + "image_caption": [ + "Figure 4: Performance of an IMP-derived sub-network of ResNet-20 on CIFAR-10 initialized to the signs at iteration 0 or $k$ and the magnitudes at iteration 0 or $k$ . Left: $k = 5 0 0$ . Right: $k = 2 0 0 0$ . " + ], + "image_footnote": [], + "bbox": [ + 209, + 390, + 784, + 510 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5 PERTURBING NEURAL NETWORKS EARLY IN TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 580, + 669, + 597 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 1 shows that the changes in the network weights over the first 500 iterations of training are essential to enable high performance at high sparsity levels. What features of this weight transformation are necessary to recover increased performance? Can they be summarized by maintaining the weight signs, but discarding their magnitudes as implied by Zhou et al. (2019)? Can they be represented distributionally? In this section, we evaluate these questions by perturbing the early state of the network in various ways. Concretely, we either add noise or shuffle the weights of IMP sub-networks of ResNet-20 across different network sub-compenents and examine the effect on the network’s ability to learn thereafter. The sub-networks derived by IMP with rewinding make it possible to understand the causal impact of perturbations on sub-networks that are as capable as the full networks but more visibly decline in performance when improperly configured. To enable comparisons between the experiments in Section 5 and provide a common frame of reference, we measure the effective standard deviation of each perturbation, i.e. stddev $( w _ { p e r t u r b } - w _ { o r i g } )$ . ", + "bbox": [ + 174, + 613, + 825, + 780 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5.1 ARE SIGNS ALL YOU NEED? ", + "text_level": 1, + "bbox": [ + 176, + 799, + 415, + 813 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Zhou et al. (2019) show that, for a set of small convolutional networks, signs alone are sufficient to capture the state of lottery ticket sub-networks. However, it is unclear whether signs are still sufficient for larger networks early in training. In Figure 4, we investigate the impact of combining the magnitudes of the weights from one time-point with the signs from another. We found that the signs at iteration 500 paired with the magnitudes from initialization (red line) or from a separate random initialization (green line) were insufficient to maintain the performance reached by using both signs and magnitudes from iteration 500 (orange line), and performance drops to that of using both magnitudes and signs from initialization (blue line). However, while using the magnitudes from iteration 500 and the signs from initialization, performance is still substantially better than initialization signs and magnitudes. In addition, the overall perturbation to the network by using the magnitudes at iteration 500 and signs from initialization (mean: 0.0, stddev: 0.033) is smaller than by using the signs at iteration 500 and the magnitudes from initialization $( 0 . 0 \\pm 0 . 0 4 2$ , mean $\\pm$ std). These results suggest that the change in weight magnitudes over the first 500 iterations of training are substantially more important than the change in the signs for enabling subsequent training. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/8047cf2a48a2e5be51317667d22db2c6655703fd386a19e3004bc37d3bf38ab8.jpg", + "image_caption": [ + "Figure 5: Performance of an IMP-derived ResNet-20 sub-network on CIFAR-10 initialized with the weights at iteration $k$ permuted within various structural elements. Left: $k = 5 0 0$ . Right: $k = 2 0 0 0$ . " + ], + "image_footnote": [], + "bbox": [ + 209, + 103, + 784, + 229 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8df733a2ecf858a504a521a4c2ece79ef9ff72c54a282b564ce7001e5b6fa5fd.jpg", + "image_caption": [ + "Figure 6: The effect of training an IMP-derived sub-network of ResNet-20 on CIFAR-10 initialized with the weights at iteration $k$ as shuffled within various structural elements where shuffling only occurs between weights with the same sign. Left: $k = 5 0 0$ . Right: $k = 2 0 0 0$ . " + ], + "image_footnote": [], + "bbox": [ + 209, + 296, + 784, + 414 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 503, + 825, + 601 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "By iteration 2000, however, pairing the iteration 2000 signs with magnitudes from initialization (red line) reaches similar performance to using the signs from initialization and the magnitudes from iteration 2000 (purple line) though not as high performance as using both from iteration 2000. This result suggests that network signs undergo important changes between iterations 500 and 2000, as only $9 \\%$ of signs change during this period. Our results also suggest that counter to the observations of Zhou et al. (2019) in shallow networks, signs are not sufficient in deeper networks. ", + "bbox": [ + 174, + 608, + 825, + 691 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.2 ARE WEIGHT DISTRIBUTIONS I.I.D.? ", + "text_level": 1, + "bbox": [ + 178, + 713, + 470, + 727 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Can the changes in weights over the first $k$ iterations be approximated distributionally? To measure this, we permuted the weights at iteration $k$ within various structural sub-components of the network (globally, within layers, and within convolutional filters). If networks are robust to these permutations, it would suggest that the weights in such sub-compenents might be approximated and sampled from. As Figure 5 shows, however, we found that performance was not robust to shuffling weights globally (green line) or within layers (red line), and drops substantially to no better than that of the original initialization (blue line) at both 500 and 2000 iterations.1 Shuffling within filters (purple line) performs slightly better, but results in a smaller overall perturbation $( 0 . 0 \\pm 0 . 0 9 2$ for $k = 5 0 0$ ) than shuffling layerwise $( 0 . 0 { \\pm } 0 . 1 4 3 )$ or globally $( 0 . 0 { \\pm } 0 . 1 4 4 )$ , suggesting that this change in perturbation strength may simply account for the difference. ", + "bbox": [ + 174, + 741, + 825, + 880 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/d765cebe2c186de4c5b3ee8cc5eadc7096326e5470eaf14de37b6bd970c9d02d.jpg", + "image_caption": [ + "Figure 7: The effect of training an IMP-derived sub-network of ResNet-20 on CIFAR-10 initialized with the weights at iteration $k$ and Gaussian noise of $n \\sigma$ , where $\\sigma$ is the standard deviation of the initialization distribution for each layer. Left: $k = 5 0 0$ . Right: $k = 2 0 0 0$ . " + ], + "image_footnote": [], + "bbox": [ + 209, + 106, + 784, + 223 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Are the signs from the rewinding iteration, $k$ , sufficient to recover the damage caused by permutation? In Figure 6, we also consider shuffling only amongst weights that have the same sign. Doing so substantially improves the performance of the filter-wise shuffle; however, it also reduces the extent of the overall perturbation $( 0 . 0 \\pm 0 . 0 4 9$ for $k = 5 0 0$ ). It also improves the performance of shuffling within layers slightly for $k = 5 0 0$ and substantially for $k = 2 0 0 0$ . We attribute the behavior for $k = 2 0 0 0$ to the signs just as in Figure 4: when the magnitudes are similar in value (Figure 4 red line) or distribution (Figure 6 red and green lines), using the signs improves performance. Reverting back to the initial signs while shuffling magnitudes within layers (brown line), however, damages the network too severely $( 0 . 0 \\pm 0 . 0 8 7$ for $k = 5 0 0 \\mathrm { \\AA }$ ) to yield any performance improvement over random noise. These results suggest that, while the signs from initialization are not sufficient for high performance at high sparsity as shown in Section 5.1, the signs from the rewinding iteration are sufficient to recover the damage caused by permutation, at least to some extent. ", + "bbox": [ + 174, + 308, + 825, + 474 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.3 IS IT ALL JUST NOISE? ", + "text_level": 1, + "bbox": [ + 176, + 492, + 370, + 506 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Some of our previous results suggested that the impact of signs and permutations may simply reduce to adding noise to the weights. To evaluate this hypothesis, we next study the effect of simply adding Gaussian noise to the network weights at iteration $k$ . To add noise appropriately for layers with different scales, the standard deviation of the noise added for each layer was normalized to a multiple of the standard deviation $\\sigma$ of the initialization distribution for that layer. In Figure 7, we see that for iteration $k = 5 0 0$ , sub-networks can tolerate $0 . 5 \\sigma$ to $1 \\sigma$ of noise before performance degrades back to that of the original initialization at higher levels of noise. For iteration $k = 2 0 0 0$ , networks are surprisingly robust to noise up to $1 \\sigma$ , and even $2 \\sigma$ exhibits nontrivial performance. ", + "bbox": [ + 174, + 517, + 825, + 628 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Figure 8, we plot the performance of each network at a fixed sparsity level as a function of the effective standard deviation of the noise imposed by each of the aforementioned perturbations. We find that the standard deviation of the effective noise explained fairly well the resultant performance $k = 5 0 0$ : $r = - 0 . 6 7 2$ , $p = 0 . 0 0 8$ ; $k = 2 0 0 0$ : $r = - 0 . 7 2 6$ , $p = 0 . 0 0 3 )$ . As expected, perturbations that preserved the performance of the network generally resulted in smaller changes to the state of the network at iteration $k$ . Interestingly, experiments that mixed signs and magnitudes from different points in training (green points) aligned least well with this pattern: the standard deviation of the perturbation is roughly similar among all of these experiments, but the accuracy of the resulting networks changes substantially. This result suggests that although the standard deviation of the noise is certainly indicative of lower accuracy, there are still specific perturbations that, while small in overall magnitude, can have a large effect on the network’s ability to learn, suggesting that the observed perturbation effects are not, in fact, just a consequence of noise. ", + "bbox": [ + 174, + 636, + 825, + 803 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 THE DATA-DEPENDENCE OF NEURAL NETWORKS EARLY IN TRAINING ", + "text_level": 1, + "bbox": [ + 173, + 821, + 797, + 838 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Section 5 suggests that the change in network behavior by iteration $k$ is not due to easilyascertainable, distributional properties of the network weights and signs. Rather, it appears that training is required to reach these network states. It is unclear, however, the extent to which various aspects of the data distribution are necessary. Mainly, is the change in weights during the early phase of training dependent on $p ( x )$ or $p ( y | \\dot { x } ) \\dot { }$ ? Here, we attempt to answer this question by measuring the extent to which we can re-create a favorable network state for sub-network training using restricted information from the training data and labels. In particular, we consider pre-training the network with techniques that ignore labels entirely (self-supervised rotation prediction, Section 6.2), provide misleading labels (training with random labels, Section 6.1), or eliminate information from examples (blurring training examples Section 6.3). ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/f475c2671eb155c5f5bfa3ab2df77732fada8935c1ae32dfdf549d2039aa9131.jpg", + "image_caption": [ + "Figure 8: The effective standard deviation of various perturbations as a function of mean evaluation accuracy (across 5 seeds) at sparsity $2 6 . 2 \\%$ . The mean of each perturbation was approximately 0. Left: $k = 5 0 0$ , $r = - 0 . 6 7 2$ , $p = 0 . 0 0 8$ ; Right: $k = 2 0 0 0$ , $r = - 0 . 7 2 6$ , $p = 0 . 0 0 3$ . " + ], + "image_footnote": [], + "bbox": [ + 209, + 104, + 785, + 321 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/8715684b756ced9cf2ac21b60d181576c6c05b8ece9df66ac1df49bf882960a3.jpg", + "image_caption": [ + "Figure 9: The effect of pre-training ResNet-20 on CIFAR-10 with random labels, self-supervised rotation, $4 \\mathbf { x }$ blurring, and $4 \\mathbf { x }$ blurring and self-supervised rotation. " + ], + "image_footnote": [], + "bbox": [ + 207, + 406, + 785, + 666 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 750, + 825, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first train a randomly-initialized, unpruned network on CIFAR-10 on the pre-training task for a set number of epochs. After pre-training, we train the network normally as if the pre-trained state were the original initialization. We then use the state of the network at the end of the pre-training phase as the “initialization” to find masks for IMP. Finally, we examine the performance of the IMPpruned sub-networks as initialized using the state after pre-training. This experiment determines the extent to which pre-training places the network in a state suitable for sub-network training as compared to using the state of the network at iteration $k$ of training on the original task. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 RANDOM LABELS ", + "text_level": 1, + "bbox": [ + 174, + 104, + 338, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To evaluate whether this phase of training is dependent on underlying structure in the data, we drew inspiration from Zhang et al. (2017) and pre-trained networks on data with randomized labels. This experiment tests whether the input distribution of the training data is sufficient to put the network in a position from which IMP with rewinding can find a sparse, trainable sub-network despite the presence of incorrect (not just missing) labels. Figure 9 (upper left) shows that pre-training on random labels for up to 10 epochs provides no improvement above rewinding to iteration 0 and that pre-training for longer begins to hurt accuracy. This result suggests that, though it is still possible that labels may not be required for learning, the presence incorrect labels is sufficient to prevent learning which approximates the early phase of training. ", + "bbox": [ + 174, + 130, + 825, + 256 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 SELF-SUPERVISED ROTATION PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 273, + 504, + 287 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "What if we remove labels entirely? Is $p ( x )$ sufficient to approximate the early phase of training? Historically, neural network training often involved two steps: a self-supervised pre-training phase followed by a supervised phase on the target task (Erhan et al., 2010). Here, we consider one such self-supervised technique: rotation prediction (Gidaris et al., 2018). During the pre-training phase, the network is presented with a training image that has randomly been rotated $9 0 n$ degrees (where $n \\in \\{ 0 , 1 , 2 , 3 \\bar \\} )$ . The network must classify examples by the value of $n$ . If self-supervised pretraining can approximate the early phase of training, it would suggest that $p ( x )$ is sufficient on its own. Indeed, as shown in Figure 9 (upper right), this pre-training regime leads to well-trainable subnetworks, though networks must be trained for many more epochs compared to supervised training (40 compared to 1.25, or a factor of $3 2 \\times 1$ ). This result suggests that the labels for the ultimate task themselves are not necessary to put the network in such a state (although explicitly misleading labels are detrimental). We emphasize that the duration of the pre-training phase required is an order of magnitude larger than the original rewinding iteration, however, suggesting that labels add important information which accelerates the learning process. ", + "bbox": [ + 174, + 299, + 825, + 494 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.3 BLURRING TRAINING EXAMPLES ", + "text_level": 1, + "bbox": [ + 178, + 512, + 442, + 526 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To probe the importance of $p ( x )$ for the early phase of training, we study the extent to which the training input distribution is necessary. Namely, we pretrain using blurred training inputs with the correct labels. Following Achille et al. (2019), we blur training inputs by downsampling by $4 \\mathbf { x }$ and then upsampling back to the full size. Figure 9 (bottom left) shows that this pre-training method succeeds: after 40 epochs of pre-training, IMP with rewinding can find sub-networks that are similar in performance to those found after training on the original task for 500 iterations (1.25 epochs). ", + "bbox": [ + 174, + 537, + 825, + 622 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Due to the success of the the rotation and blurring pre-training tasks, we explored the effect of combining these pre-training techniques. Doing so tests the extent to which we can discard both the training labels and some information from the training inputs. Figure 9 (bottom right) shows that doing so provides the network too little information: no amount of pre-training we considered makes it possible for IMP with rewinding to find sub-networks that perform tangibly better than rewinding to iteration 0. Interestingly however, as shown in Appendix B, trainable sub-networks are found for VGG-13 with this pre-training regime, suggesting that different network architectures have different sensitivities to the deprivation of labels and input content. ", + "bbox": [ + 174, + 628, + 825, + 741 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.4 SPARSE PRETRAINING ", + "text_level": 1, + "bbox": [ + 176, + 758, + 369, + 772 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Since sparse sub-networks are often challenging to train from scratch without the proper initialization (Han et al., 2015; Liu et al., 2019; Frankle & Carbin, 2019), does pre-training make it easier for sparse neural networks to learn? Doing so would serve as a rough form of curriculum learning (Bengio et al., 2009) for sparse neural networks. We experimented with training sparse sub-networks of ResNet-20 (IMP sub-networks, randomly reinitialized sub-networks, and randomly pruned subnetworks) first on self-supervised rotation and then on the main task, but found no benefit beyond rewinding to iteration 0 (Figure 10). Moreover, doing so when starting from a sub-network rewound to iteration 500 actually hurts final accuracy. This result suggests that while pre-training is sufficient to approximate the early phase of supervised training with an appropriately structured mask, it is not sufficient to do so with an inappropriate mask. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/72f8803cfddf40a06eaf44ba7dce17941a23f860928741782f1d1815af395e7b.jpg", + "image_caption": [ + "Figure 10: The effect of pretraining sparse sub-networks of Resnet-20 (rewound to iteration 500) with 40 epochs of self-supervised rotation before training on CIFAR-10. " + ], + "image_footnote": [], + "bbox": [ + 341, + 104, + 653, + 242 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 315, + 310, + 330 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we first performed extensive measurements of various statistics summarizing learning over the early part of training. Notably, we uncovered 3 sub-phases: in the very first iterations, gradient magnitudes are anomalously large and motion is rapid. Subsequently, gradients overshoot to smaller magnitudes before leveling off while performance increases rapidly. Then, learning slowly begins to decelerate. We then studied a suite of perturbations to the network state in the early phase finding that, counter to observations in smaller networks (Zhou et al., 2019), deeper networks are not robust to reinitializing with random weights with maintained signs. We also found that the weight distribution after the early phase of training is highly non-independent. Finally, we measured the data-dependence of the early phase with the surprising result that pre-training on a self-supervised task yields equivalent performance to late rewinding with IMP. ", + "bbox": [ + 174, + 347, + 825, + 486 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "These results have significant implications for the lottery ticket hypothesis. The seeming necessity of late rewinding calls into question certain interpretations of lottery tickets as well as the ability to identify sub-networks at initialization. Our observation that weights are highly non-independent at the rewinding point suggests that the weights at this point cannot be easily approximated, making approaches which attempt to “jump” directly to the rewinding point unlikely to succeed. However, our result that labels are not necessary to approximate the rewinding point suggests that the learning during this phase does not require task-specific information, suggesting that rewinding may not be necessary if networks are pre-trained appropriately. 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NetworkEpochsBatch SizeLearning RateParametersEval Accuracy
ResNet-201601280.1 (Mom 0.9)272K91.5 ± 0.2%
ResNet-561601280.1 (Mom 0.9)856K93.0 ± 0.1%
ResNet-181601280.1 (Mom 0.9)11.2M86.8 ± 0.3%
VGG-13160640.1 (Mom 0.9)9.42M93.5 ± 0.1%
WRN-16-81601280.1 (Mom 0.9)11.1M94.8 ± 0.1%
", + "bbox": [ + 210, + 137, + 787, + 237 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Table A1: Summary of the networks we study in this paper. We present ResNet-20 in the main body of the paper and the remaining networks in Appendix B. ", + "bbox": [ + 173, + 247, + 823, + 276 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Table A1 summarizes the networks. All networks follow the same training regime: we train with SGD for 160 epochs starting at learning rate 0.1 (momentum 0.9) and drop the learning rate by a factor of ten at epoch 80 and again at epoch 120. Training includes weight decay with weight 1e4. Data is augmented with normalization, random flips, and random crops up to four pixels in any direction. ", + "bbox": [ + 173, + 291, + 825, + 362 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B EXPERIMENTS FOR OTHER NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 382, + 534, + 398 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/872ddc7d0ad8582b233591a1e40741fe4f95c600a08270edc101a7423e02545a.jpg", + "image_caption": [ + "Figure A1: Rough timeline of the early phase of training for ResNet-18 on CIFAR-10, including results from previous papers. " + ], + "image_footnote": [], + "bbox": [ + 197, + 420, + 790, + 553 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/63f2f9a9f6e585e38e603d9a40fa15b8af8a6995ed2ca2c196aee9d7fbf2deb8.jpg", + "image_caption": [ + "Figure A2: The effect of IMP rewinding iteration on the accuracy of sub-networks at various levels of sparsity. Accompanies Figure 1. " + ], + "image_footnote": [], + "bbox": [ + 176, + 345, + 816, + 632 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/d8488c8ab8d7f0308f3b1974a7b9a84dca85c021322de3896d8b904f932747f4.jpg", + "image_caption": [ + "Figure A3: Basic telemetry about the state of all networks in Table A1 during the first 4000 iterations of training. Accompanies Figure 3. " + ], + "image_footnote": [], + "bbox": [ + 173, + 162, + 823, + 819 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/a34ceb4ae0c81987ecdc7f5ddfa2e60ff037795ef8cf8955eccec47d906fca5e.jpg", + "image_caption": [ + "Figure A4: The effect of training an IMP-derived sub-network initialized to the signs at iteration 0 or $k$ and the magnitudes at iteration 0 or $k$ . Accompanies Figure A4. " + ], + "image_footnote": [], + "bbox": [ + 176, + 200, + 816, + 784 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/5c33032f73bacd97dba4aac435754655fdd79b3b50ca5595eb22c73c864aca92.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 179, + 213, + 490, + 340 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/0f3e0b165de498927e7248762a6b658ac2efbd505279c8b086ad1b4808cc7041.jpg", + "image_caption": [ + "Resnet-56 (CIFAR-10) - Rewinding Iteration 500 " + ], + "image_footnote": [], + "bbox": [ + 179, + 358, + 491, + 486 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/b3d85495d764c63e5b9793c943d132939f9fb8c23934c35e70bf5a2abaaf631b.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 503, + 214, + 815, + 340 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/dbb505d3de1c365cdfb743d4eade9f62de4ba15c2fa9ed7f280990be575bdcdd.jpg", + "image_caption": [ + "Resnet-56 (CIFAR-10) - Rewinding Iteration 2000 " + ], + "image_footnote": [], + "bbox": [ + 504, + 358, + 815, + 484 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/b2ccf52edf1b0b59acfd5e765e85299de16f99b871641d074efe1cd7ee7be4f1.jpg", + "image_caption": [ + "Resnet-18 (CIFAR-10) - Rewinding Iteration 500 " + ], + "image_footnote": [], + "bbox": [ + 179, + 502, + 491, + 630 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/fb413d3205393040787190042574869ec1120adff3ee9b8638e8bf905c8c50d6.jpg", + "image_caption": [ + "Resnet-18 (CIFAR-10) - Rewinding Iteration 2000 " + ], + "image_footnote": [], + "bbox": [ + 500, + 502, + 815, + 630 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/67363ef8fb1130a5957e75416ca688f6769e3620ac64cbc18953cfe4bd3ae060.jpg", + "image_caption": [ + "WRN-16-8 (CIFAR-10) - Rewinding Iteration 50 " + ], + "image_footnote": [], + "bbox": [ + 179, + 647, + 491, + 775 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/5cdf2d289db2a8d2df68fcf74b52f0a7e0936cdc876aaadb775053cac7397f21.jpg", + "image_caption": [ + "WRN-16-8 (CIFAR-10) - Rewinding Iteration 250 ", + "Figure A5: The effect of training an IMP-derived sub-network initialized with the weights at iteration $k$ as shuffled within various structural elements. Accompanies Figure 5. " + ], + "image_footnote": [], + "bbox": [ + 503, + 647, + 815, + 775 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/2bd0fbe5e838abb5ac1e1e4022ccb7706c047b174850e2a1882f0a3c6d0501dc.jpg", + "image_caption": [ + "Figure A6: The effect of training an IMP-derived sub-network initialized with the weights at iteration $k$ as shuffled within various structural elements where shuffling only occurs between weights with the same sign. Accompanies Figure 6. " + ], + "image_footnote": [], + "bbox": [ + 178, + 186, + 818, + 773 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/d7e58a8ae01b34aebcb964fc88e4a58f41aa2ab9218ed70fd5d4343139da08a6.jpg", + "image_caption": [ + "Figure A7: The effect of training an IMP-derived sub-network initialized with the weights at iteration $k$ and Gaussian noise of $n \\sigma$ , where $\\sigma$ is the standard deviation of the initialization distribution for each layer. Accompanies Figure 7. " + ], + "image_footnote": [], + "bbox": [ + 179, + 181, + 818, + 781 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/b52dfa34b65e90016f1295f7e726d50c1dfff13edbf013fa698a5e01479a5380.jpg", + "image_caption": [ + "Figure A8: The effective standard deviation of each of the perturbations studied in Section 5 as a function of mean evaluation accuracy (across five seeds). Accompanies Figure 8. " + ], + "image_footnote": [], + "bbox": [ + 241, + 50, + 754, + 893 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/8cf6dc60b9976983ad6d59be2cc9ad9e266bb88950bb4572c5653558c98a1ec5.jpg", + "image_caption": [ + "Figure A9: The effect of pre-training CIFAR-10 with random labels. Accompanies Figure 9. " + ], + "image_footnote": [], + "bbox": [ + 178, + 145, + 818, + 429 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/06aff5a9a34d2535adc00efc1e7ba5f1559550836e2f886fb09a8a1971d49a38.jpg", + "image_caption": [ + "Figure A10: The effect of pre-training CIFAR-10 with self-supervised rotation. Accompanies Figure 9. " + ], + "image_footnote": [], + "bbox": [ + 178, + 547, + 818, + 837 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/75b300fc6a420d980deba5c945bc99ec31f9b87ff18ecf8ff04f650c76cf3135.jpg", + "image_caption": [ + "Figure A11: The effect of pre-training CIFAR-10 with $4 \\mathbf { x }$ blurring. Accompanies Figure 9. " + ], + "image_footnote": [], + "bbox": [ + 178, + 145, + 818, + 428 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/6c798e68075d87eb505781c2e6300a3e943b845fe448ebf79e2ed47aea5ed65a.jpg", + "image_caption": [ + "VGG-13 (CIFAR-10) - 4x Blurring $^ +$ Self-Superivsed Rotation " + ], + "image_footnote": [], + "bbox": [ + 179, + 560, + 491, + 689 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/7e4a1e55454f29fd7af5984c32bd2fdf38629d509dc5a36e2877375b43bd8a8e.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 504, + 561, + 815, + 688 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/79d334233a97559da6578ed0b4bf0d59b42735df0cefe987851334d4ea8e655e.jpg", + "image_caption": [ + "Resnet-18 (CIFAR-10) - 4x Blurring $^ +$ Self-Superivsed Rotation " + ], + "image_footnote": [], + "bbox": [ + 179, + 707, + 491, + 834 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/54cd29c3f583c670ce6e12169f0a37b36cdf9113cd9c041dd99964a54af20bf0.jpg", + "image_caption": [ + "WRN-16-8 (CIFAR-10) - 4x Blurring $^ +$ Self-Superivsed Rotation ", + "Figure A12: The effect of pre-training CIFAR-10 with $4 \\mathbf { x }$ blurring and self-supervised rotation. 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For exam-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "spans": [ + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "score": 1.0, + "content": "ple, sparse, trainable sub-networks emerge (Frankle et al., 2019), gradient descent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "score": 1.0, + "content": "moves into a small subspace (Gur-Ari et al., 2018), and the network undergoes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 247, + 470, + 261 + ], + "spans": [ + { + "bbox": [ + 141, + 247, + 470, + 261 + ], + "score": 1.0, + "content": "a critical period (Achille et al., 2019). Here we examine the changes that deep", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 258, + 469, + 272 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 469, + 272 + ], + "score": 1.0, + "content": "neural networks undergo during this early phase of training. We perform exten-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 270, + 469, + 282 + ], + "spans": [ + { + "bbox": [ + 141, + 270, + 469, + 282 + ], + "score": 1.0, + "content": "sive measurements of the network state during these early iterations of training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 280, + 469, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 469, + 293 + ], + "score": 1.0, + "content": "and leverage the framework of Frankle et al. 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Despite this behavior,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 335, + 469, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 469, + 348 + ], + "score": 1.0, + "content": "pre-training with blurred inputs or an auxiliary self-supervised task can approx-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 347, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 141, + 347, + 470, + 359 + ], + "score": 1.0, + "content": "imate the changes in supervised networks, suggesting that these changes are not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "score": 1.0, + "content": "inherently label-dependent, though labels significantly accelerate this process. To-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 368, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 368, + 470, + 381 + ], + "score": 1.0, + "content": "gether, these results help to elucidate the network changes occurring during this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 378, + 274, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 274, + 392 + ], + "score": 1.0, + "content": "pivotal initial period of learning.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 414, + 206, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 208, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 208, + 430 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "Over the past decade, methods for successfully training big, deep neural networks have revolution-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "ized machine learning. Yet surprisingly, the underlying reasons for the success of these approaches", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 461, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 506, + 477 + ], + "score": 1.0, + "content": "remain poorly understood, despite remarkable empirical performance (Santurkar et al., 2018; Zhang", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "et al., 2017). A large body of work has focused on understanding what happens during the later", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "stages of training (Neyshabur et al., 2019; Yaida, 2019; Chaudhuri & Soatto, 2017; Wei & Schwab,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "2019), while the initial phase has been less explored. 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The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "initial lottery ticket hypothesis, which was validated on comparatively small networks, proposed that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "small, sparse sub-networks found via pruning of converged larger models could be trained to high", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "performance provided they were initialized with the same values used in the training of the unpruned", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "model (Frankle & Carbin, 2019). 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Morcos", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 371, + 125, + 464, + 137 + ], + "spans": [ + { + "bbox": [ + 371, + 125, + 464, + 137 + ], + "score": 1.0, + "content": "Facebook AI Research", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 371, + 114, + 464, + 137 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 177, + 333, + 189 + ], + "lines": [ + { + "bbox": [ + 276, + 176, + 336, + 191 + ], + "spans": [ + { + "bbox": [ + 276, + 176, + 336, + 191 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 143, + 204, + 468, + 390 + ], + "lines": [ + { + "bbox": [ + 141, + 203, + 469, + 216 + ], + "spans": [ + { + "bbox": [ + 141, + 203, + 469, + 216 + ], + "score": 1.0, + "content": "Recent studies have shown that many important aspects of neural network learn-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 215, + 469, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 215, + 469, + 227 + ], + "score": 1.0, + "content": "ing take place within the very earliest iterations or epochs of training. For exam-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "spans": [ + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "score": 1.0, + "content": "ple, sparse, trainable sub-networks emerge (Frankle et al., 2019), gradient descent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "score": 1.0, + "content": "moves into a small subspace (Gur-Ari et al., 2018), and the network undergoes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 247, + 470, + 261 + ], + "spans": [ + { + "bbox": [ + 141, + 247, + 470, + 261 + ], + "score": 1.0, + "content": "a critical period (Achille et al., 2019). Here we examine the changes that deep", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 258, + 469, + 272 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 469, + 272 + ], + "score": 1.0, + "content": "neural networks undergo during this early phase of training. We perform exten-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 270, + 469, + 282 + ], + "spans": [ + { + "bbox": [ + 141, + 270, + 469, + 282 + ], + "score": 1.0, + "content": "sive measurements of the network state during these early iterations of training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 280, + 469, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 469, + 293 + ], + "score": 1.0, + "content": "and leverage the framework of Frankle et al. (2019) to quantitatively probe the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 292, + 469, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 292, + 469, + 303 + ], + "score": 1.0, + "content": "weight distribution and its reliance on various aspects of the dataset. We find", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 302, + 469, + 315 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 469, + 315 + ], + "score": 1.0, + "content": "that, within this framework, deep networks are not robust to reinitializing with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 312, + 469, + 326 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 469, + 326 + ], + "score": 1.0, + "content": "random weights while maintaining signs, and that weight distributions are highly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 324, + 470, + 337 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 470, + 337 + ], + "score": 1.0, + "content": "non-independent even after only a few hundred iterations. Despite this behavior,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 335, + 469, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 469, + 348 + ], + "score": 1.0, + "content": "pre-training with blurred inputs or an auxiliary self-supervised task can approx-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 347, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 141, + 347, + 470, + 359 + ], + "score": 1.0, + "content": "imate the changes in supervised networks, suggesting that these changes are not", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "score": 1.0, + "content": "inherently label-dependent, though labels significantly accelerate this process. To-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 368, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 368, + 470, + 381 + ], + "score": 1.0, + "content": "gether, these results help to elucidate the network changes occurring during this", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 378, + 274, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 274, + 392 + ], + "score": 1.0, + "content": "pivotal initial period of learning.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 17, + "bbox_fs": [ + 141, + 203, + 470, + 392 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 414, + 206, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 208, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 208, + 430 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "Over the past decade, methods for successfully training big, deep neural networks have revolution-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "ized machine learning. Yet surprisingly, the underlying reasons for the success of these approaches", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 461, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 506, + 477 + ], + "score": 1.0, + "content": "remain poorly understood, despite remarkable empirical performance (Santurkar et al., 2018; Zhang", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "et al., 2017). A large body of work has focused on understanding what happens during the later", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "stages of training (Neyshabur et al., 2019; Yaida, 2019; Chaudhuri & Soatto, 2017; Wei & Schwab,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "2019), while the initial phase has been less explored. However, a number of distinct observations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "indicate that significant and consequential changes are occurring during the most early stage of train-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "ing. These include the presence of critical periods during training (Achille et al., 2019), the dramatic", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "reshaping of the local loss landscape (Sagun et al., 2017; Gur-Ari et al., 2018), and the necessity of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "rewinding in the context of the lottery ticket hypothesis (Frankle et al., 2019). Here we perform a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 550, + 380, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 380, + 563 + ], + "score": 1.0, + "content": "thorough investigation of the state of the network in this early stage.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 441, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "To provide a unified framework for understanding the changes the network undergoes during the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "early phase, we employ the methodology of iterative magnitude pruning with rewinding (IMP), as", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "detailed below, throughout the bulk of this work (Frankle & Carbin, 2019; Frankle et al., 2019). The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "initial lottery ticket hypothesis, which was validated on comparatively small networks, proposed that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "small, sparse sub-networks found via pruning of converged larger models could be trained to high", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "performance provided they were initialized with the same values used in the training of the unpruned", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "model (Frankle & Carbin, 2019). However, follow-up work found that rewinding the weights to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "their values at some iteration early in the training of the unpruned model, rather than to their initial", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "values, was necessary to achieve good performance on deeper networks such as ResNets (Frankle", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "et al., 2019). This observation suggests that the changes in the network during this initial phase", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "are vital for the success of the training of small, sparse sub-networks. As a result, this paradigm", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "provides a simple and quantitative scheme for measuring the importance of the weights at various", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 375, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 375, + 711 + ], + "score": 1.0, + "content": "points early in training within an actionable and causal framework.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 567, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 489, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 489, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 489, + 95 + ], + "score": 1.0, + "content": "We make the following contributions, all evaluated across three different network architectures:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 126, + 104, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 129, + 103, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 129, + 103, + 504, + 118 + ], + "score": 1.0, + "content": "1. We provide an in-depth overview of various statistics summarizing learning over the early", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 117, + 207, + 129 + ], + "spans": [ + { + "bbox": [ + 141, + 117, + 207, + 129 + ], + "score": 1.0, + "content": "part of training.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 128, + 129, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 128, + 129, + 505, + 144 + ], + "score": 1.0, + "content": "2. We evaluate the impact of perturbing the state of the network in various ways during the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 142, + 142, + 287, + 154 + ], + "spans": [ + { + "bbox": [ + 142, + 142, + 287, + 154 + ], + "score": 1.0, + "content": "early phase of training, finding that:", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 147, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 147, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "(i) counter to observations in smaller networks (Zhou et al., 2019), deeper networks are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 161, + 168, + 443, + 180 + ], + "spans": [ + { + "bbox": [ + 161, + 168, + 443, + 180 + ], + "score": 1.0, + "content": "not robust to reinitializion with random weights, but maintained signs", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 145, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 145, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "(ii) the distribution of weights after the early phase of training is already highly non-i.i.d.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 161, + 192, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 161, + 192, + 504, + 204 + ], + "score": 1.0, + "content": "as permuting them dramatically harms performance, even when signs are maintained", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 141, + 204, + 506, + 218 + ], + "score": 1.0, + "content": "(iii) both of the above perturbations can roughly be approximated by simply adding noise", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 160, + 215, + 435, + 228 + ], + "spans": [ + { + "bbox": [ + 160, + 215, + 435, + 228 + ], + "score": 1.0, + "content": "to the network weights, though this effect is stronger for (ii) than (i)", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 133, + 231, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 129, + 229, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 129, + 229, + 506, + 244 + ], + "score": 1.0, + "content": "3. We measure the data-dependence of the early phase of training, finding that pre-training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 240, + 189, + 255 + ], + "score": 1.0, + "content": "using only", + "type": "text" + }, + { + "bbox": [ + 189, + 241, + 209, + 254 + ], + "score": 0.91, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "can approximate the changes that occur in the early phase of training,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 142, + 253, + 315, + 265 + ], + "score": 1.0, + "content": "though pre-training must last for far longer", + "type": "text" + }, + { + "bbox": [ + 316, + 253, + 343, + 263 + ], + "score": 0.85, + "content": "{ \\sim } 3 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "longer) and not be fed misleading labels.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 280, + 423, + 293 + ], + "lines": [ + { + "bbox": [ + 104, + 279, + 423, + 295 + ], + "spans": [ + { + "bbox": [ + 104, + 279, + 423, + 295 + ], + "score": 1.0, + "content": "2 KNOWN PHENOMENA IN THE EARLY PHASE OF TRAINING", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 296, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 297, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 297, + 317 + ], + "score": 1.0, + "content": "Lottery ticket rewinding: The original lot-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 316, + 297, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 297, + 327 + ], + "score": 1.0, + "content": "tery ticket paper (Frankle & Carbin, 2019) re-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 326, + 297, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 261, + 338 + ], + "score": 1.0, + "content": "wound weights to initialization, i.e.,", + "type": "text" + }, + { + "bbox": [ + 262, + 327, + 293, + 338 + ], + "score": 0.9, + "content": "k ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 326, + 297, + 338 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 338, + 297, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 297, + 350 + ], + "score": 1.0, + "content": "during IMP. Follow up work on larger models", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 349, + 298, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 298, + 361 + ], + "score": 1.0, + "content": "demonstrated that it is necessary to rewind to a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 360, + 297, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 297, + 372 + ], + "score": 1.0, + "content": "later point during training for IMP to succeed,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 370, + 297, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 124, + 383 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 125, + 371, + 164, + 381 + ], + "score": 0.91, + "content": "k \\ < < \\ T", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 370, + 196, + 383 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 197, + 371, + 205, + 381 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 370, + 297, + 383 + ], + "score": 1.0, + "content": "is total training itera-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 382, + 297, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 297, + 393 + ], + "score": 1.0, + "content": "tions (Frankle et al., 2019). Notably, the benefit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 392, + 297, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 297, + 405 + ], + "score": 1.0, + "content": "of rewinding to a later point in training satu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 297, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 297, + 415 + ], + "score": 1.0, + "content": "rates quickly, roughly between 500 and 2000", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 297, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 297, + 427 + ], + "score": 1.0, + "content": "iterations for ResNet-20 on CIFAR-10 (Figure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 425, + 297, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 297, + 438 + ], + "score": 1.0, + "content": "1). This timescale is strikingly similar to the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 270, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 270, + 449 + ], + "score": 1.0, + "content": "changes in the Hessian described below.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 296, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 297, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 297, + 466 + ], + "score": 1.0, + "content": "Hessian eigenspectrum: The shape of the loss", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 465, + 297, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 297, + 477 + ], + "score": 1.0, + "content": "landscape around the network state also appears", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "image", + "bbox": [ + 307, + 320, + 480, + 421 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 307, + 320, + 480, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 307, + 320, + 480, + 421 + ], + "spans": [ + { + "bbox": [ + 307, + 320, + 480, + 421 + ], + "score": 0.968, + "type": "image", + "image_path": "8c4ca4d2642f445b29e873fc950bbc21f6cd35aa36fbbe51f69c185df46c53ba.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 307, + 320, + 480, + 334.42857142857144 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 307, + 334.42857142857144, + 480, + 348.8571428571429 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 307, + 348.8571428571429, + 480, + 363.28571428571433 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 307, + 363.28571428571433, + 480, + 377.7142857142858 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 307, + 377.7142857142858, + 480, + 392.1428571428572 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 307, + 392.1428571428572, + 480, + 406.57142857142867 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 307, + 406.57142857142867, + 480, + 421.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 435, + 505, + 468 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 303, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "Figure 1: Accuracy of IMP when rewinding to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 304, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 304, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "various iterations of the early phase for ResNet-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 304, + 456, + 495, + 470 + ], + "spans": [ + { + "bbox": [ + 304, + 456, + 495, + 470 + ], + "score": 1.0, + "content": "20 sub-networks as a function of sparsity level.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "to change rapidly during the early phase of training (Sagun et al., 2017; Gur-Ari et al., 2018). At", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "initialization, the Hessian of the loss contains a number of large positive and negative eigenvalues.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "However, very rapidly the curvature is reshaped in a few marked ways: a few large eigenvalues", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "emerge, the bulk eigenvalues are close to zero, and the negative eigenvalues become very small.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Moreover, once the Hessian spectrum has reshaped, gradient descent appears to occur largely within", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "the top subspace of the Hessian (Gur-Ari et al., 2018). These results have been largely confirmed in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "large scale studies (Ghorbani et al., 2019), but note they depend to some extent on architecture and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 550, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 567 + ], + "score": 1.0, + "content": "(absence of) batch normalization (Ioffe & Szegedy, 2015). A notable exception to this consistency", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 563, + 494, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 223, + 577 + ], + "score": 1.0, + "content": "is the presence of substantial", + "type": "text" + }, + { + "bbox": [ + 223, + 564, + 236, + 574 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 563, + 494, + 577 + ], + "score": 1.0, + "content": "energy of negative eigenvalues for models trained on ImageNet.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "Critical periods in deep learning: Achille et al. (2019) found that perturbing the training process", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "by providing corrupted data early on in training can result in irrevocable damage to the final perfor-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "mance of the network. Note that the timescales over which the authors find a critical period extend", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "well beyond those we study here. However, architecture, learning rate schedule, and regularization", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "all modify the timing of the critical period, and follow-up work found that critical periods were also", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 635, + 504, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 504, + 648 + ], + "score": 1.0, + "content": "present for regularization, in particular weight decay and data augmentation (Golatkar et al., 2019).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 51.5 + }, + { + "type": "title", + "bbox": [ + 108, + 662, + 319, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 320, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 320, + 677 + ], + "score": 1.0, + "content": "3 PRELIMINARIES AND METHODOLOGY", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 55 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Networks: Throughout this paper, we study five standard convolutional neural networks for", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "CIFAR-10. These include the ResNet-20 and ResNet-56 architectures designed for CIFAR-10 (He", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "et al., 2015), the ResNet-18 architecture designed for ImageNet but commonly used on CIFAR-10", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "(He et al., 2015), the WRN-16-8 wide residual network (Zagoruyko & Komodakis, 2016), and the", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 57.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 489, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 489, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 489, + 95 + ], + "score": 1.0, + "content": "We make the following contributions, all evaluated across three different network architectures:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 489, + 95 + ] + }, + { + "type": "list", + "bbox": [ + 126, + 104, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 129, + 103, + 504, + 118 + ], + "spans": [ + { + "bbox": [ + 129, + 103, + 504, + 118 + ], + "score": 1.0, + "content": "1. We provide an in-depth overview of various statistics summarizing learning over the early", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 117, + 207, + 129 + ], + "spans": [ + { + "bbox": [ + 141, + 117, + 207, + 129 + ], + "score": 1.0, + "content": "part of training.", + "type": "text" + } + ], + "index": 2, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 129, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 128, + 129, + 505, + 144 + ], + "score": 1.0, + "content": "2. We evaluate the impact of perturbing the state of the network in various ways during the", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 142, + 287, + 154 + ], + "spans": [ + { + "bbox": [ + 142, + 142, + 287, + 154 + ], + "score": 1.0, + "content": "early phase of training, finding that:", + "type": "text" + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 147, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 147, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "(i) counter to observations in smaller networks (Zhou et al., 2019), deeper networks are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 161, + 168, + 443, + 180 + ], + "spans": [ + { + "bbox": [ + 161, + 168, + 443, + 180 + ], + "score": 1.0, + "content": "not robust to reinitializion with random weights, but maintained signs", + "type": "text" + } + ], + "index": 6, + "is_list_end_line": true + }, + { + "bbox": [ + 145, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 145, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "(ii) the distribution of weights after the early phase of training is already highly non-i.i.d.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 161, + 192, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 161, + 192, + 504, + 204 + ], + "score": 1.0, + "content": "as permuting them dramatically harms performance, even when signs are maintained", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 204, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 141, + 204, + 506, + 218 + ], + "score": 1.0, + "content": "(iii) both of the above perturbations can roughly be approximated by simply adding noise", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 160, + 215, + 435, + 228 + ], + "spans": [ + { + "bbox": [ + 160, + 215, + 435, + 228 + ], + "score": 1.0, + "content": "to the network weights, though this effect is stronger for (ii) than (i)", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + } + ], + "index": 5.5, + "bbox_fs": [ + 128, + 103, + 506, + 228 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 231, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 129, + 229, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 129, + 229, + 506, + 244 + ], + "score": 1.0, + "content": "3. We measure the data-dependence of the early phase of training, finding that pre-training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 240, + 189, + 255 + ], + "score": 1.0, + "content": "using only", + "type": "text" + }, + { + "bbox": [ + 189, + 241, + 209, + 254 + ], + "score": 0.91, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "can approximate the changes that occur in the early phase of training,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 142, + 253, + 315, + 265 + ], + "score": 1.0, + "content": "though pre-training must last for far longer", + "type": "text" + }, + { + "bbox": [ + 316, + 253, + 343, + 263 + ], + "score": 0.85, + "content": "{ \\sim } 3 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "longer) and not be fed misleading labels.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 129, + 229, + 506, + 265 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 280, + 423, + 293 + ], + "lines": [ + { + "bbox": [ + 104, + 279, + 423, + 295 + ], + "spans": [ + { + "bbox": [ + 104, + 279, + 423, + 295 + ], + "score": 1.0, + "content": "2 KNOWN PHENOMENA IN THE EARLY PHASE OF TRAINING", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 296, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 297, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 297, + 317 + ], + "score": 1.0, + "content": "Lottery ticket rewinding: The original lot-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 316, + 297, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 297, + 327 + ], + "score": 1.0, + "content": "tery ticket paper (Frankle & Carbin, 2019) re-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 326, + 297, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 261, + 338 + ], + "score": 1.0, + "content": "wound weights to initialization, i.e.,", + "type": "text" + }, + { + "bbox": [ + 262, + 327, + 293, + 338 + ], + "score": 0.9, + "content": "k ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 326, + 297, + 338 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 338, + 297, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 297, + 350 + ], + "score": 1.0, + "content": "during IMP. Follow up work on larger models", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 349, + 298, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 298, + 361 + ], + "score": 1.0, + "content": "demonstrated that it is necessary to rewind to a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 360, + 297, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 297, + 372 + ], + "score": 1.0, + "content": "later point during training for IMP to succeed,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 370, + 297, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 124, + 383 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 125, + 371, + 164, + 381 + ], + "score": 0.91, + "content": "k \\ < < \\ T", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 370, + 196, + 383 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 197, + 371, + 205, + 381 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 370, + 297, + 383 + ], + "score": 1.0, + "content": "is total training itera-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 382, + 297, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 297, + 393 + ], + "score": 1.0, + "content": "tions (Frankle et al., 2019). Notably, the benefit", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 392, + 297, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 297, + 405 + ], + "score": 1.0, + "content": "of rewinding to a later point in training satu-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 297, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 297, + 415 + ], + "score": 1.0, + "content": "rates quickly, roughly between 500 and 2000", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 297, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 297, + 427 + ], + "score": 1.0, + "content": "iterations for ResNet-20 on CIFAR-10 (Figure", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 425, + 297, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 297, + 438 + ], + "score": 1.0, + "content": "1). This timescale is strikingly similar to the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 270, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 270, + 449 + ], + "score": 1.0, + "content": "changes in the Hessian described below.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 305, + 298, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 296, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 297, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 297, + 466 + ], + "score": 1.0, + "content": "Hessian eigenspectrum: The shape of the loss", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 465, + 297, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 297, + 477 + ], + "score": 1.0, + "content": "landscape around the network state also appears", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 453, + 297, + 477 + ] + }, + { + "type": "image", + "bbox": [ + 307, + 320, + 480, + 421 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 307, + 320, + 480, + 421 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 307, + 320, + 480, + 421 + ], + "spans": [ + { + "bbox": [ + 307, + 320, + 480, + 421 + ], + "score": 0.968, + "type": "image", + "image_path": "8c4ca4d2642f445b29e873fc950bbc21f6cd35aa36fbbe51f69c185df46c53ba.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 307, + 320, + 480, + 334.42857142857144 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 307, + 334.42857142857144, + 480, + 348.8571428571429 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 307, + 348.8571428571429, + 480, + 363.28571428571433 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 307, + 363.28571428571433, + 480, + 377.7142857142858 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 307, + 377.7142857142858, + 480, + 392.1428571428572 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 307, + 392.1428571428572, + 480, + 406.57142857142867 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 307, + 406.57142857142867, + 480, + 421.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 435, + 505, + 468 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 303, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 303, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "Figure 1: Accuracy of IMP when rewinding to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 304, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 304, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "various iterations of the early phase for ResNet-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 304, + 456, + 495, + 470 + ], + "spans": [ + { + "bbox": [ + 304, + 456, + 495, + 470 + ], + "score": 1.0, + "content": "20 sub-networks as a function of sparsity level.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "to change rapidly during the early phase of training (Sagun et al., 2017; Gur-Ari et al., 2018). At", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "initialization, the Hessian of the loss contains a number of large positive and negative eigenvalues.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "However, very rapidly the curvature is reshaped in a few marked ways: a few large eigenvalues", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "emerge, the bulk eigenvalues are close to zero, and the negative eigenvalues become very small.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Moreover, once the Hessian spectrum has reshaped, gradient descent appears to occur largely within", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "the top subspace of the Hessian (Gur-Ari et al., 2018). These results have been largely confirmed in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "large scale studies (Ghorbani et al., 2019), but note they depend to some extent on architecture and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 550, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 567 + ], + "score": 1.0, + "content": "(absence of) batch normalization (Ioffe & Szegedy, 2015). A notable exception to this consistency", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 563, + 494, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 223, + 577 + ], + "score": 1.0, + "content": "is the presence of substantial", + "type": "text" + }, + { + "bbox": [ + 223, + 564, + 236, + 574 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 563, + 494, + 577 + ], + "score": 1.0, + "content": "energy of negative eigenvalues for models trained on ImageNet.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 475, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "Critical periods in deep learning: Achille et al. (2019) found that perturbing the training process", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "by providing corrupted data early on in training can result in irrevocable damage to the final perfor-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "mance of the network. Note that the timescales over which the authors find a critical period extend", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "well beyond those we study here. However, architecture, learning rate schedule, and regularization", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "all modify the timing of the critical period, and follow-up work found that critical periods were also", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 635, + 504, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 504, + 648 + ], + "score": 1.0, + "content": "present for regularization, in particular weight decay and data augmentation (Golatkar et al., 2019).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 579, + 506, + 648 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 662, + 319, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 320, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 320, + 677 + ], + "score": 1.0, + "content": "3 PRELIMINARIES AND METHODOLOGY", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 55 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Networks: Throughout this paper, we study five standard convolutional neural networks for", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "CIFAR-10. These include the ResNet-20 and ResNet-56 architectures designed for CIFAR-10 (He", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "et al., 2015), the ResNet-18 architecture designed for ImageNet but commonly used on CIFAR-10", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "(He et al., 2015), the WRN-16-8 wide residual network (Zagoruyko & Komodakis, 2016), and the", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 57.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 121, + 86, + 472, + 187 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 121, + 86, + 472, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 86, + 472, + 187 + ], + "spans": [ + { + "bbox": [ + 121, + 86, + 472, + 187 + ], + "score": 0.959, + "type": "image", + "image_path": "bbc021b67f4758bf1ffddd590865638abb4a0901c8374f61644c3639d1985c0a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 121, + 86, + 472, + 119.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 121, + 119.66666666666666, + 472, + 153.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 121, + 153.33333333333331, + 472, + 186.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 134, + 200, + 474, + 212 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 199, + 475, + 214 + ], + "spans": [ + { + "bbox": [ + 135, + 199, + 475, + 214 + ], + "score": 1.0, + "content": "Figure 2: Rough timeline of the early phase of training for ResNet-20 on CIFAR-10.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 238, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 238, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 251 + ], + "score": 1.0, + "content": "VGG-13 network (Simonyan & Zisserman (2015) as adapted by Liu et al. (2019)). Throughout the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "main body of the paper, we show ResNet-20; in Appendix B, we present the same experiments for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "score": 1.0, + "content": "the other networks. Unless otherwise stated, results were qualitatively similar across all three net-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "works. All experiments in this paper display the mean and standard deviation across five replicates", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 283, + 393, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 393, + 295 + ], + "score": 1.0, + "content": "with different random seeds. See Appendix A for further model details.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "Iterative magnitude pruning with rewinding: In order to test the effect of various hypotheses", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "about the state of sparse networks early in training, we use the Iterative Magnitude Pruning with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "rewinding (IMP) procedure of Frankle et al. (2019) to extract sub-networks from various points in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "score": 1.0, + "content": "training that could have learned on their own. The procedure involves training a network to com-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 189, + 356 + ], + "score": 1.0, + "content": "pletion, pruning the", + "type": "text" + }, + { + "bbox": [ + 189, + 343, + 209, + 354 + ], + "score": 0.86, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "of weights with the lowest magnitudes globally throughout the network,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 423, + 367 + ], + "score": 1.0, + "content": "and rewinding the remaining weights to their values from an earlier iteration", + "type": "text" + }, + { + "bbox": [ + 423, + 354, + 430, + 364 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "during the initial,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "pre-pruning training run. This process is iterated to produce networks with high sparsity levels. As", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "demonstrated in Frankle et al. (2019), IMP with rewinding leads to sparse sub-networks which can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 387, + 354, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 320, + 400 + ], + "score": 1.0, + "content": "train to high performance even at high sparsity levels", + "type": "text" + }, + { + "bbox": [ + 320, + 387, + 350, + 398 + ], + "score": 0.89, + "content": "> 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 387, + 354, + 400 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "Figure 1 shows the results of the IMP with rewinding procedure, showing the accuracy of ResNet-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 474, + 428 + ], + "score": 1.0, + "content": "20 at increasing sparsity when performing this procedure for several rewinding values of", + "type": "text" + }, + { + "bbox": [ + 475, + 415, + 481, + 425 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 414, + 506, + 428 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 143, + 437 + ], + "score": 0.9, + "content": "k \\geq 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 426, + 432, + 439 + ], + "score": 1.0, + "content": ", sub-networks can match the performance of the original network with", + "type": "text" + }, + { + "bbox": [ + 432, + 426, + 459, + 437 + ], + "score": 0.86, + "content": "1 6 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "of weights", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 455, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 168, + 450 + ], + "score": 1.0, + "content": "remaining. For", + "type": "text" + }, + { + "bbox": [ + 169, + 437, + 209, + 447 + ], + "score": 0.89, + "content": "k > 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 437, + 455, + 450 + ], + "score": 1.0, + "content": ", essentially no further improvement is observed (not shown).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 471, + 387, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 389, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 389, + 486 + ], + "score": 1.0, + "content": "4 THE STATE OF THE NETWORK EARLY IN TRAINING", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Many of the aforementioned papers refer to various points in the “early” part of training. In this", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "section, we descriptively chart the state of ResNet-20 during the earliest phase of training to provide", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "context for this related work and our subsequent experiments. We specifically focus on the first", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "4,000 iterations (10 epochs). See Figure A3 for the characterization of additional networks. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "include a summary of these results for ResNet-20 as a timeline in Figure 2, and include a broader", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 555, + 443, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 443, + 568 + ], + "score": 1.0, + "content": "timeline including results from several previous papers for ResNet-18 in Figure A1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "As shown in Figure 3, during the earliest ten iterations, the network undergoes substantial change. It", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "experiences large gradients that correspond to a rapid increase in distance from the initialization and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "a large number of sign changes of the weights. After these initial iterations, gradient magnitudes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "drop and the rate of change in each of the aforementioned quantities gradually slows through the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "remainder of the period we observe. Interestingly, gradient magnitudes reach a minimum after the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "first 200 iterations and subsequently increase to a stable level by iteration 500. Evaluation accuracy,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 215, + 650 + ], + "score": 1.0, + "content": "improves rapidly, reaching", + "type": "text" + }, + { + "bbox": [ + 216, + 638, + 235, + 649 + ], + "score": 0.86, + "content": "55 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "by the end of the first epoch (400 iterations), more than halfway to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 360, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 141, + 661 + ], + "score": 1.0, + "content": "the final", + "type": "text" + }, + { + "bbox": [ + 141, + 649, + 168, + 659 + ], + "score": 0.84, + "content": "9 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 648, + 336, + 661 + ], + "score": 1.0, + "content": ". By 2000 iterations, accuracy approaches", + "type": "text" + }, + { + "bbox": [ + 336, + 649, + 356, + 659 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 648, + 360, + 661 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "During the first 4000 iterations of training, we observe three sub-phases. 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(2019), IMP with rewinding leads to sparse sub-networks which can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 387, + 354, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 320, + 400 + ], + "score": 1.0, + "content": "train to high performance even at high sparsity levels", + "type": "text" + }, + { + "bbox": [ + 320, + 387, + 350, + 398 + ], + "score": 0.89, + "content": "> 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 387, + 354, + 400 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 300, + 505, + 400 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "Figure 1 shows the results of the IMP with rewinding procedure, showing the accuracy of ResNet-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 474, + 428 + ], + "score": 1.0, + "content": "20 at increasing sparsity when performing this procedure for several rewinding values of", + "type": "text" + }, + { + "bbox": [ + 475, + 415, + 481, + 425 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 414, + 506, + 428 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 143, + 437 + ], + "score": 0.9, + "content": "k \\geq 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 426, + 432, + 439 + ], + "score": 1.0, + "content": ", sub-networks can match the performance of the original network with", + "type": "text" + }, + { + "bbox": [ + 432, + 426, + 459, + 437 + ], + "score": 0.86, + "content": "1 6 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "of weights", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 437, + 455, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 168, + 450 + ], + "score": 1.0, + "content": "remaining. For", + "type": "text" + }, + { + "bbox": [ + 169, + 437, + 209, + 447 + ], + "score": 0.89, + "content": "k > 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 437, + 455, + 450 + ], + "score": 1.0, + "content": ", essentially no further improvement is observed (not shown).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 404, + 506, + 450 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 471, + 387, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 389, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 389, + 486 + ], + "score": 1.0, + "content": "4 THE STATE OF THE NETWORK EARLY IN TRAINING", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Many of the aforementioned papers refer to various points in the “early” part of training. In this", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "section, we descriptively chart the state of ResNet-20 during the earliest phase of training to provide", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "context for this related work and our subsequent experiments. We specifically focus on the first", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "4,000 iterations (10 epochs). See Figure A3 for the characterization of additional networks. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "include a summary of these results for ResNet-20 as a timeline in Figure 2, and include a broader", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 555, + 443, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 443, + 568 + ], + "score": 1.0, + "content": "timeline including results from several previous papers for ResNet-18 in Figure A1.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 500, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "As shown in Figure 3, during the earliest ten iterations, the network undergoes substantial change. It", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "experiences large gradients that correspond to a rapid increase in distance from the initialization and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "a large number of sign changes of the weights. After these initial iterations, gradient magnitudes", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "drop and the rate of change in each of the aforementioned quantities gradually slows through the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "remainder of the period we observe. Interestingly, gradient magnitudes reach a minimum after the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "first 200 iterations and subsequently increase to a stable level by iteration 500. Evaluation accuracy,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 215, + 650 + ], + "score": 1.0, + "content": "improves rapidly, reaching", + "type": "text" + }, + { + "bbox": [ + 216, + 638, + 235, + 649 + ], + "score": 0.86, + "content": "55 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "by the end of the first epoch (400 iterations), more than halfway to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 360, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 141, + 661 + ], + "score": 1.0, + "content": "the final", + "type": "text" + }, + { + "bbox": [ + 141, + 649, + 168, + 659 + ], + "score": 0.84, + "content": "9 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 648, + 336, + 661 + ], + "score": 1.0, + "content": ". By 2000 iterations, accuracy approaches", + "type": "text" + }, + { + "bbox": [ + 336, + 649, + 356, + 659 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 648, + 360, + 661 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 572, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 680 + ], + "score": 1.0, + "content": "During the first 4000 iterations of training, we observe three sub-phases. In the first phase, lasting", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "only the initial few iterations, gradient magnitudes are very large and, consequently, the network", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "changes rapidly. In the second phase, lasting about 500 iterations, performance quickly improves,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "weight magnitudes quickly increase, sign differences from initialization quickly increase, and gra-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "dient magnitudes reach a minimum before settling at a stable level. Finally, in the third phase, all of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 432, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 432, + 733 + ], + "score": 1.0, + "content": "these quantities continue to change in the same direction, but begin to decelerate.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 664, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 123, + 81, + 488, + 226 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 123, + 81, + 488, + 226 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 81, + 488, + 226 + ], + "spans": [ + { + "bbox": [ + 123, + 81, + 488, + 226 + ], + "score": 0.977, + "type": "image", + "image_path": "536f691a12fa237fb72a5290ddf148134e074daf7b9eb9951f39a422f822647b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 123, + 81, + 488, + 129.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 123, + 129.33333333333334, + 488, + 177.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 123, + 177.66666666666669, + 488, + 226.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 239, + 505, + 294 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 239, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 504, + 251 + ], + "score": 1.0, + "content": "Figure 3: Basic telemetry about the state of ResNet-20 during the first 4000 iterations (10 epochs).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "Top row: evaluation accuracy/loss; average weight magnitude; percentage of weights that change", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "sign from initialization; the values of ten randomly-selected weights. Bottom row: gradient magni-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "tude; L2 distance of weights from their initial values and final values at the end of training; cosine", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 281, + 440, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 440, + 296 + ], + "score": 1.0, + "content": "similarity of weights from their initial values and final values at the end of training.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "image", + "bbox": [ + 128, + 309, + 480, + 404 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 309, + 480, + 404 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 128, + 309, + 480, + 404 + ], + "spans": [ + { + "bbox": [ + 128, + 309, + 480, + 404 + ], + "score": 0.97, + "type": "image", + "image_path": "9b938cd5121d512198fefd1d83a99771476d10b7fe225ff94604b45fa851893d.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 128, + 309, + 480, + 340.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 128, + 340.6666666666667, + 480, + 372.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 128, + 372.33333333333337, + 480, + 404.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 417, + 505, + 440 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "Figure 4: Performance of an IMP-derived sub-network of ResNet-20 on CIFAR-10 initialized to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 428, + 489, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 193, + 441 + ], + "score": 1.0, + "content": "signs at iteration 0 or", + "type": "text" + }, + { + "bbox": [ + 193, + 429, + 200, + 438 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 428, + 344, + 441 + ], + "score": 1.0, + "content": "and the magnitudes at iteration 0 or", + "type": "text" + }, + { + "bbox": [ + 345, + 429, + 352, + 438 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 428, + 378, + 441 + ], + "score": 1.0, + "content": ". Left:", + "type": "text" + }, + { + "bbox": [ + 378, + 429, + 413, + 439 + ], + "score": 0.9, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 428, + 445, + 441 + ], + "score": 1.0, + "content": ". Right:", + "type": "text" + }, + { + "bbox": [ + 446, + 429, + 486, + 439 + ], + "score": 0.9, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 428, + 489, + 441 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + } + ], + "index": 10.25 + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 410, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 412, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 412, + 475 + ], + "score": 1.0, + "content": "5 PERTURBING NEURAL NETWORKS EARLY IN TRAINING", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "Figure 1 shows that the changes in the network weights over the first 500 iterations of training are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "essential to enable high performance at high sparsity levels. What features of this weight transfor-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "mation are necessary to recover increased performance? Can they be summarized by maintaining", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "the weight signs, but discarding their magnitudes as implied by Zhou et al. (2019)? Can they be", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "represented distributionally? In this section, we evaluate these questions by perturbing the early", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 541, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 553 + ], + "score": 1.0, + "content": "state of the network in various ways. Concretely, we either add noise or shuffle the weights of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "IMP sub-networks of ResNet-20 across different network sub-compenents and examine the effect", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "on the network’s ability to learn thereafter. The sub-networks derived by IMP with rewinding make", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "it possible to understand the causal impact of perturbations on sub-networks that are as capable as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "the full networks but more visibly decline in performance when improperly configured. To enable", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "comparisons between the experiments in Section 5 and provide a common frame of reference, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 606, + 478, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 397, + 621 + ], + "score": 1.0, + "content": "measure the effective standard deviation of each perturbation, i.e. stddev", + "type": "text" + }, + { + "bbox": [ + 397, + 607, + 473, + 619 + ], + "score": 0.86, + "content": "( w _ { p e r t u r b } - w _ { o r i g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 606, + 478, + 621 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 633, + 254, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 257, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 257, + 646 + ], + "score": 1.0, + "content": "5.1 ARE SIGNS ALL YOU NEED?", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Zhou et al. (2019) show that, for a set of small convolutional networks, signs alone are sufficient", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "to capture the state of lottery ticket sub-networks. However, it is unclear whether signs are still", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "sufficient for larger networks early in training. In Figure 4, we investigate the impact of combining", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "the magnitudes of the weights from one time-point with the signs from another. 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This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "result suggests that network signs undergo important changes between iterations 500 and 2000, as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 126, + 538 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 126, + 526, + 141, + 536 + ], + "score": 0.83, + "content": "9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "of signs change during this period. Our results also suggest that counter to the observations", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 537, + 449, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 449, + 550 + ], + "score": 1.0, + "content": "of Zhou et al. (2019) in shallow networks, signs are not sufficient in deeper networks.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 109, + 565, + 288, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 290, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 290, + 578 + ], + "score": 1.0, + "content": "5.2 ARE WEIGHT DISTRIBUTIONS I.I.D.?", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 277, + 600 + ], + "score": 1.0, + "content": "Can the changes in weights over the first", + "type": "text" + }, + { + "bbox": [ + 277, + 587, + 285, + 597 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 585, + 506, + 600 + ], + "score": 1.0, + "content": "iterations be approximated distributionally? 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This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "result suggests that network signs undergo important changes between iterations 500 and 2000, as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 126, + 538 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 126, + 526, + 141, + 536 + ], + "score": 0.83, + "content": "9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "of signs change during this period. Our results also suggest that counter to the observations", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 537, + 449, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 449, + 550 + ], + "score": 1.0, + "content": "of Zhou et al. (2019) in shallow networks, signs are not sufficient in deeper networks.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 482, + 506, + 550 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 565, + 288, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 290, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 290, + 578 + ], + "score": 1.0, + "content": "5.2 ARE WEIGHT DISTRIBUTIONS I.I.D.?", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 277, + 600 + ], + "score": 1.0, + "content": "Can the changes in weights over the first", + "type": "text" + }, + { + "bbox": [ + 277, + 587, + 285, + 597 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 585, + 506, + 600 + ], + "score": 1.0, + "content": "iterations be approximated distributionally? To mea-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 299, + 610 + ], + "score": 1.0, + "content": "sure this, we permuted the weights at iteration", + "type": "text" + }, + { + "bbox": [ + 299, + 599, + 306, + 608 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "within various structural sub-components of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "network (globally, within layers, and within convolutional filters). If networks are robust to these", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "permutations, it would suggest that the weights in such sub-compenents might be approximated and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 630, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 645 + ], + "score": 1.0, + "content": "sampled from. 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In Figure 6, we also consider shuffling only amongst weights that have the same sign. Doing so", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "substantially improves the performance of the filter-wise shuffle; however, it also reduces the extent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 276, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 214, + 291 + ], + "score": 1.0, + "content": "of the overall perturbation", + "type": "text" + }, + { + "bbox": [ + 215, + 277, + 265, + 288 + ], + "score": 0.87, + "content": "( 0 . 0 \\pm 0 . 0 4 9", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 276, + 280, + 291 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 280, + 277, + 316, + 288 + ], + "score": 0.89, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 276, + 506, + 291 + ], + "score": 1.0, + "content": "). It also improves the performance of shuffling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 210, + 301 + ], + "score": 1.0, + "content": "within layers slightly for", + "type": "text" + }, + { + "bbox": [ + 210, + 289, + 248, + 299 + ], + "score": 0.9, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 289, + 336, + 301 + ], + "score": 1.0, + "content": "and substantially for", + "type": "text" + }, + { + "bbox": [ + 336, + 289, + 379, + 299 + ], + "score": 0.9, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 289, + 506, + 301 + ], + "score": 1.0, + "content": ". We attribute the behavior for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 149, + 309 + ], + "score": 0.89, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "to the signs just as in Figure 4: when the magnitudes are similar in value (Figure 4 red", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "line) or distribution (Figure 6 red and green lines), using the signs improves performance. Reverting", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 320, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 335 + ], + "score": 1.0, + "content": "back to the initial signs while shuffling magnitudes within layers (brown line), however, damages", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 212, + 345 + ], + "score": 1.0, + "content": "the network too severely", + "type": "text" + }, + { + "bbox": [ + 212, + 332, + 263, + 343 + ], + "score": 0.9, + "content": "( 0 . 0 \\pm 0 . 0 8 7", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 331, + 280, + 345 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 280, + 332, + 320, + 343 + ], + "score": 0.88, + "content": "k = 5 0 0 \\mathrm { \\AA }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 331, + 505, + 345 + ], + "score": 1.0, + "content": ") to yield any performance improvement over", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "random noise. These results suggest that, while the signs from initialization are not sufficient for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "high performance at high sparsity as shown in Section 5.1, the signs from the rewinding iteration", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 366, + 440, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 440, + 377 + ], + "score": 1.0, + "content": "are sufficient to recover the damage caused by permutation, at least to some extent.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 390, + 227, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 228, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 228, + 402 + ], + "score": 1.0, + "content": "5.3 IS IT ALL JUST NOISE?", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 410, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "Some of our previous results suggested that the impact of signs and permutations may simply reduce", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "to adding noise to the weights. To evaluate this hypothesis, we next study the effect of simply", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 341, + 444 + ], + "score": 1.0, + "content": "adding Gaussian noise to the network weights at iteration", + "type": "text" + }, + { + "bbox": [ + 341, + 433, + 348, + 442 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 432, + 505, + 444 + ], + "score": 1.0, + "content": ". To add noise appropriately for layers", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "with different scales, the standard deviation of the noise added for each layer was normalized to a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 245, + 466 + ], + "score": 1.0, + "content": "multiple of the standard deviation", + "type": "text" + }, + { + "bbox": [ + 245, + 456, + 253, + 464 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "of the initialization distribution for that layer. In Figure 7, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 191, + 477 + ], + "score": 1.0, + "content": "see that for iteration", + "type": "text" + }, + { + "bbox": [ + 192, + 465, + 229, + 475 + ], + "score": 0.9, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 466, + 340, + 477 + ], + "score": 1.0, + "content": ", sub-networks can tolerate", + "type": "text" + }, + { + "bbox": [ + 340, + 465, + 361, + 476 + ], + "score": 0.84, + "content": "0 . 5 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 466, + 373, + 477 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 374, + 466, + 386, + 475 + ], + "score": 0.8, + "content": "1 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 466, + 505, + 477 + ], + "score": 1.0, + "content": "of noise before performance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 460, + 488 + ], + "score": 1.0, + "content": "degrades back to that of the original initialization at higher levels of noise. For iteration", + "type": "text" + }, + { + "bbox": [ + 461, + 476, + 501, + 487 + ], + "score": 0.88, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 476, + 505, + 488 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 488, + 491, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 294, + 500 + ], + "score": 1.0, + "content": "networks are surprisingly robust to noise up to", + "type": "text" + }, + { + "bbox": [ + 294, + 488, + 306, + 497 + ], + "score": 0.81, + "content": "1 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 488, + 347, + 500 + ], + "score": 1.0, + "content": ", and even", + "type": "text" + }, + { + "bbox": [ + 348, + 488, + 360, + 497 + ], + "score": 0.82, + "content": "2 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 488, + 491, + 500 + ], + "score": 1.0, + "content": "exhibits nontrivial performance.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "In Figure 8, we plot the performance of each network at a fixed sparsity level as a function of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "score": 1.0, + "content": "effective standard deviation of the noise imposed by each of the aforementioned perturbations. We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "find that the standard deviation of the effective noise explained fairly well the resultant performance", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 110, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 110, + 537, + 145, + 547 + ], + "score": 0.83, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 537, + 149, + 549 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 149, + 537, + 199, + 548 + ], + "score": 0.83, + "content": "r = - 0 . 6 7 2", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 537, + 202, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 203, + 537, + 245, + 548 + ], + "score": 0.87, + "content": "p = 0 . 0 0 8", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 537, + 248, + 549 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 249, + 537, + 289, + 547 + ], + "score": 0.83, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 537, + 293, + 549 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 293, + 537, + 343, + 547 + ], + "score": 0.8, + "content": "r = - 0 . 7 2 6", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 537, + 346, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 347, + 537, + 390, + 548 + ], + "score": 0.83, + "content": "p = 0 . 0 0 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 537, + 505, + 549 + ], + "score": 1.0, + "content": ". As expected, perturbations", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 559 + ], + "score": 1.0, + "content": "that preserved the performance of the network generally resulted in smaller changes to the state of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 559, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 200, + 571 + ], + "score": 1.0, + "content": "the network at iteration", + "type": "text" + }, + { + "bbox": [ + 200, + 559, + 207, + 569 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 559, + 506, + 571 + ], + "score": 1.0, + "content": ". Interestingly, experiments that mixed signs and magnitudes from different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "points in training (green points) aligned least well with this pattern: the standard deviation of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 580, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 506, + 595 + ], + "score": 1.0, + "content": "perturbation is roughly similar among all of these experiments, but the accuracy of the resulting", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "networks changes substantially. This result suggests that although the standard deviation of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "noise is certainly indicative of lower accuracy, there are still specific perturbations that, while small", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "in overall magnitude, can have a large effect on the network’s ability to learn, suggesting that the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 625, + 401, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 401, + 636 + ], + "score": 1.0, + "content": "observed perturbation effects are not, in fact, just a consequence of noise.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 106, + 651, + 488, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 490, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 490, + 666 + ], + "score": 1.0, + "content": "6 THE DATA-DEPENDENCE OF NEURAL NETWORKS EARLY IN TRAINING", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 403, + 689 + ], + "score": 1.0, + "content": "Section 5 suggests that the change in network behavior by iteration", + "type": "text" + }, + { + "bbox": [ + 404, + 677, + 411, + 687 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "is not due to easily-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "ascertainable, distributional properties of the network weights and signs. Rather, it appears that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "training is required to reach these network states. It is unclear, however, the extent to which various", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "aspects of the data distribution are necessary. Mainly, is the change in weights during the early", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 234, + 734 + ], + "score": 1.0, + "content": "phase of training dependent on", + "type": "text" + }, + { + "bbox": [ + 234, + 721, + 253, + 732 + ], + "score": 0.92, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 719, + 266, + 734 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 267, + 720, + 295, + 732 + ], + "score": 0.89, + "content": "p ( y | \\dot { x } ) \\dot { }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "? Here, we attempt to answer this question by mea-", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 128, + 84, + 480, + 177 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 84, + 480, + 177 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 84, + 480, + 177 + ], + "spans": [ + { + "bbox": [ + 128, + 84, + 480, + 177 + ], + "score": 0.971, + "type": "image", + "image_path": "d765cebe2c186de4c5b3ee8cc5eadc7096326e5470eaf14de37b6bd970c9d02d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 128, + 84, + 480, + 115.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 128, + 115.0, + 480, + 146.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 128, + 146.0, + 480, + 177.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 191, + 505, + 225 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 203 + ], + "score": 1.0, + "content": "Figure 7: The effect of training an IMP-derived sub-network of ResNet-20 on CIFAR-10 initialized", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 203, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 222, + 213 + ], + "score": 1.0, + "content": "with the weights at iteration", + "type": "text" + }, + { + "bbox": [ + 223, + 203, + 230, + 212 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 203, + 323, + 213 + ], + "score": 1.0, + "content": "and Gaussian noise of", + "type": "text" + }, + { + "bbox": [ + 324, + 204, + 337, + 212 + ], + "score": 0.76, + "content": "n \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 203, + 368, + 213 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 369, + 204, + 376, + 212 + ], + "score": 0.8, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 203, + 505, + 213 + ], + "score": 1.0, + "content": "is the standard deviation of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 213, + 401, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 289, + 225 + ], + "score": 1.0, + "content": "initialization distribution for each layer. Left:", + "type": "text" + }, + { + "bbox": [ + 290, + 213, + 325, + 223 + ], + "score": 0.9, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 213, + 357, + 225 + ], + "score": 1.0, + "content": ". 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In Figure 6, we also consider shuffling only amongst weights that have the same sign. Doing so", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "substantially improves the performance of the filter-wise shuffle; however, it also reduces the extent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 276, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 214, + 291 + ], + "score": 1.0, + "content": "of the overall perturbation", + "type": "text" + }, + { + "bbox": [ + 215, + 277, + 265, + 288 + ], + "score": 0.87, + "content": "( 0 . 0 \\pm 0 . 0 4 9", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 276, + 280, + 291 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 280, + 277, + 316, + 288 + ], + "score": 0.89, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 276, + 506, + 291 + ], + "score": 1.0, + "content": "). It also improves the performance of shuffling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 210, + 301 + ], + "score": 1.0, + "content": "within layers slightly for", + "type": "text" + }, + { + "bbox": [ + 210, + 289, + 248, + 299 + ], + "score": 0.9, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 289, + 336, + 301 + ], + "score": 1.0, + "content": "and substantially for", + "type": "text" + }, + { + "bbox": [ + 336, + 289, + 379, + 299 + ], + "score": 0.9, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 289, + 506, + 301 + ], + "score": 1.0, + "content": ". We attribute the behavior for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 149, + 309 + ], + "score": 0.89, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "to the signs just as in Figure 4: when the magnitudes are similar in value (Figure 4 red", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "line) or distribution (Figure 6 red and green lines), using the signs improves performance. Reverting", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 320, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 335 + ], + "score": 1.0, + "content": "back to the initial signs while shuffling magnitudes within layers (brown line), however, damages", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 212, + 345 + ], + "score": 1.0, + "content": "the network too severely", + "type": "text" + }, + { + "bbox": [ + 212, + 332, + 263, + 343 + ], + "score": 0.9, + "content": "( 0 . 0 \\pm 0 . 0 8 7", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 331, + 280, + 345 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 280, + 332, + 320, + 343 + ], + "score": 0.88, + "content": "k = 5 0 0 \\mathrm { \\AA }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 331, + 505, + 345 + ], + "score": 1.0, + "content": ") to yield any performance improvement over", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "random noise. These results suggest that, while the signs from initialization are not sufficient for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "high performance at high sparsity as shown in Section 5.1, the signs from the rewinding iteration", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 366, + 440, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 440, + 377 + ], + "score": 1.0, + "content": "are sufficient to recover the damage caused by permutation, at least to some extent.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 245, + 506, + 377 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 390, + 227, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 228, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 228, + 402 + ], + "score": 1.0, + "content": "5.3 IS IT ALL JUST NOISE?", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 410, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "Some of our previous results suggested that the impact of signs and permutations may simply reduce", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "to adding noise to the weights. To evaluate this hypothesis, we next study the effect of simply", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 341, + 444 + ], + "score": 1.0, + "content": "adding Gaussian noise to the network weights at iteration", + "type": "text" + }, + { + "bbox": [ + 341, + 433, + 348, + 442 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 432, + 505, + 444 + ], + "score": 1.0, + "content": ". To add noise appropriately for layers", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "with different scales, the standard deviation of the noise added for each layer was normalized to a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 245, + 466 + ], + "score": 1.0, + "content": "multiple of the standard deviation", + "type": "text" + }, + { + "bbox": [ + 245, + 456, + 253, + 464 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "of the initialization distribution for that layer. In Figure 7, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 191, + 477 + ], + "score": 1.0, + "content": "see that for iteration", + "type": "text" + }, + { + "bbox": [ + 192, + 465, + 229, + 475 + ], + "score": 0.9, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 466, + 340, + 477 + ], + "score": 1.0, + "content": ", sub-networks can tolerate", + "type": "text" + }, + { + "bbox": [ + 340, + 465, + 361, + 476 + ], + "score": 0.84, + "content": "0 . 5 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 466, + 373, + 477 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 374, + 466, + 386, + 475 + ], + "score": 0.8, + "content": "1 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 466, + 505, + 477 + ], + "score": 1.0, + "content": "of noise before performance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 460, + 488 + ], + "score": 1.0, + "content": "degrades back to that of the original initialization at higher levels of noise. For iteration", + "type": "text" + }, + { + "bbox": [ + 461, + 476, + 501, + 487 + ], + "score": 0.88, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 476, + 505, + 488 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 488, + 491, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 294, + 500 + ], + "score": 1.0, + "content": "networks are surprisingly robust to noise up to", + "type": "text" + }, + { + "bbox": [ + 294, + 488, + 306, + 497 + ], + "score": 0.81, + "content": "1 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 488, + 347, + 500 + ], + "score": 1.0, + "content": ", and even", + "type": "text" + }, + { + "bbox": [ + 348, + 488, + 360, + 497 + ], + "score": 0.82, + "content": "2 \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 488, + 491, + 500 + ], + "score": 1.0, + "content": "exhibits nontrivial performance.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 410, + 506, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "In Figure 8, we plot the performance of each network at a fixed sparsity level as a function of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "score": 1.0, + "content": "effective standard deviation of the noise imposed by each of the aforementioned perturbations. We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "find that the standard deviation of the effective noise explained fairly well the resultant performance", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 110, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 110, + 537, + 145, + 547 + ], + "score": 0.83, + "content": "k = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 537, + 149, + 549 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 149, + 537, + 199, + 548 + ], + "score": 0.83, + "content": "r = - 0 . 6 7 2", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 537, + 202, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 203, + 537, + 245, + 548 + ], + "score": 0.87, + "content": "p = 0 . 0 0 8", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 537, + 248, + 549 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 249, + 537, + 289, + 547 + ], + "score": 0.83, + "content": "k = 2 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 537, + 293, + 549 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 293, + 537, + 343, + 547 + ], + "score": 0.8, + "content": "r = - 0 . 7 2 6", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 537, + 346, + 549 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 347, + 537, + 390, + 548 + ], + "score": 0.83, + "content": "p = 0 . 0 0 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 537, + 505, + 549 + ], + "score": 1.0, + "content": ". As expected, perturbations", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 549, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 559 + ], + "score": 1.0, + "content": "that preserved the performance of the network generally resulted in smaller changes to the state of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 559, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 200, + 571 + ], + "score": 1.0, + "content": "the network at iteration", + "type": "text" + }, + { + "bbox": [ + 200, + 559, + 207, + 569 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 559, + 506, + 571 + ], + "score": 1.0, + "content": ". Interestingly, experiments that mixed signs and magnitudes from different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "points in training (green points) aligned least well with this pattern: the standard deviation of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 580, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 506, + 595 + ], + "score": 1.0, + "content": "perturbation is roughly similar among all of these experiments, but the accuracy of the resulting", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "networks changes substantially. 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Figure 9 (upper left) shows that pre-training on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "random labels for up to 10 epochs provides no improvement above rewinding to iteration 0 and that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "pre-training for longer begins to hurt accuracy. This result suggests that, though it is still possible", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "that labels may not be required for learning, the presence incorrect labels is sufficient to prevent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 333, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 333, + 205 + ], + "score": 1.0, + "content": "learning which approximates the early phase of training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 217, + 309, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 311, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 311, + 229 + ], + "score": 1.0, + "content": "6.2 SELF-SUPERVISED ROTATION PREDICTION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 265, + 251 + ], + "score": 1.0, + "content": "What if we remove labels entirely? Is", + "type": "text" + }, + { + "bbox": [ + 265, + 237, + 285, + 250 + ], + "score": 0.92, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "sufficient to approximate the early phase of training?", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "Historically, neural network training often involved two steps: a self-supervised pre-training phase", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "followed by a supervised phase on the target task (Erhan et al., 2010). Here, we consider one such", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "score": 1.0, + "content": "self-supervised technique: rotation prediction (Gidaris et al., 2018). During the pre-training phase,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 423, + 294 + ], + "score": 1.0, + "content": "the network is presented with a training image that has randomly been rotated", + "type": "text" + }, + { + "bbox": [ + 423, + 282, + 441, + 292 + ], + "score": 0.8, + "content": "9 0 n", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "degrees (where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 107, + 292, + 174, + 304 + ], + "score": 0.92, + "content": "n \\in \\{ 0 , 1 , 2 , 3 \\bar \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 292, + 398, + 305 + ], + "score": 1.0, + "content": ". The network must classify examples by the value of", + "type": "text" + }, + { + "bbox": [ + 399, + 294, + 406, + 302 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 292, + 505, + 305 + ], + "score": 1.0, + "content": ". If self-supervised pre-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 409, + 316 + ], + "score": 1.0, + "content": "training can approximate the early phase of training, it would suggest that", + "type": "text" + }, + { + "bbox": [ + 409, + 303, + 429, + 316 + ], + "score": 0.92, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "is sufficient on its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "own. Indeed, as shown in Figure 9 (upper right), this pre-training regime leads to well-trainable sub-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "networks, though networks must be trained for many more epochs compared to supervised training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 253, + 348 + ], + "score": 1.0, + "content": "(40 compared to 1.25, or a factor of", + "type": "text" + }, + { + "bbox": [ + 253, + 336, + 273, + 347 + ], + "score": 0.86, + "content": "3 2 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 337, + 505, + 348 + ], + "score": 1.0, + "content": "). This result suggests that the labels for the ultimate task", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "themselves are not necessary to put the network in such a state (although explicitly misleading labels", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "are detrimental). We emphasize that the duration of the pre-training phase required is an order of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "magnitude larger than the original rewinding iteration, however, suggesting that labels add important", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 380, + 313, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 313, + 393 + ], + "score": 1.0, + "content": "information which accelerates the learning process.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 109, + 406, + 271, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 273, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 273, + 419 + ], + "score": 1.0, + "content": "6.3 BLURRING TRAINING EXAMPLES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 220, + 439 + ], + "score": 1.0, + "content": "To probe the importance of", + "type": "text" + }, + { + "bbox": [ + 221, + 426, + 240, + 439 + ], + "score": 0.92, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "for the early phase of training, we study the extent to which the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "training input distribution is necessary. Namely, we pretrain using blurred training inputs with the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 492, + 462 + ], + "score": 1.0, + "content": "correct labels. Following Achille et al. (2019), we blur training inputs by downsampling by", + "type": "text" + }, + { + "bbox": [ + 492, + 449, + 504, + 459 + ], + "score": 0.56, + "content": "4 \\mathbf { x }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "and then upsampling back to the full size. Figure 9 (bottom left) shows that this pre-training method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "succeeds: after 40 epochs of pre-training, IMP with rewinding can find sub-networks that are similar", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 492, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 492, + 494 + ], + "score": 1.0, + "content": "in performance to those found after training on the original task for 500 iterations (1.25 epochs).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "Due to the success of the the rotation and blurring pre-training tasks, we explored the effect of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "combining these pre-training techniques. Doing so tests the extent to which we can discard both the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "score": 1.0, + "content": "training labels and some information from the training inputs. Figure 9 (bottom right) shows that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "doing so provides the network too little information: no amount of pre-training we considered makes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "it possible for IMP with rewinding to find sub-networks that perform tangibly better than rewinding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "to iteration 0. Interestingly however, as shown in Appendix B, trainable sub-networks are found for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "VGG-13 with this pre-training regime, suggesting that different network architectures have different", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 576, + 338, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 338, + 588 + ], + "score": 1.0, + "content": "sensitivities to the deprivation of labels and input content.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 601, + 226, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 227, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 227, + 613 + ], + "score": 1.0, + "content": "6.4 SPARSE PRETRAINING", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Since sparse sub-networks are often challenging to train from scratch without the proper initializa-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "tion (Han et al., 2015; Liu et al., 2019; Frankle & Carbin, 2019), does pre-training make it easier", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 506, + 658 + ], + "score": 1.0, + "content": "for sparse neural networks to learn? Doing so would serve as a rough form of curriculum learning", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "(Bengio et al., 2009) for sparse neural networks. We experimented with training sparse sub-networks", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "of ResNet-20 (IMP sub-networks, randomly reinitialized sub-networks, and randomly pruned sub-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "networks) first on self-supervised rotation and then on the main task, but found no benefit beyond", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "rewinding to iteration 0 (Figure 10). Moreover, doing so when starting from a sub-network rewound", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "to iteration 500 actually hurts final accuracy. This result suggests that while pre-training is sufficient", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "to approximate the early phase of supervised training with an appropriately structured mask, it is not", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 293, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 293, + 734 + ], + "score": 1.0, + "content": "sufficient to do so with an inappropriate mask.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 83, + 207, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 208, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 208, + 95 + ], + "score": 1.0, + "content": "6.1 RANDOM LABELS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 103, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "To evaluate whether this phase of training is dependent on underlying structure in the data, we drew", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "inspiration from Zhang et al. (2017) and pre-trained networks on data with randomized labels. This", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "experiment tests whether the input distribution of the training data is sufficient to put the network", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "in a position from which IMP with rewinding can find a sparse, trainable sub-network despite the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "presence of incorrect (not just missing) labels. Figure 9 (upper left) shows that pre-training on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "random labels for up to 10 epochs provides no improvement above rewinding to iteration 0 and that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "pre-training for longer begins to hurt accuracy. This result suggests that, though it is still possible", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "that labels may not be required for learning, the presence incorrect labels is sufficient to prevent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 333, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 333, + 205 + ], + "score": 1.0, + "content": "learning which approximates the early phase of training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 103, + 505, + 205 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 217, + 309, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 311, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 311, + 229 + ], + "score": 1.0, + "content": "6.2 SELF-SUPERVISED ROTATION PREDICTION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 265, + 251 + ], + "score": 1.0, + "content": "What if we remove labels entirely? Is", + "type": "text" + }, + { + "bbox": [ + 265, + 237, + 285, + 250 + ], + "score": 0.92, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "sufficient to approximate the early phase of training?", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "Historically, neural network training often involved two steps: a self-supervised pre-training phase", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "followed by a supervised phase on the target task (Erhan et al., 2010). Here, we consider one such", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 284 + ], + "score": 1.0, + "content": "self-supervised technique: rotation prediction (Gidaris et al., 2018). During the pre-training phase,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 423, + 294 + ], + "score": 1.0, + "content": "the network is presented with a training image that has randomly been rotated", + "type": "text" + }, + { + "bbox": [ + 423, + 282, + 441, + 292 + ], + "score": 0.8, + "content": "9 0 n", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "degrees (where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 107, + 292, + 174, + 304 + ], + "score": 0.92, + "content": "n \\in \\{ 0 , 1 , 2 , 3 \\bar \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 292, + 398, + 305 + ], + "score": 1.0, + "content": ". The network must classify examples by the value of", + "type": "text" + }, + { + "bbox": [ + 399, + 294, + 406, + 302 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 292, + 505, + 305 + ], + "score": 1.0, + "content": ". If self-supervised pre-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 409, + 316 + ], + "score": 1.0, + "content": "training can approximate the early phase of training, it would suggest that", + "type": "text" + }, + { + "bbox": [ + 409, + 303, + 429, + 316 + ], + "score": 0.92, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "is sufficient on its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "own. Indeed, as shown in Figure 9 (upper right), this pre-training regime leads to well-trainable sub-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "networks, though networks must be trained for many more epochs compared to supervised training", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 253, + 348 + ], + "score": 1.0, + "content": "(40 compared to 1.25, or a factor of", + "type": "text" + }, + { + "bbox": [ + 253, + 336, + 273, + 347 + ], + "score": 0.86, + "content": "3 2 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 337, + 505, + 348 + ], + "score": 1.0, + "content": "). This result suggests that the labels for the ultimate task", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "themselves are not necessary to put the network in such a state (although explicitly misleading labels", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "are detrimental). We emphasize that the duration of the pre-training phase required is an order of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "magnitude larger than the original rewinding iteration, however, suggesting that labels add important", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 380, + 313, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 313, + 393 + ], + "score": 1.0, + "content": "information which accelerates the learning process.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 237, + 506, + 393 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 406, + 271, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 273, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 273, + 419 + ], + "score": 1.0, + "content": "6.3 BLURRING TRAINING EXAMPLES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 220, + 439 + ], + "score": 1.0, + "content": "To probe the importance of", + "type": "text" + }, + { + "bbox": [ + 221, + 426, + 240, + 439 + ], + "score": 0.92, + "content": "p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "for the early phase of training, we study the extent to which the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "training input distribution is necessary. Namely, we pretrain using blurred training inputs with the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 492, + 462 + ], + "score": 1.0, + "content": "correct labels. Following Achille et al. (2019), we blur training inputs by downsampling by", + "type": "text" + }, + { + "bbox": [ + 492, + 449, + 504, + 459 + ], + "score": 0.56, + "content": "4 \\mathbf { x }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "and then upsampling back to the full size. Figure 9 (bottom left) shows that this pre-training method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "succeeds: after 40 epochs of pre-training, IMP with rewinding can find sub-networks that are similar", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 492, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 492, + 494 + ], + "score": 1.0, + "content": "in performance to those found after training on the original task for 500 iterations (1.25 epochs).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 426, + 505, + 494 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 498, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "Due to the success of the the rotation and blurring pre-training tasks, we explored the effect of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "combining these pre-training techniques. Doing so tests the extent to which we can discard both the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "score": 1.0, + "content": "training labels and some information from the training inputs. Figure 9 (bottom right) shows that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "doing so provides the network too little information: no amount of pre-training we considered makes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "it possible for IMP with rewinding to find sub-networks that perform tangibly better than rewinding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "to iteration 0. Interestingly however, as shown in Appendix B, trainable sub-networks are found for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "VGG-13 with this pre-training regime, suggesting that different network architectures have different", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 576, + 338, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 338, + 588 + ], + "score": 1.0, + "content": "sensitivities to the deprivation of labels and input content.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 498, + 506, + 588 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 601, + 226, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 227, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 227, + 613 + ], + "score": 1.0, + "content": "6.4 SPARSE PRETRAINING", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "Since sparse sub-networks are often challenging to train from scratch without the proper initializa-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "tion (Han et al., 2015; Liu et al., 2019; Frankle & Carbin, 2019), does pre-training make it easier", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 506, + 658 + ], + "score": 1.0, + "content": "for sparse neural networks to learn? Doing so would serve as a rough form of curriculum learning", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "(Bengio et al., 2009) for sparse neural networks. We experimented with training sparse sub-networks", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "of ResNet-20 (IMP sub-networks, randomly reinitialized sub-networks, and randomly pruned sub-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "networks) first on self-supervised rotation and then on the main task, but found no benefit beyond", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "rewinding to iteration 0 (Figure 10). Moreover, doing so when starting from a sub-network rewound", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "to iteration 500 actually hurts final accuracy. This result suggests that while pre-training is sufficient", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "to approximate the early phase of supervised training with an appropriately structured mask, it is not", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 293, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 293, + 734 + ], + "score": 1.0, + "content": "sufficient to do so with an inappropriate mask.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 621, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 209, + 83, + 400, + 192 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 209, + 83, + 400, + 192 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 209, + 83, + 400, + 192 + ], + "spans": [ + { + "bbox": [ + 209, + 83, + 400, + 192 + ], + "score": 0.971, + "type": "image", + "image_path": "72f8803cfddf40a06eaf44ba7dce17941a23f860928741782f1d1815af395e7b.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 83, + 400, + 96.625 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 209, + 96.625, + 400, + 110.25 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 209, + 110.25, + 400, + 123.875 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 209, + 123.875, + 400, + 137.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 209, + 137.5, + 400, + 151.125 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 209, + 151.125, + 400, + 164.75 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 209, + 164.75, + 400, + 178.375 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 209, + 178.375, + 400, + 192.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 207, + 505, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "Figure 10: The effect of pretraining sparse sub-networks of Resnet-20 (rewound to iteration 500)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 219, + 396, + 230 + ], + "spans": [ + { + "bbox": [ + 107, + 219, + 396, + 230 + ], + "score": 1.0, + "content": "with 40 epochs of self-supervised rotation before training on CIFAR-10.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 6.0 + }, + { + "type": "title", + "bbox": [ + 108, + 250, + 190, + 262 + ], + "lines": [ + { + "bbox": [ + 104, + 248, + 192, + 265 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 192, + 265 + ], + "score": 1.0, + "content": "7 DISCUSSION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 104, + 273, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 505, + 289 + ], + "score": 1.0, + "content": "In this paper, we first performed extensive measurements of various statistics summarizing learning", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 284, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 505, + 299 + ], + "score": 1.0, + "content": "over the early part of training. Notably, we uncovered 3 sub-phases: in the very first iterations, gra-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "dient magnitudes are anomalously large and motion is rapid. Subsequently, gradients overshoot to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "smaller magnitudes before leveling off while performance increases rapidly. Then, learning slowly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "begins to decelerate. We then studied a suite of perturbations to the network state in the early phase", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "finding that, counter to observations in smaller networks (Zhou et al., 2019), deeper networks are not", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "robust to reinitializing with random weights with maintained signs. We also found that the weight", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "distribution after the early phase of training is highly non-independent. Finally, we measured the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "data-dependence of the early phase with the surprising result that pre-training on a self-supervised", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 359, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 359, + 386 + ], + "score": 1.0, + "content": "task yields equivalent performance to late rewinding with IMP.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "These results have significant implications for the lottery ticket hypothesis. The seeming necessity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "of late rewinding calls into question certain interpretations of lottery tickets as well as the ability to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "identify sub-networks at initialization. Our observation that weights are highly non-independent at", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "the rewinding point suggests that the weights at this point cannot be easily approximated, making", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "approaches which attempt to “jump” directly to the rewinding point unlikely to succeed. 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URL https://arxiv.org/abs/1710.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 116, + 606, + 153, + 619 + ], + "spans": [ + { + "bbox": [ + 116, + 606, + 153, + 619 + ], + "score": 1.0, + "content": "11029.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 109, + 626, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 107, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 107, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "Dumitru Erhan, Yoshua Bengio, Aaron Courville, Pierre-Antoine Manzagol, Pascal Vincent, and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 116, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 116, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "Samy Bengio. Why does unsupervised pre-training help deep learning? 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Finally, we measured the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "data-dependence of the early phase with the surprising result that pre-training on a self-supervised", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 359, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 359, + 386 + ], + "score": 1.0, + "content": "task yields equivalent performance to late rewinding with IMP.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 273, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "These results have significant implications for the lottery ticket hypothesis. The seeming necessity", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "of late rewinding calls into question certain interpretations of lottery tickets as well as the ability to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "identify sub-networks at initialization. Our observation that weights are highly non-independent at", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "the rewinding point suggests that the weights at this point cannot be easily approximated, making", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "approaches which attempt to “jump” directly to the rewinding point unlikely to succeed. However,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "our result that labels are not necessary to approximate the rewinding point suggests that the learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "score": 1.0, + "content": "during this phase does not require task-specific information, suggesting that rewinding may not be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 468, + 313, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 313, + 480 + ], + "score": 1.0, + "content": "necessary if networks are pre-trained appropriately.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 389, + 506, + 480 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 495, + 175, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 176, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 176, + 509 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 504, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "score": 1.0, + "content": "Alessandro Achille, Matteo Rovere, and Stefano Soatto. Critical learning periods in deep neural", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 115, + 525, + 185, + 536 + ], + "spans": [ + { + "bbox": [ + 115, + 525, + 185, + 536 + ], + "score": 1.0, + "content": "networks. 2019.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 512, + 505, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "Yoshua Bengio, Jer´ ome Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In ˆ", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 115, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "Proceedings of the 26th annual international conference on machine learning, pp. 41–48. 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NetworkEpochsBatch SizeLearning RateParametersEval Accuracy
ResNet-201601280.1 (Mom 0.9)272K91.5 ± 0.2%
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NetworkEpochsBatch SizeLearning RateParametersEval Accuracy
ResNet-201601280.1 (Mom 0.9)272K91.5 ± 0.2%
ResNet-561601280.1 (Mom 0.9)856K93.0 ± 0.1%
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VGG-13160640.1 (Mom 0.9)9.42M93.5 ± 0.1%
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+1,523 @@ +# GEOMETRY-AWARE GRADIENT ALGORITHMS FOR NEURAL ARCHITECTURE SEARCH + +Liam $\mathbf { L } \mathbf { i } ^ { * 1 }$ , Mikhail Khodak∗2, Maria-Florina Balcan2, and Ameet Talwalkar1,2 1 Determined AI, 2 Carnegie Mellon University, ∗ denotes equal contribution me@liamcli.com,khodak@cmu.edu,ninamf@cs.cmu.edu,talwalkar@cmu.edu + +# ABSTRACT + +Recent state-of-the-art methods for neural architecture search (NAS) exploit gradient-based optimization by relaxing the problem into continuous optimization over architectures and shared-weights, a noisy process that remains poorly understood. We argue for the study of single-level empirical risk minimization to understand NAS with weight-sharing, reducing the design of NAS methods to devising optimizers and regularizers that can quickly obtain high-quality solutions to this problem. Invoking the theory of mirror descent, we present a geometry-aware framework that exploits the underlying structure of this optimization to return sparse architectural parameters, leading to simple yet novel algorithms that enjoy fast convergence guarantees and achieve state-of-the-art accuracy on the latest NAS benchmarks in computer vision. Notably, we exceed the best published results for both CIFAR and ImageNet on both the DARTS search space and NAS-Bench201; on the latter we achieve near-oracle-optimal performance on CIFAR-10 and CIFAR-100. Together, our theory and experiments demonstrate a principled way to co-design optimizers and continuous relaxations of discrete NAS search spaces. + +# 1 INTRODUCTION + +Neural architecture search has become an important tool for automating machine learning (ML) but can require hundreds of thousands of GPU-hours to train. Recently, weight-sharing approaches have achieved state-of-the-art performance while drastically reducing the computational cost of NAS to just that of training a single shared-weights network (Pham et al., 2018; Liu et al., 2019). Methods such as DARTS (Liu et al., 2019), GDAS (Dong & Yang, 2019), and many others (Pham et al., 2018; Zheng et al., 2019; Yang et al., 2020; Xie et al., 2019; Liu et al., 2018; Laube & Zell, 2019; Cai et al., 2019; Akimoto et al., 2019; Xu et al., 2020) combine weight-sharing with a continuous relaxation of the discrete search space to allow cheap gradient updates, enabling the use of popular optimizers. However, despite some empirical success, weight-sharing remains poorly understood and has received criticism due to (1) rank-disorder (Yu et al., 2020; Zela et al., 2020b; Zhang et al., 2020; Pourchot et al., 2020), where the shared-weights performance is a poor surrogate of standalone performance, and (2) poor results on recent benchmarks (Dong & Yang, 2020; Zela et al., 2020a). + +Motivated by the challenge of developing simple and efficient methods that achieve state-of-the-art performance, we study how to best handle the goals and optimization objectives of NAS. We start by observing that weight-sharing subsumes architecture hyperparameters as another set of learned parameters of the shared-weights network, in effect extending the class of functions being learned. This suggests that a reasonable approach towards obtaining high-quality NAS solutions is to study how to regularize and optimize the empirical risk over this extended class. While many regularization approaches have been implicitly proposed in recent NAS efforts, we focus instead on the question of optimizing architecture parameters, which may not be amenable to standard procedures such as SGD that work well for standard neural network weights. In particular, to better-satisfy desirable properties such as generalization and sparsity of architectural decisions, we propose to constrain architecture parameters to the simplex and update them using exponentiated gradient, which has favorable convergence properties due to the underlying problem structure. Theoretically, we draw upon the mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) to give convergence guarantees when using any of a broad class of such geometry-aware gradient methods to optimize the weight-sharing objective; empirically, we show that our solution leads to strong improvements on several NAS benchmarks. We summarize these contributions below: + +1. We argue for studying NAS with weight-sharing as a single-level objective over a structured function class in which architectural decisions are treated as learned parameters rather than hyperparameters. Our setup clarifies recent concerns about rank disorder and makes clear that proper regularization and optimization of this objective is critical to obtaining high-quality solutions. + +2. Focusing on optimization, we propose to improve existing NAS algorithms by re-parameterizing architecture parameters over the simplex and updating them using exponentiated gradient, a variant of mirror descent that converges quickly over this domain and enjoys favorable sparsity properties. This simple modification—which we call the Geometry-Aware Exponentiated Algorithm (GAEA)—is easily applicable to numerous methods, including first-order DARTS Liu et al. (2019), GDAS Dong & Yang (2019), and PC-DARTS (Xu et al., 2020). + +3. To show correctness and efficiency of our scheme, we prove polynomial-time stationary-point convergence of block-stochastic mirror descent—a family of geometry-aware gradient algorithms that includes GAEA—over a continuous relaxation of the single-level NAS objective. To the best of our knowledge these are the first finite-time convergence guarantees for gradient-based NAS. + +4. We demonstrate that GAEA improves upon state-of-the-art methods on three of the latest NAS benchmarks for computer vision. Specifically, we beat the current best results on NAS-Bench-201 (Dong & Yang, 2020) by $0 . 1 8 \%$ on CIFAR-10, $1 . 5 9 \%$ on CIFAR-100, and $0 . 8 2 \%$ on ImageNet16-120; we also outperform the state-of-the-art on the DARTS search space Liu et al. (2019), for both CIFAR-10 and ImageNet, and match it on NAS-Bench-1Shot1 (Zela et al., 2020a).1 + +Related Work. Most optimization analyses of NAS show monotonic improvement (Akimoto et al., 2019), asymptotic guarantees (Yao et al., 2020), or bounds on auxiliary quantities disconnected from any objective (Noy et al., 2019; Nayman et al., 2019; Carlucci et al., 2019). In contrast, we prove polynomial-time stationary-point convergence on a single-level objective for weight-sharing NAS, so far only studied empirically (Xie et al., 2019; Li et al., 2019). Our results draw upon the mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) and extend recent nonconvex convergence results Zhang & He (2018) to handle alternating descent. While there exist related results (Dang & Lan, 2015) the associated guarantees do not hold for the algorithms we propose. Finally, we note that a variant of GAEA that modifies first-order DARTS is related to XNAS (Nayman et al., 2019), whose update also involves exponentiated gradient; however, GAEA is simpler and easier to implement.2 Furthermore, the regret guarantees for XNAS do not relate to any meaningful performance measure for NAS such as speed or accuracy, whereas we guarantee convergence on the ERM objective. + +# 2 THE WEIGHT-SHARING OPTIMIZATION PROBLEM + +In supervised ML we have a dataset $T$ of labeled pairs $( x , y )$ drawn from a distribution $\mathcal { D }$ over input/output spaces $X$ and $Y$ . The goal is to use $T$ to search a function class $H$ for $h _ { \mathbf { w } } : X \mapsto Y$ parameterized by $\mathbf { w } \in \mathbb { R } ^ { d }$ that has low expected test loss $\ell ( h _ { \mathbf { w } } ( x ) , y )$ when using $x$ to predict the associated $y$ on unseen samples drawn from $D$ , as measured by some loss $\ell : Y \times Y \mapsto [ 0 , \infty )$ . A common way to do so is by approximate (regularized) empirical risk minimization (ERM), i.e. finding $\mathbf { w } \in \mathbb { R } ^ { \breve { d } }$ with the smallest average loss over $T$ , via some iterative method $\mathtt { A l g }$ , e.g. SGD. + +# 2.1 THE BENEFITS AND CRITICISMS OF WEIGHT-SHARING FOR NAS + +NAS is often viewed as hyperparameter optimization on top of $\mathtt { A l g }$ , with each architecture $a \in { \mathcal { A } }$ corresponding to a function class $H _ { a } = \{ h _ { \mathbf { w } , a } : X \mapsto Y , \mathbf { w } \in \mathbb { R } ^ { d } \}$ to be selected by using validation data $V \subset X \times Y$ to evaluate the predictor obtained by fixing $a$ and doing approximate ERM over $T$ : + +$$ +\operatorname* { m i n } _ { a \in A } \quad \sum _ { ( x , y ) \in V } \ell ( h _ { \mathbf { w } _ { a } , a } ( x ) , y ) \qquad \mathrm { s . t . } \qquad \mathbf { w } _ { a } = \mathrm { A l } \boldsymbol { \mathrm { g } } ( T , a ) +$$ + +Since training individual sets of weights for any sizeable number of architectures is prohibitive, weight-sharing methods instead use a single set of shared weights to obtain validation signal about many architectures at once. In its most simple form, RS-WS (Li & Talwalkar, 2019), these weights are trained to minimize a non-adaptive objective, $\begin{array} { r } { \operatorname* { m i n } _ { \mathbf { w } \in \mathbb { R } ^ { d } } \mathbb { E } _ { a } \sum _ { ( x , y ) \in T } \ell ( h _ { \mathbf { w } _ { a } , a } ( x ) , y ) } \end{array}$ , where the expectation is over a fixed distribution over architectures $\mathcal { A }$ . The final architecture $a$ is then chosen to maximize the outer (validation) objective in (1) subject to ${ \bf w } _ { a } = { \bf w }$ . More frequently used is a bilevel objective over some continuous relaxation $\Theta$ of the architecture space $\mathcal { A }$ , after which a valid architecture is obtained via a discretization step $\mathsf { M a p } : \Theta \mapsto \mathcal { A }$ (Pham et al., 2018; Liu et al., 2019): + +$$ +\operatorname* { m i n } _ { \theta \in \Theta } \quad \sum _ { ( x , y ) \in V } \ell ( h _ { \mathbf { w } , \theta } ( x ) , y ) \qquad \mathrm { s . t . } \qquad \mathbf { w } \in \underset { \mathbf { u } \in \mathbb { R } ^ { d } } { \arg \operatorname* { m i n } } \sum _ { ( x , y ) \in T } \ell ( h _ { \mathbf { u } , \theta } ( x ) , y ) +$$ + +This objective is not significantly different from (2), since $\mathbb { A } \bot \mathbb { g } ( T , a )$ approximately minimizes the empirical risk w.r.t. $T$ ; the difference is replacing discrete architectures with relaxed architecture parameters $\theta \in \Theta$ , w.r.t. which we can take derivatives of the outer objective. This allows (2) to be approximated via alternating gradient updates w.r.t. w and $\theta$ . Relaxations can be stochastic, so that $\operatorname { M a p } ( \theta )$ is a sample from a $\theta$ -parameterized distribution (Pham et al., 2018; Dong & Yang, 2019), or a mixture, in which case $\operatorname { M a p } ( \theta )$ selects architectural decisions with the highest weight in a convex combination given by $\theta$ (Liu et al., 2019). We overview this in more detail in Appendix A. + +While weight-sharing significantly shortens search (Pham et al., 2018), it draws two main criticisms: + +• Rank disorder: this describes when the rank of an architecture $a$ according to the validation risk evaluated with fixed shared weights w is poorly correlated with the one using “standalone” weights $\mathbf { w } _ { a } = \mathbb { A } \mathbb { 1 } \mathtt { g } ( T , a )$ . This causes suboptimal architectures to be selected after shared weights search (Yu et al., 2020; Zela et al., 2020b; Zhang et al., 2020; Pourchot et al., 2020). + +• Poor performance: weight-sharing can converge to degenerate architectures (Zela et al., 2020a) and is outperformed by regular hyperparameter tuning on NAS-Bench-201 (Dong & Yang, 2020). + +# 2.2 SINGLE-LEVEL NAS AS A BASELINE OBJECT OF STUDY + +Why are we able to apply weight-sharing to NAS? The key is that, unlike regular hyperparameters such as step-size, architectural hyperparameters directly affect the loss function without requiring a dependent change in the model weights w. Thus we can distinguish architectures without retraining simply by changing architectural decisions. Besides enabling weight-sharing, this point reveals that the goal of NAS is perhaps better viewed as a regular learning problem over an extended class ${ \cal H } _ { \cal A } = \bigcup _ { a \in { \cal A } } { \cal H } _ { a } = \{ h _ { \mathbf { w } , a } : \hat { X } \mapsto Y , \mathbf { w } \in \mathbb { R } ^ { d } , a \in \mathcal { A } \}$ that subsumes the architectural decisions as parameters of a larger model class, an unrelaxed “supernet.” The natural approach to solving this is by approximate empirical risk minimization, e.g. by approximating continuous objective below on the right using a gradient algorithm and passing the output $\theta$ through Map to obtain a valid architecture: + +$$ +\operatorname* { m i n } _ { \mathbf { w } \in \mathbb { R } ^ { d } , a \in \mathcal { A } } \quad \sum _ { ( x , y ) \in T } \ell ( h _ { \mathbf { w } , a } ( x ) , y ) \qquad \operatorname* { m i n } _ { \mathbf { w } \in \mathbb { R } ^ { d } , \theta \in \Theta } \quad \sum _ { ( x , y ) \in T } \ell ( h _ { \mathbf { w } , \theta } ( x ) , y ) +$$ + +Several works have optimized this single-level objective as an alternative to bilevel (2) (Xie et al., 2019; Li et al., 2019). We argue for its use as the baseline object of study in NAS for three reasons: + +1. As discussed above, it is the natural first approach to solving the statistical objective of NAS: finding a good predictor $h _ { \mathbf { w } , a } \in H _ { \cal A }$ in the extended function class over architectures and weights. +2. The common alternating gradient approach to the bilevel problem (2) is in practice very similar to alternating block approaches to ERM (3); as we will see, there are established ways of analyzing such methods for the latter objective, while for the former convergence is known only under very strong assumptions such as uniqueness of the inner minimum (Franceschi et al., 2018). +3. While less frequently used in practice than bilevel, single-level optimization can be very effective: we use it to achieve new state-of-the-art results on NAS-Bench-201 (Dong & Yang, 2020). + +Understanding NAS as single-level optimization—the usual deep learning setting—makes weightsharing a natural, not surprising, approach. Furthermore, for methods—both single-level and bilevel— that adapt architecture parameters during search, it suggests that we need not worry about rank disorder as long as we can use optimization to find a single feasible point that generalizes well; we explicitly do not need a ranking. Non-adaptive methods such as RS-WS still do require rank correlation to select good architectures after search, but they are explicitly not changing $\theta$ and so have no variant solving (3). The single-level formulation thus reduces search method design to well-studied questions of how to best regularize and optimize ERM. While there are many techniques for regularizing weight-sharing—including partial channels (Xu et al., 2020) and validation Hessian penalization (Zela et al., 2020a)—we focus on the second question of optimization. + +# 3 GEOMETRY-AWARE GRADIENT ALGORITHMS + +We seek to minimize the (possibly regularized) empirical risk $\begin{array} { r } { f ( \mathbf { w } , \theta ) = \frac { 1 } { | T | } \sum _ { ( x , y ) \in T } \ell ( h _ { \mathbf { w } , \theta } ( x ) , y ) } \end{array}$ over shared-weights $\mathbf { w } \in \mathbb { R } ^ { d }$ and architecture parameters $\theta \in \Theta$ . Assuming we have noisy gradients of $f$ w.r.t. w or $\theta$ at any point $( \mathbf { w } , \theta ) \in \mathbb { R } ^ { d } \times \Theta .$ —i.e. $\tilde { \nabla } _ { \mathbf { w } } f ( \mathbf { w } , \theta )$ or $\tilde { \nabla } _ { \boldsymbol { \theta } } f ( \mathbf { w } , \boldsymbol { \theta } )$ satisfying $\mathbb { E } \tilde { \nabla } _ { \mathbf { w } } f ( \mathbf { w } , \theta ) = \nabla _ { \mathbf { w } } f ( \mathbf { w } , \theta )$ or $\mathbb { E } \tilde { \nabla } _ { \theta } f ( \mathbf { w } , \theta ) = \nabla _ { \theta } f ( \mathbf { w } , \theta )$ , respectively—our goal is a point where $f$ , or at least its gradient, is small, while taking as few gradients as possible. Our main complication is that architecture parameters lie in a constrained, non-Euclidean domain $\Theta$ . Most search spaces $\mathcal { A }$ are product sets of categorical decisions—which operation $o \in O$ to use at edge $e \in E$ —so the natural relaxation is a product of $| E | \mid O |$ -simplices. However, NAS methods often re-parameterize $\Theta$ to be unconstrained using a softmax and then SGD or Adam (Kingma & Ba, 2015). Is there a better parameterization-algorithm co-design? We consider a geometry-aware approach that uses mirror descent to design NAS methods with better properties depending on the domain; a key desirable property is to return sparse architectural parameters to reduce loss from post-search discretization. + +# 3.1 BACKGROUND ON MIRROR DESCENT + +Mirror descent has many formulations (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003; ShalevShwartz, 2011); the proximal starts by noting that, in the unconstrained case, an SGD update at a point $\theta \in \Theta = \mathbb { R } ^ { k }$ given gradient estimate $\tilde { \nabla } \bar { f } ( \theta )$ with step-size $\eta > 0$ is equivalent to + +$$ +\theta - \eta \tilde { \nabla } f ( \theta ) \quad = \quad \begin{array} { c c c c } { \underset { \mathbf { u } \in \mathbb { R } ^ { k } } { \arg \operatorname* { m i n } } } & { \eta \tilde { \nabla } f ( \theta ) \cdot \mathbf { u } } & { + } & { \frac { 1 } { 2 } \| \mathbf { u } - \theta \| _ { 2 } ^ { 2 } } \end{array} +$$ + +Here the first term aligns the output with the gradient while the second (proximal) term regularizes for closeness to the previous point as measured by the Euclidean distance. While the SGD update has been found to work well for unconstrained high-dimensional optimization, e.g. deep nets, this choice of proximal regularization may be sub-optimal over a constrained space with sparse solutions. The canonical such setting is optimization over the unit simplex, i.e. when $\Theta = \{ \theta \in [ 0 , 1 ] ^ { k } : \| \theta \| _ { 1 } = 1 \}$ . Replacing the $\ell _ { 2 }$ -regularizer in Equation 4 by the relative entropy $\mathbf { u } \cdot ( \log \mathbf { u } - \log \theta )$ , i.e. the KL-divergence, yields the exponentiated gradient (EG) update ( $\odot$ denotes element-wise product): + +$$ +\begin{array} { r l r l r } { \theta \odot \dot { \exp ( - \eta \tilde { \nabla } f ( \theta ) ) } } & { { } \propto } & { \underset { \mathbf { u } \in \Theta } { \operatorname { a r g m i n } } } & { \eta \tilde { \nabla } f ( \theta ) \cdot \mathbf { u } } & { { } + } & { \mathbf { u } \cdot ( \log \mathbf { u } - \log \theta ) } \end{array} +$$ + +Note that the full EG update is obtained by $\ell _ { 1 }$ -normalizing the l.h.s. It is well-known that EG over the $k$ -dimensional simplex requires only $\mathcal { O } ( \log k ) / \varepsilon ^ { 2 }$ iterations to achieve a function value $\varepsilon$ -away from optimal (Beck & Teboulle, 2003, Theorem 5.1), compared to the ${ \cal O } ( k / \varepsilon ^ { 2 } )$ guarantee of gradient descent. This nearly dimension-independent iteration complexity is achieved by choosing a regularizer—the KL divergence—well-suited to the underlying geometry—the simplex. More generally, mirror descent is specified by a distance-generating function (DGF) $\phi$ that is stronglyconvex w.r.t. some norm. $\phi$ induces a Bregman divergence $D _ { \phi } ( { \mathbf { u } } | | { \mathbf { v } } ) = \phi ( { \mathbf { u } } ) - \phi ( { \mathbf { v } } ) - \nabla \phi ( { \mathbf { v } } ) \cdot ( { \mathbf { u } } - { \mathbf { v } } )$ (Bregman, 1967), a notion of distance on $\Theta$ that acts as a regularizer in the mirror descent update: + +$$ +\begin{array} { r l r } { \underset { \mathbf { u } \in \Theta } { \mathrm { a r g m i n } } } & { { } \eta \tilde { \nabla } f ( \boldsymbol { \theta } ) \cdot \mathbf { u } } & { + \quad D _ { \boldsymbol { \phi } } ( \mathbf { u } \vert \vert \boldsymbol { \theta } ) } \end{array} +$$ + +For example, to recover SGD (4) we set $\begin{array} { r } { \phi ( \mathbf { u } ) = \frac 1 2 \lVert \mathbf { u } \rVert _ { 2 } ^ { 2 } } \end{array}$ , which is strongly-convex w.r.t. the Euclidean norm, while EG (5) is recovered by setting $\phi ( \mathbf { u } ) \bar { \mathbf { \Psi } } = \mathbf { u } \cdot \mathrm { l o g } \mathbf { u }$ , strongly-convex w.r.t. the $\ell _ { 1 }$ -norm. + +# 3.2 BLOCK-STOCHASTIC MIRROR DESCENT + +In the previous section we saw how mirror descent can perform better over certain geometries such as the simplex. However, in weight-sharing we are interested in optimizing over a hybrid geometry containing both the shared weights in an unconstrained Euclidean space and the architecture parameters in a non-Euclidean domain. Thus we focus on optimization over two blocks: shared weights $\mathbf { w } \in \mathbb { R } ^ { d }$ and architecture parameters $\theta \in \Theta$ , the latter associated with a DGF $\phi$ that is strongly-convex w.r.t. some norm $\| \cdot \|$ . In NAS a common approach is to perform alternating gradient steps on each domain; for example, both ENAS (Pham et al., 2018) and first-order DARTS (Liu et al., 2019) alternate between SGD on the shared weights and Adam on architecture parameters. This approach is encapsulated in the block-stochastic algorithm described in Algorithm 1, which at each step chooses one block at random to update using mirror descent (recall that SGD is a variant) and after $T$ steps returns a random iterate. Algorithm 1 generalizes the single-level variant of both ENAS and first-order DARTS if SGD is used to update $\theta$ instead of Adam, with some mild caveats: in practice blocks are picked cyclically and the algorithm returns the last iterate, not a a random one. To analyze the convergence of Algorithm 1 we first state some regularity assumptions on the function: + +# Algorithm 1: Block-stochastic mirror descent optimization of a function $f : \mathbb { R } ^ { d } \times \Theta \mapsto \mathbb { R }$ . + +
for iteration t=1,...,TdoInput: nitialization(w(1),(1)）∈Rd×,strongly-convex DGF𝜙:→R,numberofiterations T≥1,step-size n>0
sample bt~Unif{w,0} if block bt = w then//randomly select update block
w(t+1) ←w(t)-nwf(w(t),0(t))// SGD update to shared weights
θ(t+1)←0(t)// no update to architecture params
else w(t+1）←w(t)
// no update to shared weights
θ(t+1)← argminuεe nθf(w(t),θ(t))·u + D(ul|θ(t))// update architecture params
Output:(w(r),θ(r)) for r~ Unif{1,...,T}
+ +Assumption 1. Suppose $\phi$ is strongly-convex w.r.t. some norm $\| \cdot \|$ on a convex set $\Theta$ and the objective function $f : \mathbb { R } ^ { d } \times \Theta \mapsto [ 0 , \infty )$ satisfies the following: + +1. $\gamma$ -relatively-weak-convexity: $f ( \mathbf { w } , \theta ) + \gamma \phi ( \theta )$ is convex on $\mathbb { R } ^ { d } \times \Theta$ for some $\gamma > 0$ . + +2. gradient bound: $\mathbb { E } \| \tilde { \nabla } _ { \mathbf { w } } f ( \mathbf { w } , \theta ) \| _ { 2 } ^ { 2 } \leq G _ { \mathbf { w } } ^ { 2 }$ and $\mathbb { E } \| \tilde { \nabla } _ { \theta } f ( \mathbf { w } , \theta ) \| _ { * } ^ { 2 } \leq G _ { \theta } ^ { 2 }$ for some $G _ { \mathbf { w } } , G _ { \theta } \geq 0 .$ + +The second assumption is a standard bound on the gradient norm while the first is a generalization of smoothness that allows all smooth and some non-smooth non-convex functions (Zhang & He, 2018). + +Our aim will be to show (first-order) $\varepsilon$ -stationary-point convergence of Algorithm 1, a standard metric indicating that it has reached a point with no feasible descent direction, up to error $\varepsilon$ ; for example, in the unconstrained Euclidean case an $\varepsilon$ -stationary-point is simply one where the gradient has squared-norm $\leq \varepsilon$ . The number of steps required to obtain such a point thus measures how fast a first-order method terminates. Stationarity is also significant as a necessary condition for optimality. + +In our case $\Theta$ may be constrained and so the gradient may never be small, thus necessitating a measure other than gradient norm. We use Bregman stationarity (Zhang & He, 2018, Equation 2.11), which measures stationary at a point $( \mathbf { w } , \theta )$ using the Bregman divergence between the point and its proximal map $\begin{array} { r } { \mathrm { p r o x } _ { \lambda } ( \mathbf { w } , \theta ) = \arg \operatorname* { m i n } _ { \mathbf { u } \in \mathbb { R } ^ { d } \times \Theta } \lambda f ( \mathbf { u } ) + D _ { \ell _ { 2 } , \phi } ( \mathbf { u } | | \mathbf { w } , \theta ) } \end{array}$ for some $\lambda > 0$ : + +$$ +\Delta _ { \lambda } ( \mathbf { w } , \boldsymbol { \theta } ) = \frac { D _ { \ell _ { 2 } , \phi } ( \mathbf { w } , \boldsymbol { \theta } | | \mathrm { p r o x } _ { \lambda } ( \mathbf { w } , \boldsymbol { \theta } ) ) + D _ { \ell _ { 2 } , \phi } ( \mathrm { p r o x } _ { \lambda } ( \mathbf { w } , \boldsymbol { \theta } ) | | \mathbf { w } , \boldsymbol { \theta } ) } { \lambda ^ { 2 } } +$$ + +Here $\begin{array} { r } { \lambda = \frac { 1 } { 2 \gamma } } \end{array}$ and the Bregman divergence $D _ { \ell _ { 2 } , \phi }$ is that of the DGF $\frac { 1 } { 2 } \| \mathbf { w } \| _ { 2 } ^ { 2 } + \phi ( { \theta } )$ that encodes the geometry of the joint optimization domain over w $\in \mathbb { R } ^ { d }$ and $\theta$ ; note that the dependence of the stationarity measure on $\gamma$ is standard (Dang $\&$ Lan, 2015; Zhang & He, 2018). + +To understand why reaching a point $( \mathbf { w } , \theta )$ with small Bregman stationarity is a reasonable goal, note that the proximal operator $\operatorname { p r o x } _ { \lambda }$ has the property that its fixed points, i.e. those satisfying $( \mathbf { w } , \theta ) = \mathrm { p r o x } _ { \lambda } ( \mathbf { w } , \theta )$ , correspond to points where $f$ has no feasible descent direction. Thus measuring how close $( \mathbf { w } , \theta )$ is to being a fixed point of $\operatorname { p r o x } _ { \lambda }$ —as is done using the Bregman divergence in (7)—is a good measure of how far away the point is from being a stationary point of $f$ . Finally, note that if $f$ is smooth, $\phi$ is Euclidean, and $\Theta$ is unconstrained—i.e. if we are running SGD over architecture parameters as well—then $\Delta _ { \frac { 1 } { 2 \gamma } } \leq \varepsilon$ implies an $O ( \varepsilon )$ -bound on the squared gradient norm, recovering the standard definition of $\varepsilon$ -stationarity. More intuition on proximal operators can be found in Parikh & Boyd (2013, Section 1.2), while further details on Bregman stationarity and how it relates to other notions of convergence can be found in Zhang & He (2018, Section 2.3). + +The following result shows that Algorithm 1 needs polynomially many iterations to finds a point $( \mathbf { w } , \theta )$ with $\varepsilon$ -small Bregman stationarity in-expectation: + +Theorem 1. Let $F = f ( \mathbf { w } ^ { ( 1 ) } , \theta ^ { ( 1 ) } )$ be the value of $f$ at initialization. Under Assumption 1, if we run Algorithm $I$ for $\begin{array} { r } { T = \frac { 1 6 \gamma F } { \varepsilon ^ { 2 } } ( G _ { \mathbf { w } } ^ { 2 } + G _ { \theta } ^ { 2 } ) } \end{array}$ iterations with step-size $\begin{array} { r } { \eta = \sqrt { \frac { 4 F } { \gamma ( G _ { \mathbf { w } } ^ { 2 } + G _ { \theta } ^ { 2 } ) T } } } \end{array}$ then $\begin{array} { r } { \mathbb { E } \Delta _ { \frac { 1 } { 2 \gamma } } \big ( \mathbf { w } ^ { ( r ) } , \theta ^ { ( r ) } \big ) \leq \varepsilon . } \end{array}$ . Here the expectation is over the randomness of the algorithm and gradients. + +The proof in the appendix follows from single-block analysis (Zhang & He, 2018, Theorem 3.1) and in fact holds for the general case of any number of blocks associated to any set of strongly-convex DGFs. Although there are prior results for the multi-block case (Dang & Lan, 2015), they do not hold for nonsmooth Bregman divergences such as the KL divergence needed for exponentiated gradient. + +![](images/d648ab607dd40f81f37634764c44572a01953b0b33302d329a29127ff5492113.jpg) +Figure 1: Sparsity: Evolution over search phase epochs of the average entropy of the operationweights for GAEA and approaches it modifies when run on the DARTS search space (left), NASBench-1Shot1 Search Space 1 (middle), and NASBench-201 on CIFAR-10 (right). GAEA reduces entropy much more quickly, allowing it to quickly obtain sparse architecture weights. This leads to both faster convergence to a single architecture and a lower loss when pruning at the end of search. + +Thus Algorithm 1 returns an $\varepsilon$ -stationary-point given $T = \mathcal { O } ( G _ { \mathbf { w } } ^ { 2 } + G _ { \theta } ^ { 2 } ) / \varepsilon ^ { 2 }$ iterations, where $G _ { \mathbf { w } } ^ { 2 }$ bounds the squared $\ell _ { 2 }$ -norm of the shared-weights gradient $\tilde { \nabla } _ { \mathbf { w } }$ and $G _ { \theta } ^ { 2 }$ bounds the squared magnitude of the architecture gradient $\tilde { \nabla } _ { \theta }$ , as measured by the dual norm $\| \cdot \| _ { * }$ of $\| \cdot \|$ . Only the last term $G _ { \theta }$ is affected by our choice of DGF $\phi$ . The DGF of SGD is strongly-convex w.r.t. the $\ell _ { 2 }$ -norm, which is its own dual, so $G _ { \mathbf { w } } ^ { 2 }$ is defined via $\ell _ { 2 }$ . However, for EG the DGF $\phi ( \mathbf { u } ) = \mathbf { u } \cdot \log \mathbf { u }$ is strongly-convex√ w.r.t. the $\ell _ { 1 }$ -norm, whose dual is $\ell _ { \infty }$ . Since the $\ell _ { 2 }$ -norm of a $k$ -dimensional vector can be $\sqrt { k }$ times its $\ell _ { \infty }$ -norm, picking this DGF can lead to better bound on $G _ { \theta }$ and thus on the number of iterations. + +# 3.3 GAEA: A GEOMETRY-AWARE EXPONENTIATED ALGORITHM + +Equipped with these single-level guarantees, we turn to designing methods that can in-principle be applied to both the single-level and bilevel objectives, seeking parameterizations and algorithms that converge quickly and encourage favorable properties; in particular, we focus on returning architecture parameters that are sparse to reduce loss due to post-search discretization. EG is often considered to converge quickly to sparse solutions over the simplex (Bradley & Bagnell, 2008; Bubeck, 2019), which makes it a natural choice for the architecture update. We thus propose GAEA, a GeometryAware Exponentiated Algorithm in which operation weights on each edge are constrained to the simplex and trained using EG; as in DARTS, the shared weights w are trained using SGD. GAEA can be used as a simple, principled modification to the many NAS methods that treat architecture parameters $\theta \in \Theta = \mathbb { R } ^ { | E | \times | O | }$ as real-valued “logits” to be passed through a softmax to obtain mixture weights or probabilities for simplices over the operations $O$ . Such methods include DARTS, PC-DARTS ( $\mathrm { X u }$ et al., 2020), and GDAS (Dong & Yang, 2019). To apply GAEA, first re-parameterize $\Theta$ to be the product set of $| E |$ simplices, each associated to an edge $( i , j ) \in E$ ; thus $\theta _ { i , j , o }$ corresponds directly to the weight or probability of operation $o \in O$ for edge $( i , j )$ , not a logit. Then, given a stochastic gradient $\tilde { \nabla } _ { \boldsymbol { \theta } } f ( \mathbf { w } ^ { ( t ) } , \boldsymbol { \theta } ^ { ( t ) } )$ and step-size $\eta > 0$ , replace the architecture update by EG: + +$$ +\begin{array} { r l r } & { \tilde { \theta } ^ { ( t + 1 ) } \gets \theta ^ { ( t ) } \odot \mathrm { e x p } \left( - \eta \tilde { \nabla } _ { \theta } f \big ( \mathbf { w } ^ { ( t ) } , \theta ^ { ( t ) } \big ) \right) } & { \mathrm { ( m u l t i p l i c a t i v e ~ u p d a t e ) } } \\ & { \theta _ { i , j , o } ^ { ( t + 1 ) } \gets \frac { \tilde { \theta } _ { i , j , o } ^ { ( t + 1 ) } } { \sum _ { o ^ { \prime } \in O } \tilde { \theta } _ { i , j , o ^ { \prime } } ^ { ( t + 1 ) } } \ \forall o \in O , \forall \ ( i , j ) \in E } & { \mathrm { ( s i m p l e x ~ p r o j e c t i o n ) } } \end{array} +$$ + +These two simple modifications, $r e$ -parameterization and exponentiation, suffice to obtain state-ofthe-art results on several NAS benchmarks, as shown in Section 4. Note that to obtain a bilevel algorithm we simply replace the gradient w.r.t. $\theta$ of the training loss with that of the validation loss. + +GAEA is equivalent to Algorithm 1 with $\begin{array} { r } { \phi ( \theta ) = \sum _ { ( i , j ) \in E } \sum _ { o \in O } \theta _ { i , j , o } \log \theta _ { i , j , o } } \end{array}$ , which is stronglyconvex w.r.t. $\| \cdot \| _ { 1 } / { \sqrt { | E | } }$ over the product of $| E | \mid O |$ -simplices. The dual is ${ \sqrt { | E | } } \| \cdot \| _ { \infty }$ , so if $G _ { \mathbf { w } }$ bounds the shared-weights gradient and we have an entry-wise bound on the architecture gradient then GAEA reach $\varepsilon$ -stationarity in $\mathcal { O } ( G _ { \mathbf { w } } ^ { 2 } + | E | ) / \varepsilon ^ { 2 }$ iterations. This can be up to a factor $| O |$ improvement over SGD, either over the simplex or the logit space. In addition, GAEA encourages sparsity in the architecture weights by using a multiplicative update over simplices and not an additive update over $\mathbb { R } ^ { | E | \times | O | }$ . Obtaining sparse architecture parameters is critical for good performance, both for the mixture relaxation, where it alleviates the effect of discretization on the validation loss, and for the stochastic relaxation, where it reduces noise when sampling architectures. + +Table 1: DARTS: Comparison with SOTA NAS methods on the DARTS search space, plus three results on different search spaces with a similar number of parameters reported at the top for comparison. All evaluations and reported performances of models found on the DARTS search space use similar training routines; this includes auxiliary towers and cutout but no other modifications, e.g. label smoothing (Muller et al., 2019), AutoAugment (Cubuk et al., 2019), Swish (Ramachandran ¨ et al., 2017), Squeeze & Excite (Hu et al., 2018), etc. The specific training procedure we use is that of PC-DARTS, which differs slightly from the DARTS routine by a small change to the drop-path probability; PDARTS tunes both this and batch-size. Our results are averaged over 10 random seeds. Search cost is hardware-dependent; we used Tesla V100 GPUs. For more details see Tables 4 & 5. + +
Search Method (source)CIFAR-10 ErrorSearch Cost (GPU Days)ImageNet Error top-1Search Cost (GPU Days)method
BestAveragetop-5
NASNet-A* (Zoph et al.,2018)-2.65200026.08.41800RL
AmoebaNet-B* (Real et al.,2019)12.55± 0.05315024.37.63150evolution
ProxylessNAS*(Cai et al.,2019)2.08424.97.58.3gradient (WS)
ENAS (Pham et al., 2018)2.890.5--1RL (WS)
RS-Wst (Li& Talwalkar,2019)2.712.85±0.080.7-1-random (WS)
ASNG (Akimoto et al.,2019)12.83 ± 0.140.1--gradient (WS)
SNAS (Xie et al., 2019)=2.85±0.021.527.39.21.5gradient (WS)
DARTS (1st)† (Liu et al.,2019)=3.00 ±0.140.4---gradient (WS)
DARTS (2nd)† (Liu et al., 2019)2.76 ±0.09126.78.74.0gradient (WS)
PDARTS (Chen et al.,2019)2.50=0.324.47.40.3gradient (WS)
PC-DARTSt (Xu et al.,2020)-2.57 ±0.070.124.27.33.8gradient (WS)
GAEA PC-DARTS† (ours)2.392.50 ±0.060.124.07.33.8gradient (WS)
PC-DARTSt (Xu et al.,2020)(search on CIFAR-1O,train on ImageNet)25.17.80.1gradient (WS)
GAEA PC-DARTS† (ours)(search on CIFAR-10, train on ImageNet)24.37.30.1gradient (WS)
+ +Search space/backbone differ from the DARTS setting; we show results for networks with a comparable number of parameters. For fair comparison to other work, we show the search cost for training the shared-weights network with a single initialization. + +# 4 EMPIRICAL RESULTS USING GAEA + +We evaluate GAEA on three different computer vision benchmarks: the large and heavily studied search space from DARTS (Liu et al., 2019) and two smaller oracle evaluation benchmarks, NASBench-1Shot1 (Zela et al., 2020a), and NAS-Bench-201 (Dong & Yang, 2020). NAS-Bench-1Shot1 differs from the others by applying operations per node instead of per edge, while NAS-Bench-201 differs by not requiring edge-pruning. Since GAEA can modify a variety of methods, e.g. DARTS, PC-DARTS (Xu et al., 2020), and GDAS (Dong & Yang, 2019), on each benchmark we start by evaluating the GAEA variant of the current best method on that benchmark. We show that despite the diversity of search spaces, GAEA improves upon this state-of-the-art across all three. Note that we use the same step-size for GAEA variants of DARTS/PC-DARTS and do not require weight-decay on architecture parameters. We defer experimental details and hyperparameter settings to the appendix and release all code, hyperparameters, and random seeds for reproducibility. + +# 4.1 CONVERGENCE AND SPARSITY OF GAEA + +We first examine the impact of GAEA on convergence and sparsity. Figure 1 shows the entropy of the operation weights averaged across nodes for a GAEA-variant and its base method across the three benchmarks, demonstrating that it decreases much faster for GAEA-modified approaches. This validates our expectation that GAEA encourages sparse architecture parameters, which should alleviate the mismatch between the continuously relaxed architecture parameters and the discrete architecture returned. Indeed, we find that post-search discretization on the DARTS search space causes the validation accuracy of the PC-DARTS supernet to drop from $7 2 . 1 7 \%$ to $1 5 . 2 7 \%$ , while for GAEA PC-DARTS the drop is only $7 5 . 0 7 \%$ to $3 3 . 2 3 \%$ ; note that this is shared-weights accuracy, obtained without retraining the final network. The numbers demonstrate that GAEA both (1) achieves better supernet optimization of the weight-sharing objective and (2) suffers less due to discretization. + +# 4.2 GAEA ON THE DARTS SEARCH SPACE + +Here we evaluate GAEA on the task of designing CNN cells for CIFAR-10 (Krizhevksy, 2009) and ImageNet (Russakovsky et al., 2015) by using it to modify PC-DARTS (Xu et al., 2020), the current state-of-the-art method. We follow the same three stage process used by both DARTS and RS-WS for search and evaluation. Table 1 displays results on both datasets and demonstrates that GAEA’s parameterization and optimization scheme improves upon PC-DARTS. In fact, GAEA PC-DARTS outperforms all search methods except ProxylessNAS, which uses 1.5 times as many parameters on a different search space. Thus we improve the state-of-the-art on the DARTS search space. To meet a higher bar for reproducibility on CIFAR-10, in Appendix C we report “broad reproducibility” (Li & Talwalkar, 2019) by repeating our pipeline with new seeds. While GAEA PC-DARTS consistently finds good networks when selecting the best of four independent trials, multiple trials are required due to sensitivity to initialization, as is true for many approaches (Liu et al., 2019; Xu et al., 2020). + +![](images/256fe57190a403501ca5f7fd2ddac27d71516438ba9a18b29b2b46a4467b7320.jpg) +Figure 2: NAS-Bench-1Shot1: Online comparison of PC-DARTS and GAEA PC-DARTS in terms of the test regret at each epoch of shared-weights training, i.e. the difference between the ground truth test error of the proposed architecture and that of the best architecture in the search space. The dark lines indicate the mean of four random trials and the light colored bands $\pm$ one standard deviation. The dashed line is the final regret of the best weight-sharing method according to Zela et al. (2020b); note that in our reproduction PC-DARTS performed better than their evaluation on spaces 1 and 3. + +On ImageNet, we follow $\mathrm { X u }$ et al. (2020) by using subsamples containing $1 0 \%$ and $2 . 5 \%$ of the training images from ILSVRC-2012 (Russakovsky et al., 2015) as training and validation sets, respectively. We fix architecture parameters for the first 35 epochs, then run GAEA PC-DARTS with step-size 0.1. All other hyperparameters match those of $\mathrm { X u }$ et al. (2020). Table 1 shows the final performance of both the architecture found by GAEA PC-DARTS on CIFAR-10 and the one found directly on ImageNet when trained from scratch for 250 epochs using the same settings as $\mathrm { X u }$ et al. (2020). GAEA PC-DARTS achieves a top-1 test error of $2 4 . 0 \%$ , which is state-of-the-art performance in the mobile setting when excluding additional training modifications, e.g. those in the caption. Additionally, the architecture found by GAEA PC-DARTS for CIFAR-10 and transferred achieves a test error of $2 4 . 2 \%$ , comparable to the $2 4 . 2 \%$ error of the one found by PC-DARTS directly on ImageNet. Top architectures found by GAEA PC-DARTS are depicted in Figure 3 in the appendix. + +# 4.3 GAEA ON NAS-BENCH-1SHOT1 + +NAS-Bench-1Shot1 (Zela et al., 2020a) is a subset of NAS-Bench-101 (Ying et al., 2019) that allows benchmarking weight-sharing methods on three search spaces over CIFAR-10 that differ in the number of nodes considered and the number of input edges per node. Of the weight-sharing methods benchmarked by Zela et al. (2020a), we found that PC-DARTS achieves the best performance on 2 of 3 search spaces, so we again evaluate GAEA PC-DARTS here. Figure 2 shows that GAEA PC-DARTS consistently finds better architectures on average than PC-DARTS and thus exceeds the performance of the best method from Zela et al. (2020a) on 2 of 3 search spaces. We hypothesize that the benefits of GAEA are limited here due to the near-saturation of NAS methods. In particular, existing methods obtain within $1 \%$ test error of the top network in each space, while the latters’ test errors when evaluated with different initializations are $0 . 3 7 \%$ , $0 . 2 3 \%$ and $0 . 1 9 \%$ , respectively. + +# 4.4 GAEA ON NAS-BENCH-201 + +NAS-Bench-201 has one search space on three datasets—CIFAR-10, CIFAR-100, and ImageNet16-120—that includes 4-node architectures with an operation from $O = \{ \mathrm { n o n e }$ , skip connect, 1x1 convolution, 3x3 convolution, $3 \mathrm { x } 3 \mathrm { a v g } \mathrm { p o o l } \}$ on each edge, yielding 15625 possible networks. Dong & Yang (2020) report results for several algorithms in the transfer NAS setting, where search is conducted on CIFAR-10 and the resulting networks are trained on a possibly different target dataset. Table 2 reports a subset of these results alongside evaluations of our implementation of several existing and GAEA-modified NAS methods in both the transfer and direct setting. Both the results from Dong & Yang (2020) and our reproductions show that GDAS is the best previous weight-sharing method; we evaluate GAEA GDAS and find that it achieves better results on CIFAR-100 and similar results on the other two datasets. + +Table 2: NAS-Bench-201: Results are separated into traditional hyperparameter optimization algorithms with search run on CIFAR-10 (top block), weight-sharing methods with search run on CIFAR-10 (middle block), and weight-sharing methods run directly on the dataset used for training (bottom block). The use of transfer NAS follows the evaluations conducted by Dong & Yang (2020); unless otherwise stated all non-GAEA results are from their paper. The best results in the transfer and direct settings on each dataset are bolded. + +
Search*(seconds)CIFAR-10 (test)CIFAR-100 (test)ImageNet-16-120 (test)
RegularREAN/A93.92 ± 0.3071.84 ± 0.9945.54 ± 1.03
HO,RSN/A93.70±0.3671.04 ± 1.0844.57 ±1.25
search onREINFORCEN/A93.85 ± 0.3771.71 ± 1.0945.25 ± 1.18
CIFAR-10BOHBN/A93.61 ±0.5270.85 ±1.2844.42 ± 1.49
RSPS758787.66 ± 1.6958.33 ± 4.3431.14 ± 3.88
DARTS (bilevel)3578154.30 ±0.0015.61 ±0.0016.32 ± 0.00
WeightSETN3413987.64 ± 0.0059.05 ± 0.2432.52 ±0.21
sharing,GDAS3160993.61 ±0.0970.70 ± 0.3041.71 ± 0.98
search onDARTS‡ (bilevel)10683†54.30 ±0.0015.32 ± 0.0016.38 ± 0.00
CIFAR-10GAEA DARTS (bilevel)7930+91.63 ± 2.5768.39 ± 4.4741.59 ± 4.20
DARTS* (ERM)18112t84.39 ±3.8254.81 ± 7.0831.82 ± 4.78
GAEA DARTS (ERM)9061†94.10±0.2972.60± 0.8945.81 ± 0.51
WeightGDAS27923†93.52 ± 0.1567.52 ± 0.1540.91 ± 0.12
GAEA GDAS16754t93.55 ± 0.1370.47 ± 0.4740.91 ±0.12
sharing,DARTS‡ (bilevel)10683†54.30 ±0.0015.32 ±0.0028.96 ±10.22
directGAEA DARTS (bilevel)7930t91.63 ± 2.5771.87 ± 0.5745.69 ± 0.56
searchDARTS* (ERM)18112†84.39 ±3.8251.26 ±6.1431.35 ± 7.46
GAEA DARTS (ERM)9061†94.10±0.2973.43 ± 0.1346.36 ±0.00
ResNetN/A93.9770.8643.63
OptimalN/A94.3773.5147.31
+ +Search cost reported for running the search algorithm on CIFAR-10. Search cost measured on NVIDIA P100 GPUs. ‡ Our reproduction or implementation of a non-GAEA method. + +Since we are interested in improving upon not only GAEA GDAS but also upon traditional hyperparameter optimization methods, we also investigate the performance of GAEA applied to first-order DARTS. We evaluate GAEA DARTS with both single-level (ERM) and bilevel optimization; recall that in the latter case we optimize architecture parameters w.r.t. the validation loss and the shared weights w.r.t. the training loss, whereas in ERM there is no data split. GAEA DARTS (ERM) achieves state-of-the-art performance on all three datasets in both the transfer and direct setting, exceeding the test accuracy of both weight-sharing and traditional hyperparameter tuning by a wide margin. GAEA DARTS (bilevel) performs worse but still exceeds all other methods on CIFAR-100 and ImageNet-16-120 in the direct search setting. The result thus also confirms the relevance of studying the single-level case to understand NAS; notably, the DARTS (ERM) baseline also improves substantially upon the DARTS (bilevel) baseline. + +# 5 CONCLUSION + +In this paper we take an optimization-based view of NAS, arguing that the design of good NAS algorithms is largely a matter of successfully optimizing and regularizing the supernet. In support of this, we develop GAEA, a simple modification of gradient-based NAS that attains state-of-the-art performance on several computer vision benchmarks while enjoying favorable speed and sparsity properties. We believe that obtaining high-performance NAS algorithms for a wide variety of applications will continue to require a similar co-design of search space parameterizations and optimization methods, and that our geometry-aware framework can help accelerate this process. In particular, most modern NAS algorithms search over products of categorical decision spaces, to which our approach is directly applicable. More generally, as the field moves towards more ambitious search spaces, e.g. full-network topologies or generalizations of operations such as convolution or attention, these developments may result in new architecture domains for which our work can inform the design of appropriate, geometry-aware optimization methods. + +# ACKNOWLEDGMENTS + +We thank Jeremy Cohen, Jeffrey Li, and Nicholas Roberts for helpful feedback. This work was supported in part by DARPA under cooperative agreements FA875017C0141 and HR0011202000, NSF grants CCF-1535967, CCF-1910321, IIS-1618714, IIS-1705121, IIS-1838017, IIS-1901403, and IIS-2046613, a Microsoft Research Faculty Fellowship, a Bloomberg Data Science research grant, an Amazon Research Award, an AWS Machine Learning Research Award, a Facebook Faculty Research Award, funding from Booz Allen Hamilton Inc., a Block Center Grant, a Carnegie Bosch Institute Research Award, and a Two Sigma Fellowship Award. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA, NSF, or any other funding agency. + +# REFERENCES + +Youhei Akimoto, Shinichi Shirakawa, Nozomu Noshinari, Kento Uchida, Shota Saito, and Kouhei Nishida. Adaptive stochastic natural gradient method for one-shot neural architecture search. 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On the convergence rate of stochastic mirror descent for nonsmooth nonconvex optimization. arXiv, 2018. + +Yuge Zhang, Zejun Lin, Junyang Jiang, Quanlu Zhang, Yujing Wang, Hui Xue, Chen Zhang, and Yaming Yang. Deeper insights into weight sharing in neural architecture search. arXiv, 2020. + +Xiawu Zheng, Rongrong Ji, Lang Tang, Baochang Zhang, Jianzhuang Liu, and Qi Tian. Multinomial distribution learning for effective neural architecture search. In Proceedings of the IEEE International Conference on Computer Vision, 2019. + +Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018. + +# A BACKGROUND ON NAS WITH WEIGHT-SHARING + +Here we review the NAS setup motivating our work. Weight-sharing methods almost exclusively use micro cell-based search spaces for their tractability and additional structure (Pham et al., 2018; Liu et al., 2019). These search spaces can be represented as directed acyclic graphs (DAGs) with a set of ordered nodes $N$ and edges $E$ . Each node $x ^ { ( i ) } \in N$ is a feature representation and each edge $( i , j ) \in E$ is associated with an operation on the feature of node $j$ passed to node $i$ and aggregated with other inputs to form $x ^ { ( j ) }$ , with the restriction that a given node $j$ can only receive edges from prior nodes as input. Hence, the feature at node $i$ is $\begin{array} { r } { x ^ { ( i ) } = \sum _ { j < i } \overset { \cdot } { o } ^ { ( i , j ) } ( x ^ { ( j ) } ) } \end{array}$ . Search spaces are specified by the number of nodes, the number of edges per node, and the set of operations $O$ that can be applied at each edge. Thus for NAS, $\mathcal { A } \subset \{ 0 , \bar { 1 } \} ^ { | E | \times | O | }$ is the set of all valid architectures for encoded by edge and operation decisions. Treating both the shared weights $\mathbf { w } \in \mathbb { R } ^ { d }$ and architecture decisions $a \in { \mathcal { A } }$ as parameters, weight-sharing methods train a single network subsuming all possible functions within the search space. + +Gradient-based weight-sharing methods apply continuous relaxations to the architecture space $\mathcal { A }$ in order to compute gradients in a continuous space $\Theta$ . Methods like DARTS (Liu et al., 2019) and its variants (Chen et al., 2019; Laube & Zell, 2019; Hundt et al., 2019; Liang et al., 2019; Noy et al., 2019; Nayman et al., 2019) relax the search space by considering a mixture of operations per edge. For example, we will consider a relaxation where the architecture space $\mathcal { A } = \bar { \{ 0 , 1 \} } ^ { | E | \times | O \bar { 1 } }$ is relaxed into $\Theta \bar { = } [ 0 , 1 ] ^ { | E | \times | O | }$ with the constraint that $\textstyle \sum _ { o \in O } \theta _ { i , j , o } = 1$ , i.e. the operation weights on each edge sum to 1. The feature at node $i$ is then $\begin{array} { r } { x ^ { ( i ) } = \sum _ { j < i } \sum _ { o \in O } \theta _ { i , j , o } o ( x ^ { ( j ) } ) } \end{array}$ . To get a valid architecture $a \in { \mathcal { A } }$ from a mixture $\theta$ , rounding and pruning are typically employed after the search phase. + +An alternative, stochastic approach, such as that used by GDAS (Dong & Yang, 2019), instead uses $\Theta$ -parameterized distributions $p _ { \theta }$ over $\mathcal { A }$ to sample architectures (Pham et al., 2018; Xie et al., 2019; Akimoto et al., 2019; Cai et al., 2019); unbiased gradients w.r.t. $\theta \in \Theta$ can be computed using Monte Carlo sampling. The goal of all these relaxations is to use simple gradient-based approaches to approximately optimize (1) over $a \in { \mathcal { A } }$ by optimizing (2) over $\theta \in \Theta$ instead. However, both the relaxation and the optimizer critically affect the convergence speed and solution quality. We next present a principled approach for understanding both mixture and stochastic methods. + +# B OPTIMIZATION + +This section contains proofs and generalizations of the non-convex optimization results in Section 3. Throughout this section, $\mathbb { V }$ denotes a finite-dimensional real vector space with Euclidean inner product $\langle \cdot , \cdot \rangle$ , $\mathbb { R } _ { + }$ denotes the set of nonnegative real numbers, and $\overline { { \mathbb { R } } }$ denotes the set of extended real numbers $\mathbb { R } \cup \{ \pm \infty \}$ . + +# B.1 PRELIMINARIES + +Definition 1. Consider a closed and convex subset $\mathcal { X } \subset \mathbb { V } .$ . For any $\alpha > 0$ and norm $\| \cdot \| : \mathcal { X } \mapsto \mathbb { R } _ { + }$ an everywhere-subdifferentiable function $f : \mathcal { X } \mapsto \overline { { \mathbb { R } } }$ is called $\alpha$ -strongly-convex w.r.t. $| | \cdot | | i f$ $\forall \mathbf { x } , \mathbf { y } \in \mathcal { X }$ we have + +$$ +f ( \mathbf { y } ) \geq f ( \mathbf { x } ) + \langle \nabla f ( \mathbf { x } ) , \mathbf { y } - \mathbf { x } \rangle + { \frac { \alpha } { 2 } } \| \mathbf { y } - \mathbf { x } \| ^ { 2 } +$$ + +Definition 2. Consider a closed and convex subset $\mathcal { X } \subset \mathbb { V } .$ . For any $\beta > 0$ and norm $\| \cdot \| : \mathcal { X } \mapsto \mathbb { R } _ { + }$ an continuously-differentiable function $f : \mathcal { X } \mapsto \mathbb { R }$ is called $\beta$ -strongly-smooth w.r.t. $\| \cdot \| i f$ $\forall \mathbf { x } , \mathbf { y } \in \mathcal { X }$ we have + +$$ +f ( \mathbf { y } ) \leq f ( \mathbf { x } ) + \langle \nabla f ( \mathbf { x } ) , \mathbf { y } - \mathbf { x } \rangle + { \frac { \beta } { 2 } } \| \mathbf { y } - \mathbf { x } \| ^ { 2 } +$$ + +Definition 3. Let $\mathcal { X }$ be a closed and convex subset of $\mathbb { V }$ . The Bregman divergence induced by $a$ strictly convex, continuously-differentiable distance-generating function (DGF) $\phi : \mathcal { X } \mapsto \overline { { \mathbb { R } } }$ is + +$$ +D _ { \phi } ( \mathbf { x } | | \mathbf { y } ) = \phi ( \mathbf { x } ) - \phi ( \mathbf { y } ) - \left. \nabla \phi ( \mathbf { y } ) , \mathbf { x } - \mathbf { y } \right. \forall \mathbf { x } , \mathbf { y } \in \mathcal { X } +$$ + +By definition, the Bregman divergence satisfies the following properties: + +$$ +D _ { \phi } ( \mathbf { x } | | \mathbf { y } ) \geq 0 \forall x , y \in \mathcal { X } a n d D _ { \phi } ( \mathbf { x } | | \mathbf { y } ) = 0 \iff x = y . +$$ + +2. If $\phi$ is $\alpha$ -strongly-convex w.r.t. norm $\| \cdot \|$ then so is $D _ { \phi } ( \cdot | | \mathbf { y } ) \ \forall \ \mathbf { y } \in \mathcal { X }$ . Furthermore, $\begin{array} { r } { D _ { \phi } ( \mathbf { x } | | \mathbf { y } ) \geq \frac { \alpha } { 2 } \| \mathbf { x } - \mathbf { y } \| ^ { 2 } \forall \mathbf { x } , \mathbf { y } \in \mathcal { X } . } \end{array}$ . + +3. If $\phi$ is $\beta$ -strongly-smooth w.r.t. norm $\| \cdot \|$ then so is $D _ { \phi } ( \cdot | | \mathbf { y } ) \ \forall \ \mathbf { y } \in \mathcal { X }$ . Furthermore, $\begin{array} { r } { D _ { \phi } ( \mathbf { x } | | \mathbf { y } ) \leq \frac { \beta } { 2 } \| \mathbf { x } - \mathbf { y } \| ^ { 2 } \forall \mathbf { x } , \mathbf { y } \in \mathcal { X } } \end{array}$ . + +Definition 4. (Zhang & He, 2018, Definition 2.1) Consider a closed and convex subset $\mathcal { X } \subset \mathbb { V } .$ . For any $\gamma > 0$ and ${ \overset { \vartriangle } { \phi } } : { \boldsymbol { \mathcal { X } } } \mapsto { \overline { { \mathbb { R } } } }$ an everywhere-subdifferentiable function $f : \mathcal { X } \mapsto \mathbb { R }$ is called $\gamma$ -relatively-weakly-convex ( $\langle \gamma$ -RWC) w.r.t. $\phi$ if $f ( \cdot ) + \gamma \phi ( \cdot )$ is convex on $\mathcal { X }$ . + +Definition 5. (Zhang & He, 2018, Definition 2.3) Consider a closed and convex subset $\mathcal { X } \subset \mathbb { V } .$ . For any $\lambda > 0$ , function $f : \mathcal { X } \mapsto \mathbb { R } ,$ , and D $G F \phi : \mathcal { X } \mapsto \overline { { \mathbb { R } } }$ the Bregman proximal operator of $f$ is + +$$ +\operatorname { p r o x } _ { \lambda } ( \mathbf { x } ) = \underset { \mathbf { u } \in \mathcal { X } } { \arg \operatorname* { m i n } } \lambda f ( \mathbf { u } ) + D _ { \phi } ( \mathbf { u } | | \mathbf { x } ) +$$ + +Definition 6. (Zhang & He, 2018, Equation 2.11) Consider a closed and convex subset $\mathcal { X } \subset \mathbb { V } .$ . For any $\lambda > 0$ , function $f : \mathcal { X } \mapsto \mathbb { R }$ , and DGF ${ \vec { \mathbf { \tau } } } \phi : \mathcal { X } \mapsto \overline { { \mathbb { R } } }$ the Bregman stationarity of $f$ at any point $\mathbf { x } \in \mathcal { X }$ is + +$$ +\Delta _ { \lambda } ( { \bf x } ) = \frac { D _ { \phi } ( { \bf x } | | \mathrm { p r o x } _ { \lambda } ( { \bf x } ) ) + D _ { \phi } ( \mathrm { p r o x } _ { \lambda } ( { \bf x } ) | | { \bf x } ) } { \lambda ^ { 2 } } +$$ + +# B.2 RESULTS + +Throughout this subsection let $\mathbb { V } = \times _ { i = 1 } ^ { b } \mathbb { V } _ { i }$ be a product space of $b$ finite-dimensional real vector spaces $\mathbb { V } _ { i }$ , each with an associated norm $\| \cdot \| _ { i } : \mathbb { V } _ { i } \mapsto \mathbb { R } _ { + }$ , and $\chi = \times _ { i = 1 } ^ { b } \chi _ { i }$ be a product set of $b$ subsets $\mathcal { X } _ { i } \subset \mathbb { V } _ { i }$ , each with an associated 1-strongly-convex DGF $\phi _ { i } : \mathcal { X } _ { i } \mapsto \overline { { \mathbb { R } } }$ w.r.t. $\| \cdot \| _ { i }$ . For each $i \in [ b ]$ will use $\| \cdot \| _ { i , * }$ to denote the dual norm of $\| \cdot \| _ { i }$ and for any element $\mathbf { x } \in \mathcal { X }$ we will use $\mathbf { x } _ { i }$ to denote its component in block $i$ and $\mathbf { x } _ { - i }$ to denote the component across all blocks other than $i$ . Define the functions $\| \cdot \| : \mathbb { V } \mapsto \mathbb { R } _ { + }$ and $\| \cdot \| _ { * } \mathbb { V } \mapsto \mathbb { R } _ { + }$ for any $\mathbf { x } \in \mathbb { V }$ by $\begin{array} { r } { \| \mathbf { x } \| ^ { 2 } = \sum _ { i = 1 } ^ { b } \| \mathbf { x } _ { i } \| _ { i } ^ { 2 } } \end{array}$ and $\begin{array} { r } { \| \mathbf { x } \| _ { * } ^ { 2 } = \sum _ { i = 1 } ^ { b } \| \mathbf { x } _ { i } \| _ { i , * } ^ { 2 } } \end{array}$ , respectively, and the function $\phi : \mathcal { X } \mapsto \overline { { \mathbb { R } } }$ for any $\mathbf { x } \in \mathcal { X }$ by $\begin{array} { r } { \phi ( \mathbf { x } ) = \sum _ { i = 1 } ^ { b } \phi _ { i } ( \mathbf { x } ) } \end{array}$ . Finally, for any $n \in \mathbb { N }$ we will use $[ n ]$ to denote the set $\{ 1 , \ldots , n \}$ . Setting 1. For some fixed constants $\gamma _ { i } , L _ { i } > 0$ for each $i \in [ b ]$ we have the following: + +1. $f : \mathcal { X } \mapsto \mathbb { R }$ is everywhere-subdifferentiable with minimum $f ^ { * } > - \infty$ and for all $\mathbf { x } \in \mathcal { X }$ and each $i \in [ b ]$ the restriction $f ( \cdot , \mathbf { x } _ { - i } )$ is $\gamma _ { i }$ -RWC w.r.t. $\phi _ { i }$ . +2. For each $i \in [ b ]$ there exists a stochastic oracle $G _ { i }$ that for input $\mathbf { x } \in \mathcal { X }$ outputs a random vector $G _ { i } ( \mathbf { x } , \boldsymbol { \xi } )$ s.t. $\mathbb { E } _ { \xi } G _ { i } ( \mathbf { x } , \xi ) \in \partial _ { i } f ( \mathbf { x } )$ , where $\partial _ { i } f ( \mathbf { x } )$ is the subdifferential set of the restriction $f ( \cdot , \mathbf { x } _ { - i } )$ at $\mathbf { x } _ { i }$ . Moreover, $\mathbb { E } _ { \xi } \| G _ { i } ( \mathbf { x } , \xi ) \| _ { i , * } ^ { 2 } \le L _ { i } ^ { 2 }$ . + +Define $\gamma = \operatorname* { m a x } _ { i \in [ b ] } \gamma _ { i }$ and $\begin{array} { r } { L ^ { 2 } = \sum _ { i = 1 } ^ { b } L _ { i } ^ { 2 } } \end{array}$ + +Claim 1. $\| \cdot \|$ is a norm on $\mathbb { V }$ . + +Proof. Positivity and homogeneity are trivial. For the triangle inequality, note that for any $\lambda \in [ 0 , 1 ]$ and any $\mathbf { x } , \mathbf { y } \in { \mathcal { X } }$ we have that + +$$ +\begin{array} { r l } { \| \lambda \mathbf { x } + ( 1 - \lambda ) \mathbf { y } \| = \displaystyle \sqrt { \displaystyle \sum _ { i = 1 } ^ { b } \| \lambda \mathbf { x } _ { i } + ( 1 - \lambda ) \mathbf { y } _ { i } \| _ { i } ^ { 2 } } } & { } \\ { \leq \sqrt { \displaystyle \sum _ { i = 1 } ^ { b } ( \lambda \| \mathbf { x } _ { i } \| _ { i } + ( 1 - \lambda ) \| \mathbf { y } _ { i } \| _ { i } ) ^ { 2 } } } & { } \\ { \leq \lambda \sqrt { \displaystyle \sum _ { i = 1 } ^ { b } \| \mathbf { x } _ { i } \| _ { i } ^ { 2 } } + ( 1 - \lambda ) \sqrt { \displaystyle \sum _ { i = 1 } ^ { b } \| \mathbf { y } _ { i } \| _ { i } ^ { 2 } } } & { } \\ { = \lambda \| \mathbf { x } \| + ( 1 - \lambda ) \| \mathbf { y } \| } & { } \end{array} +$$ + +where the first inequality follows by convexity of the norms $\| \cdot \| _ { i } \forall i \in [ b ]$ and the fact that the Euclidean norm on $\mathbb { R } ^ { b }$ is nondecreasing in each argument, while the second inequality follows by convexity of the Euclidean norm on $\mathbb { R } ^ { \bar { b } }$ . Setting $\begin{array} { r } { \lambda = \frac { 1 } { 2 } } \end{array}$ and multiplying both sides by 2 yields the triangle inequality. □ + +Algorithm 2: Block-stochastic mirror descent over $\mathcal { X } = \times _ { i = 1 } ^ { b } \mathcal { X } _ { i }$ given associated DGFs $\phi _ { i } : \mathcal { X } _ { i } \mapsto \overline { { \mathbb { R } } }$ . + +Input: initialization $\mathbf { x } ^ { ( 1 ) } \in \mathcal { X }$ , number of steps $T \geq 1$ , step-size sequence $\{ \eta _ { t } \} _ { t = 1 } ^ { T }$ + +r iteration $t \in [ T ]$ do +sample $i \sim \mathrm { U n i f } [ b ]$ +set x−i $\mathbf { x } _ { - i } ^ { ( t + 1 ) } = \mathbf { x } _ { - i } ^ { ( t ) }$ +get $\mathbf { g } = G _ { i } ( \mathbf { x } ^ { ( t ) } , \xi _ { t } )$ +set xi $\begin{array} { r } { \mathbf { x } _ { i } ^ { ( t + 1 ) } = \arg \operatorname* { m i n } _ { \mathbf { u } \in \mathcal { X } _ { i } } \eta _ { t } \langle \mathbf { g } , \mathbf { u } \rangle + D _ { \phi _ { i } } ( \mathbf { u } | | \mathbf { x } _ { i } ^ { ( t ) } ) } \end{array}$ + +Claim 2. $\frac { 1 } { 2 } \| \cdot \| _ { * } ^ { 2 }$ is the convex conjugate of ${ \frac { 1 } { 2 } } \| \cdot \| ^ { 2 }$ . + +Proof. Consider any $\mathbf { u } \in \mathbb { V }$ . To upper-bound the convex conjugate note that + +$$ +\begin{array} { l } { \displaystyle \operatorname* { s u p } _ { \mathbf { x } \in \mathbb { V } } \langle \mathbf { u } , \mathbf { x } \rangle - \frac { \| \mathbf { x } \| ^ { 2 } } { 2 } = \displaystyle \operatorname* { s u p } _ { \mathbf { x } \in \mathbb { V } } \displaystyle \sum _ { i = 1 } ^ { b } \langle \mathbf { u } _ { i } , \mathbf { x } _ { i } \rangle - \frac { \| \mathbf { x } _ { i } \| _ { i } ^ { 2 } } { 2 } } \\ { \displaystyle \quad \leq \operatorname* { s u p } _ { \mathbf { x } \in \mathbb { V } } \displaystyle \sum _ { i = 1 } ^ { b } \| \mathbf { u } _ { i } \| _ { i , * } \| \mathbf { x } _ { i } \| _ { i } - \frac { \| \mathbf { x } _ { i } \| _ { i } ^ { 2 } } { 2 } } \\ { \displaystyle \quad = \frac { 1 } { 2 } \displaystyle \sum _ { i = 1 } ^ { b } \| \mathbf { u } _ { i } \| _ { i , * } ^ { 2 } } \\ { \displaystyle \quad = \frac { \| \mathbf { u } \| _ { * } ^ { 2 } } { 2 } } \end{array} +$$ + +where the first inequality follows by definition of a dual norm and the second by maximizing each term w.r.t. $\| \mathbf { x } _ { i } \| _ { i }$ . For the lower bound, pick $\mathbf { x } \in \mathbb { V }$ s.t. $\langle \mathbf { u } _ { i } , \mathbf { x } _ { i } \rangle = \| \mathbf { u } _ { i } \| _ { i , * } \| \mathbf { x } _ { i } \| _ { i }$ and $\| \mathbf { x } _ { i } \| _ { i } =$ $\| \mathbf { u } _ { i } \| _ { i , * } \forall i \in [ b ]$ , which must exist by the definition of a dual norm. Then + +$$ +\langle \mathbf { u } , \mathbf { x } \rangle - \frac { \| \mathbf { x } \| ^ { 2 } } { 2 } = \sum _ { i = 1 } ^ { b } \langle \mathbf { u } _ { i } , \mathbf { x } _ { i } \rangle - \frac { \| \mathbf { x } _ { i } \| _ { i } ^ { 2 } } { 2 } = \frac { 1 } { 2 } \sum _ { i = 1 } ^ { b } \frac { \| \mathbf { u } _ { i } \| _ { i , * } ^ { 2 } } { 2 } = \frac { \| \mathbf { u } \| _ { * } ^ { 2 } } { 2 } +$$ + +so $\begin{array} { r } { \operatorname* { s u p } _ { \mathbf { x } \in \mathbb { V } } \langle \mathbf { u } , \mathbf { x } \rangle - \frac { 1 } { 2 } \| \mathbf { x } \| ^ { 2 } \geq \frac { 1 } { 2 } \| \mathbf { u } \| _ { * } ^ { 2 } } \end{array}$ , completing the proof. + +Theorem 2. Let $\hat { x }$ be the output of Algorithm 2 after $T$ iterations with non-increasing step-size sequence $\{ \eta _ { t } \} _ { t = 1 } ^ { T }$ . Then under Setting $^ { l }$ , for any $\hat { \gamma } > \gamma$ we have that + +$$ +\mathbb { E } \Delta _ { \frac { 1 } { \hat { \mathbf { \xi } } } } ( \hat { \mathbf { x } } ) \leq \frac { \hat { \gamma } b } { \hat { \gamma } - \gamma } \frac { \operatorname* { m i n } _ { \mathbf { u } \in \mathcal { X } } f ( \mathbf { u } ) + \hat { \gamma } D _ { \phi } ( \mathbf { u } | | \mathbf { x } ^ { ( 1 ) } ) - f ^ { * } + \frac { \hat { \gamma } L ^ { 2 } } { 2 b } \sum _ { t = 1 } ^ { T } \eta _ { t } ^ { 2 } } { \sum _ { t = 1 } ^ { T } \eta _ { t } } +$$ + +where the expectation is w.r.t. $\xi _ { t }$ and the randomness of the algorithm. + +Proof. Define transforms be a stochastic oracle tha $\mathbf { U } _ { i } , i \in \ [ b ]$ $\mathbf { U } _ { i } ^ { T } \mathbf { x } = \mathbf { x } _ { i }$ $\begin{array} { r } { \mathbf { x } = \sum _ { i = 1 } ^ { b } \mathbf { U } _ { i } \mathbf { x } _ { i } \ \forall \textbf { x } \in \mathcal { X } } \end{array}$ . Let impli $G$ $\textbf { x } \in { \mathcal { X } }$ $G ( { \bf x } , i , \xi ) = b { \bf U } _ { i } G _ { i } ( { \bf x } , \xi )$ $\begin{array} { r } { \mathbb { E } _ { i , \xi } G ( \mathbf { x } , i , \xi ) = \frac { 1 } { b } \sum _ { i = 1 } ^ { b } b \mathbf { U } _ { i } \mathbb { E } _ { \xi } G _ { i } ( \mathbf { x } , \xi ) \in \sum _ { i = 1 } ^ { b } \mathbf { U } _ { i } \partial _ { i } f ( \mathbf { x } ) = \partial f ( \mathbf { x } ) } \end{array}$ and $\mathbb { E } _ { i , \xi } \Vert G ( \mathbf { x } , i , \xi ) \Vert _ { * } ^ { 2 } =$ $\begin{array} { r } { \frac { 1 } { b } \sum _ { i = 1 } ^ { b } b ^ { 2 } \mathbb { E } _ { \xi } \| \mathbf { U } _ { i } G _ { i } ( \mathbf { x } , \boldsymbol { \xi } ) \| _ { i , * } ^ { 2 } \leq b \sum _ { i = 1 } ^ { b } L _ { i } ^ { 2 } = b L ^ { 2 } . \mathrm { \bf { T } } } \end{array}$ hen + +$$ +\begin{array} { r l } & { \mathbf { x } ^ { ( t + 1 ) } = \mathbf { U } _ { i } \left[ \underset { \mathbf { u } \in \mathcal { X } _ { i } } { \arg \operatorname* { m i n } } \eta _ { t } \langle g , u \rangle + D _ { \phi _ { i } } ( \mathbf { u } | | \mathbf { x } _ { i } ^ { ( t ) } ) \right] } \\ & { \quad \quad \quad = \mathbf { U } _ { i } \mathbf { U } _ { i } ^ { T } \left[ \underset { \mathbf { u } \in \mathcal { X } } { \arg \operatorname* { m i n } } \eta _ { t } \langle \mathbf { U } _ { i } G _ { i } ( \mathbf { x } ^ { ( t ) } , \xi _ { t } ) , \mathbf { u } \rangle + \underset { i = 1 } { \overset { b } { \sum } } D _ { \phi _ { i } } ( \mathbf { u } _ { i } | | \mathbf { x } _ { i } ^ { ( t ) } ) \right] } \\ & { \quad \quad \quad = \underset { \mathbf { u } \in \mathcal { X } } { \arg \operatorname* { m i n } } \frac { \eta _ { t } } { b } \langle G ( \mathbf { x } , i , \xi _ { t } ) , \mathbf { u } \rangle + D _ { \phi } ( \mathbf { u } | | \mathbf { x } ^ { ( t ) } ) } \end{array} +$$ + +Thus Algorithm 2 is equivalent to Zhang & He (2018, Algorithm 1) with stochastic oracle $G ( \mathbf { x } , i , \xi )$ , step-size sequence $\{ \eta _ { t } / b \} _ { t = 1 } ^ { T }$ , and no regularizer. Note that $\phi$ is 1-strongly-convex w.r.t. $\| \cdot \|$ and $f$ is $\gamma$ -RWC w.r.t. $\phi$ , so in light of Claims 1 and 2 our setup satisfies Assumption 3.1 of Zhang & He (2018). The result then follows from Theorem 3.1 of the same. □ + +Corollary 1. Under Setting $^ { l }$ let ˆx be the output of Algorithm 2 with constant step-size $\eta _ { t } ~ =$ q 2b(f(1)−f∗)γL2T ∀ t ∈ [T ], where f (1) = f (x(1)). Then we have + +$$ +\mathbb { E } \Delta _ { \frac { 1 } { 2 \gamma } } \big ( \hat { \mathbf { x } } \big ) \leq 2 L \sqrt { \frac { 2 b \gamma \big ( f ^ { ( 1 ) } - f ^ { * } \big ) } { T } } +$$ + +where the expectation is w.r.t. $\xi _ { t }$ and the randomness of the algorithm. Equivalently, we can reach a point ˆx satisfying $\begin{array} { r } { \mathbb { E } \Delta _ { \frac { 1 } { 2 \gamma } } ( \hat { \mathbf { x } } ) \leq \varepsilon i n \frac { 8 \gamma b L ^ { 2 } ( f ^ { ( 1 ) } - f ^ { * } ) } { \varepsilon ^ { 2 } } } \end{array}$ stochastic oracle calls. + +B.3 A SINGLE-LEVEL ANALYSIS OF ENAS AND DARTS + +In this section we apply our analysis to understanding two existing NAS algorithms, ENAS (Pham et al., 2018) and DARTS (Liu et al., 2019). For simplicity, we assume objectives induced by architectures in the relaxed search space are $\gamma$ -smooth, which excludes components such as ReLU. However, such cases can be smoothed via Gaussian convolution, i.e. adding noise to every gradient; thus given the noisiness of SGD training we believe the following analysis is still informative (Kleinberg et al., 2018). + +ENAS continuously relaxes $\mathcal { A }$ via a neural controller that samples architectures $a \in { \mathcal { A } }$ , so $\Theta =$ $\mathbb { R } ^ { \mathcal { O } ( h ^ { 2 } ) }$ , where $h$ is the number of hidden units. The controller is trained with Monte Carlo gradients. On the other hand, first-order DARTS uses a mixture relaxation similar to the one in Section A but using a softmax instead of constraining parameters to the simplex. Thus $\Theta = \mathbb { R } ^ { | E | \times | O | }$ for $E$ the set of learnable edges and $O$ the set of possible operations. If we assume that both algorithms use SGD for the architecture parameters then to compare them we are interested in their respective values of $G _ { \theta }$ , which we will refer to as $G _ { \mathrm { E N A S } }$ and $G _ { \mathrm { D A R T S } }$ . Before proceeding, we note again that our theory holds only for the single-level objective and when using SGD as the architecture optimizer, whereas both algorithms specify the bilevel objective and Adam (Kingma & Ba, 2015), respectively. + +At a very high level, the Monte Carlo gradients used by ENAS are known to be high-variance, so $G _ { \mathrm { E N A S } }$ may be much larger than $G _ { \mathrm { D A R T S } }$ , yielding faster convergence for DARTS, which is reflected in practice (Liu et al., 2019). We can also do a simple low-level analysis under the assumption that all architecture gradients are bounded entry-wise, i.e. in $\ell _ { \infty }$ -norm, by some constant; then since the squared $\ell _ { 2 }$ -norm is bounded by the product of the dimension and the squared $\ell _ { \infty }$ -norm we have $G _ { \mathrm { E N A S } } ^ { 2 } = \mathcal { O } ( h ^ { 2 } )$ while $G _ { \mathrm { D A R T S } } ^ { 2 } = \mathcal { O } ( | \bar { E } | | { \cal O } | )$ . Since ENAS uses a hidden state size of $h = 1 0 0$ and the DARTS search space has $| E | = 1 4$ edges and $| O | = 7$ operations, this also points to DARTS needing fewer iterations to converge. + +# C EXPERIMENTAL DETAILS + +We provide additional detail on the experimental setup and hyperparameter settings used for each benchmark studied in Section 4. We also provide a more detailed discussion of how XNAS differs from GAEA, along with empirical results for XNAS on the NAS-Bench-201 benchmark. + +# C.1 DARTS SEARCH SPACE + +We consider the same search space as DARTS (Liu et al., 2019), which has become one of the standard search spaces for CNN cell search (Xie et al., 2019; Nayman et al., 2019; Chen et al., 2019; Noy et al., 2019; Liang et al., 2019). Following the evaluation procedure used in Liu et al. (2019) and Xu et al. (2020) , our evaluation of GAEA PC-DARTS consists of three stages: + +• Stage 1: In the search phase, we run GAEA PC-DARTS with 5 random seeds to reduce variance from different initialization of the shared-weights network.3 +• Stage 2: We evaluate the best architecture identified by each search run by training from scratch. +• Stage 3: We perform a more thorough evaluation of the best architecture from stage 2 by training with ten different random seed initializations. + +For completeness, we describe the convolutional neural network search space considered. A cell consists of 2 input nodes and 4 intermediate nodes for a total of 6 nodes. The nodes are ordered and subsequent nodes can receive the output of prior nodes as input so for a given node $k$ , there are $k - 1$ possible input edges to node $k$ . Therefore, there are a total of $2 + 3 + 4 + 5 = 1 4$ edges in the weight-sharing network. + +An architecture is defined by selecting 2 input edges per intermediate node and also selecting a single operation per edge from the following 8 operations: (1) $3 \times 3$ separable convolution, (2) $5 \times 5$ separable convolution, (3) $3 \times 3$ dilated convolution, (4) $5 \times 5$ dilated convolution, (5) max pooling, (6) average pooling, (7) identity (8) zero. We use the same search space to design a “normal” cell and a “reduction” cell; the normal cells have stride 1 operations that do not change the dimension of the input, while the reduction cells have stride 2 operations that half the length and width dimensions of the input. In the experiments, for both cell types, , after which the output of all intermediate nodes are concatenated to form the output of the cell. + +# C.1.1 STAGE 1: ARCHITECTURE SEARCH + +For stage 1, as is done by DARTS and PC-DARTS, we use GAEA PC-DARTS to update architecture parameters for a smaller shared-weights network in the search phase with 8 layers and 16 initial channels. All hyperparameters for training the weight-sharing network are the same as that used by PC-DARTS: + +train: scheduler: cosine lr_anneal_cycles: 1 smooth_cross_entropy: false batch_size: 256 learning_rate: 0.1 learning_rate_min: 0.0 momentum: 0.9 weight_decay: 0.0003 init_channels: 16 layers: 8 autoaugment: false cutout: false auxiliary: false drop_path_prob: 0 grad_clip: 5 + +For GAEA PC-DARTS, we initialize the architecture parameters with equal weight across all options (equal weight across all operations per edge and equal weight across all input edges per node). Then, we train the shared-weights network for 10 epochs without performing any architecture updates to warmup the weights. Then, we use a learning rate of 0.1 for the exponentiated gradient update for GAEA PC-DARTS. + +# C.1.2 STAGE 2 AND 3: ARCHITECTURE EVALUATION + +For stages 2 and 3, we train each architecture for 600 epochs with the same hyperparameter settings as PC-DARTS: + +train: scheduler: cosine lr_anneal_cycles: 1 smooth_cross_entropy: false batch_size: 128 learning_rate: 0.025 learning_rate_min: 0.0 momentum: 0.9 weight_decay: 0.0003 init_channels: 36 layers: 20 autoaugment: false cutout: true cutout_length: 16 auxiliary: true auxiliary_weight: 0.4 drop_path_prob: 0.3 grad_clip: 5 + +# C.1.3 BROAD REPRODUCIBILITY + +Our ‘broad reproducibility’ results in Table 3 show the final stage 3 evaluation performance of GAEA PC-DARTS for 2 additional sets of random seeds from stage 1 search. The performance of GAEA PC-DARTS for one set is similar to that reported in Table 1, while the other is on par with the performance reported for PC-DARTS in Xu et al. (2020). We do observe non-negligible variance in the performance of the architecture found by different random seed initializations of the shared-weights network, necessitating running multiple searches before selecting an architecture. We also found that it was possible to identify and eliminate poor performing architectures in just 20 epochs of training during stage 2 intermediate evaluation, thereby reducing the total training cost by over $7 5 \%$ (we only trained 3 out of 10 architectures for the entire 600 epochs). + +Stage 3 Evaluation +Table 3: GAEA PC-DARTS Stage 3 Evaluation for 3 sets of random seeds. + +
Set 1 (Reported)Set2Set3
2.50 ± 0.072.50±0.092.60±0.09
+ +We depict the top architectures found by GAEA PC-DARTS for CIFAR-10 and ImageNet in Figure 3 and detailed results in Tables 4 and 5. + +
ArchitectureTest Error AverageParams (M)Search Cost (GPU Days)Comparable Search SpaceSearch Method
Shake-Shake(DeVries&Taylor,2017)Best N/A2.5626.2--manual
PyramidNet (Yamada et al.,2018)2.31N/A261-manual
NASNet-A* (Zoph et al.,2018)N/A2.653.32000NRL
AmoebaNet-B*(Real et al.,2019)N/A2.55± 0.052.83150Nevolution
ProxylessNAS* (Cai et al.,2019)2.08N/A5.74Ngradient
ENAS (Pham et al.,2018)2.89N/A4.60.5YRL
Random search Wst (Li& Talwalkar,2019)2.712.85 ±0.083.80.7Yrandom
ASNG-NAS (Akimoto et al.,2019)N/A2.83 ± 0.143.90.1Ygradient
SNAS (Xie et al.,2019)N/A2.85 ±0.022.81.5Ygradient
DARTS (lst-order)†(Liu et al.,2019)N/A3.00 ±0.143.30.4Ygradient
DARTS (2nd-order)†(Liu et al.,2019)N/A2.76±0.093.31Ygradient
PDARTS# (Chen et al., 2019)2.50N/A3.40.3Ygradient
PC-DARTS† (Xu et al.,2020)N/A2.57 ±0.073.60.1Ygradient
GAEA PC-DARTS† (Ours)2.392.50±0.063.70.1Ygradient
+ +We show results for networks with a comparable number of parameters. For fair comparison to other work, we show the search cost for training the shared-weights network with a single initialization. ‡ Search space and backbone architecture (PyramidNet) differ from the DARTS setting. PDARTS results not reported for multiple seeds. Additionally, PDARTS uses deeper weight-sharing networks during search, on which PC-DARTS has also been shown to improve performance (Xu et al., 2020), so we GAEA PC-DARTS to further improve if modified similarly. + +Table 4: DARTS (CIFAR-10): Comparison with manually designed networks and those found by SOTA NAS methods, mainly on the DARTS search space (Liu et al., 2019). Results grouped by the type of search method: manually designed, full-evaluation NAS, and weight-sharing NAS. All test errors are for models trained with auxiliary towers and cutout (parameter counts exclude auxiliary weights). Test errors we report are averaged over 10 seeds. “-” indicates that the field does not apply while “N/A” indicates unknown. Note that search cost is hardware-dependent; our results used Tesla V100 GPUs. + +
Test ErrorParams (M)Search Cost (GPU Days)Search Method
ArchitectureSourcetop-1top-5
MobileNet(Howard et al.,2017)29.410.54.2-manual
ShuffleNet V2 2x(Ma et al.,2018)25.1N/A~5-manual
NASNet-A*(Zoph et al.,2018)26.08.45.31800RL
AmoebaNet-C*(Real et al.,2019)24.37.66.43150evolution
DARTS(Liu et al.,2019)26.78.74.74.0gradient
SNAS(Xie et al.,2019)27.39.24.31.5gradient
ProxylessNAS‡(Cai et al.,2019)24.97.57.18.3gradient
PDARTS#(Chen et al., 2019)24.47.44.90.3gradient
PC-DARTS (CIFAR-10)+(Xu et al.,2020)25.17.85.30.1gradient
PC-DARTS (ImageNet)t(Xu et al.,2020)24.27.35.33.8gradient
GAEA PC-DARTS (CIFAR-10)†Ours24.37.35.30.1gradient
GAEA PC-DARTS (ImageNet)†Ours24.07.35.63.8gradient
+ +∗,† ,‡ ,# See notes in Table 4. + +Table 5: DARTS (ImageNet): Comparison with manually designed networks and those found by SOTA NAS methods, mainly on the DARTS search space (Liu et al., 2019). Results are grouped by the type of search method: manually designed, full-evaluation NAS, and weight-sharing NAS. All test errors are for models trained with auxiliary towers and cutout but no other modifications, e.g. label smoothing (Muller et al., 2019), AutoAugment (Cubuk ¨ et al., 2019), Swish (Ramachandran et al., 2017), squeeze and excite modules (Hu et al., 2018), etc. “-” indicates that the field does not apply while “N/A” indicates unknown. Note that search cost is hardware-dependent; our results used Tesla V100 GPUs. + +![](images/63ae622cf29eb0f9ab780252106da6d0cf794f22106cbaf1aa1ef10d1b7f747b.jpg) +(a) CIFAR-10: Normal Cell + +![](images/3569ea92af070f0ff5181fc6a9eb008f2cb5fae43d928684d613531cc3d4a3c7.jpg) +(b) CIFAR-10: Reduction Cell + +![](images/148a34e424faf3bd2389b857129d4478f7e86f99095c5089b79873d477d6b55d.jpg) +(c) ImageNet: Normal Cell + +![](images/97ad583bc5680581ac1635958f01f41737c5cabe214a5c5fd5f49240ac4f5fba.jpg) +Figure 3: The best normal and reduction cells found by GAEA PC-DARTS on CIFAR-10 (top) and ImageNet (bottom). + +![](images/a9695b181215ef2198739bbcc3c51586579237fa3521e3b6eb9418b8493325f8.jpg) +Figure 4: NAS-Bench-201: Learning Curves. Evolution over search phase epochs of the best architecture according to the NAS method. DARTS (first-order) converges to nearly all skip connections while GAEA is able to suppress overfitting to the mixture relaxation by encouraging sparsity in operation weights. + +# C.2 NAS-BENCH-1SHOT1 + +The NAS-Bench-1Shot1 benchmark (Zela et al., 2020a) contains 3 different search spaces that are subsets of the NAS-Bench-101 search space. The search spaces differ in the number of nodes and the number of input edges selected per node. We refer the reader to (Zela et al., 2020a) for details about each individual search space. + +Of the NAS methods evaluated in Zela et al. (2020a), PC-DARTS had the most robust performance across the three search spaces and converged to the best architecture in search spaces 1 and 3. GDAS, a probabilistic gradient NAS method, achieved the best performance on search space 2. Hence, we focused on applying a geometry-aware approach to PC-DARTS. We implemented GAEA PCDARTS within the repository provided by the authors of Zela et al. (2020a) available at https: //github.com/automl/nasbench-1shot1. We used the same hyperparameter settings for training the weight-sharing network as that used by Zela et al. (2020a) for PC-DARTS. Similar to the previous benchmark, we initialize architecture parameters to allocate equal weight to all options. For the architecture updates, the only hyperparameter for GAEA PC-DARTS is the learning rate for exponentiated gradient, which we set to 0.1. + +As mentioned in Section 4, the search spaces considered in this benchmark differ in that operations are applied after aggregating all edge inputs to a node instead of per edge input as in the DARTS and NAS-Bench-201 search spaces. This structure inherently limits the size of the weight-sharing network to scale with the number of nodes instead of the number of edges $( \mathcal { O } ( | \mathrm { n o d e s } | ^ { \bar { 2 } } ) ;$ , thereby limiting the degree of overparameterization. Understanding the impact of overparameterization on the performance of weight-sharing NAS methods is a direction for future study. + +# C.3 NAS-BENCH-201 + +The NAS-Bench-201 benchmark (Dong & Yang, 2020) evaluates a single search space across 3 datasets: CIFAR-10, CIFAR-100, and a miniature version of ImageNet (ImageNet-16-120). ImageNet-16-120 is a downsampled version of ImageNet with $1 6 \times 1 6$ images and 120 classes for a total of $1 5 1 . 7 \mathrm { k }$ training images, 3k validation images, and 3k test images. The authors of Dong & Yang (2020) evaluated the architecture search performance of multiple weight-sharing methods and traditional hyperparameter optimization methods on all three datasets. According to the results from Dong & Yang (2020), GDAS outperformed other weight-sharing methods by a large margin. Hence, we first evaluated the performance of GAEA GDAS on each of the three datasets. Our implementation of GAEA GDAS uses an architecture learning rate of 0.1, which matches the learning rate used for GAEA approaches in the previous two benchmarks. Additionally, we run GAEA GDAS for 150 epochs instead of 250 epochs used for GDAS in the original benchmarked results; this is why the search cost is lower for GAEA GDAS. All other hyperparameter settings are the same. Our results for GAEA GDAS is comparable to the reported results for GDAS on CIFAR-10 and CIFAR-100 but slightly lower on ImageNet-16-120. Compared to our reproduced results for GDAS, GAEA GDAS outperforms GDAS on CIFAR-100 and matches it on CIFAR-10 and ImageNet-16-120. + +Next, to see if we can use GAEA to further improve the performance of weight-sharing methods, we evaluated GAEA DARTS (first order) applied to both the single-level (ERM) and bi-level optimization problems. Again, we used a learning rate of 0.1 and trained GAEA DARTS for 25 epochs on each dataset. The one additional modification we made was to exclude the zero operation, which limits GAEA DARTS to a subset of the search space. To isolate the impact of this modification, we also evaluated first-order DARTS with this modification. Similar to (Dong & Yang, 2020), we observe DARTS with this modification to also converge to architectures with nearly all skip connections, resulting in similar performance as that reported in Dong & Yang (2020). We present the learning curves of the oracle architecture recommended by DARTS and GAEA DARTS (when excluding zero operation) over the training horizon for 4 different runs in Figure 4. For GAEA GDAS and GAEA DARTS, we train the weight-sharing network with the following hyperparameters: + +train: scheduler: cosine lr_anneal_cycles: 1 smooth_cross_entropy: false batch_size: 64 learning_rate: 0.025 learning_rate_min: 0.001 momentum: 0.9 weight_decay: 0.0003 init_channels: 16 layers: 5 autoaugment: false cutout: false auxiliary: false auxiliary_weight: 0.4 drop_path_prob: 0 grad_clip: 5 + +Surprisingly, we observe single-level optimization to yield better performance than solving the bilevel problem with GAEA DARTS on this search space. In fact, the performance of GAEA DARTS (ERM) not only exceeds that of GDAS, but also outperforms traditional hyperparameter optimization approaches on all three datasets, nearly reaching the optimal accuracy on all three datasets. In contrast, GAEA DARTS (bi-level) outperforms GDAS on CIFAR-100 and ImageNet-16-120 but underperforms slightly on CIFAR-10. The single-level results on this benchmark provides concrete support for our convergence analysis, which only applies to the ERM problem. As noted in Section 4, the search space considered in this benchmark differs from the prior two in that there is no subsequent edge pruning. Additionally, the search space is fairly small with only 3 nodes for which architecture decisions must be made. The success of GAEA DARTS (ERM) on this benchmark indicate the need for a better understanding of when single-level optimization should be used in favor of the default bi-level optimization problem and how the search space impacts the decision. + +![](images/80d0caa75d9919dc7f9a33b41a5523e5c17f1bbf9e12492fe2d2427fdc69679c.jpg) +Figure 5: NAS-Bench-201: XNAS Learning Curves. Evolution over search phase epochs of the best architecture according 4 runs of XNAS. XNAS exhibits the same behavior as DARTS and converges to nearly all skip connections. + +# C.4 COMPARISON TO XNAS + +As discussed in Section 1, XNAS is similar to GAEA in that it uses an exponentiated gradient update but is motivated from a regret minimization perspective. Nayman et al. (2019) provides regret bounds for XNAS relative to the observed sequence of validation losses, however, this is not equivalent to the regret relative to the best architecture in the search space, which would have generated a different sequence of validation losses. + +XNAS also differs in its implementation in two ways: (1) a wipeout routine is used to zero out operations that cannot recover to exceed the current best operation within the remaining number of iterations and (2) architecture gradient clipping is applied per data point before aggregating to form the update. These differences are motivated from the regret analysis and meaningfully increase the complexity of the algorithm. Unfortunately, the authors do not provide the code for architecture search in their code release at https://github.com/NivNayman/XNAS. Nonetheless, we implemented XNAS for the NAS-Bench-201 search space to provide a point of comparison to GAEA. + +Our results shown in Figure 5 demonstrate that XNAS exhibits much of the same behavior as DARTS in that the operations all converge to skip connections. We hypothesize that this is due to the gradient clipping, which obscures the signal kept by GAEA in favor of convolutional operations. \ No newline at end of file diff --git a/parse/train/MuSYkd1hxRP/MuSYkd1hxRP_content_list.json b/parse/train/MuSYkd1hxRP/MuSYkd1hxRP_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..0314c640d52dcbbb6f08aacd3348bd7899b7cb64 --- /dev/null +++ b/parse/train/MuSYkd1hxRP/MuSYkd1hxRP_content_list.json @@ -0,0 +1,2646 @@ +[ + { + "type": "text", + "text": "GEOMETRY-AWARE GRADIENT ALGORITHMS FOR NEURAL ARCHITECTURE SEARCH ", + "text_level": 1, + "bbox": [ + 176, + 99, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Liam $\\mathbf { L } \\mathbf { i } ^ { * 1 }$ , Mikhail Khodak∗2, Maria-Florina Balcan2, and Ameet Talwalkar1,2 1 Determined AI, 2 Carnegie Mellon University, ∗ denotes equal contribution me@liamcli.com,khodak@cmu.edu,ninamf@cs.cmu.edu,talwalkar@cmu.edu ", + "bbox": [ + 183, + 169, + 823, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 452, + 242, + 544, + 257 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent state-of-the-art methods for neural architecture search (NAS) exploit gradient-based optimization by relaxing the problem into continuous optimization over architectures and shared-weights, a noisy process that remains poorly understood. We argue for the study of single-level empirical risk minimization to understand NAS with weight-sharing, reducing the design of NAS methods to devising optimizers and regularizers that can quickly obtain high-quality solutions to this problem. Invoking the theory of mirror descent, we present a geometry-aware framework that exploits the underlying structure of this optimization to return sparse architectural parameters, leading to simple yet novel algorithms that enjoy fast convergence guarantees and achieve state-of-the-art accuracy on the latest NAS benchmarks in computer vision. Notably, we exceed the best published results for both CIFAR and ImageNet on both the DARTS search space and NAS-Bench201; on the latter we achieve near-oracle-optimal performance on CIFAR-10 and CIFAR-100. Together, our theory and experiments demonstrate a principled way to co-design optimizers and continuous relaxations of discrete NAS search spaces. ", + "bbox": [ + 233, + 268, + 766, + 474 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 501, + 336, + 517 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural architecture search has become an important tool for automating machine learning (ML) but can require hundreds of thousands of GPU-hours to train. Recently, weight-sharing approaches have achieved state-of-the-art performance while drastically reducing the computational cost of NAS to just that of training a single shared-weights network (Pham et al., 2018; Liu et al., 2019). Methods such as DARTS (Liu et al., 2019), GDAS (Dong & Yang, 2019), and many others (Pham et al., 2018; Zheng et al., 2019; Yang et al., 2020; Xie et al., 2019; Liu et al., 2018; Laube & Zell, 2019; Cai et al., 2019; Akimoto et al., 2019; Xu et al., 2020) combine weight-sharing with a continuous relaxation of the discrete search space to allow cheap gradient updates, enabling the use of popular optimizers. However, despite some empirical success, weight-sharing remains poorly understood and has received criticism due to (1) rank-disorder (Yu et al., 2020; Zela et al., 2020b; Zhang et al., 2020; Pourchot et al., 2020), where the shared-weights performance is a poor surrogate of standalone performance, and (2) poor results on recent benchmarks (Dong & Yang, 2020; Zela et al., 2020a). ", + "bbox": [ + 173, + 529, + 825, + 694 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Motivated by the challenge of developing simple and efficient methods that achieve state-of-the-art performance, we study how to best handle the goals and optimization objectives of NAS. We start by observing that weight-sharing subsumes architecture hyperparameters as another set of learned parameters of the shared-weights network, in effect extending the class of functions being learned. This suggests that a reasonable approach towards obtaining high-quality NAS solutions is to study how to regularize and optimize the empirical risk over this extended class. While many regularization approaches have been implicitly proposed in recent NAS efforts, we focus instead on the question of optimizing architecture parameters, which may not be amenable to standard procedures such as SGD that work well for standard neural network weights. In particular, to better-satisfy desirable properties such as generalization and sparsity of architectural decisions, we propose to constrain architecture parameters to the simplex and update them using exponentiated gradient, which has favorable convergence properties due to the underlying problem structure. Theoretically, we draw upon the mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) to give convergence guarantees when using any of a broad class of such geometry-aware gradient methods to optimize the weight-sharing objective; empirically, we show that our solution leads to strong improvements on several NAS benchmarks. We summarize these contributions below: ", + "bbox": [ + 174, + 702, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1. We argue for studying NAS with weight-sharing as a single-level objective over a structured function class in which architectural decisions are treated as learned parameters rather than hyperparameters. Our setup clarifies recent concerns about rank disorder and makes clear that proper regularization and optimization of this objective is critical to obtaining high-quality solutions. ", + "bbox": [ + 176, + 103, + 825, + 160 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2. Focusing on optimization, we propose to improve existing NAS algorithms by re-parameterizing architecture parameters over the simplex and updating them using exponentiated gradient, a variant of mirror descent that converges quickly over this domain and enjoys favorable sparsity properties. This simple modification—which we call the Geometry-Aware Exponentiated Algorithm (GAEA)—is easily applicable to numerous methods, including first-order DARTS Liu et al. (2019), GDAS Dong & Yang (2019), and PC-DARTS (Xu et al., 2020). ", + "bbox": [ + 174, + 162, + 825, + 246 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3. To show correctness and efficiency of our scheme, we prove polynomial-time stationary-point convergence of block-stochastic mirror descent—a family of geometry-aware gradient algorithms that includes GAEA—over a continuous relaxation of the single-level NAS objective. To the best of our knowledge these are the first finite-time convergence guarantees for gradient-based NAS. ", + "bbox": [ + 174, + 247, + 825, + 303 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "4. We demonstrate that GAEA improves upon state-of-the-art methods on three of the latest NAS benchmarks for computer vision. Specifically, we beat the current best results on NAS-Bench-201 (Dong & Yang, 2020) by $0 . 1 8 \\%$ on CIFAR-10, $1 . 5 9 \\%$ on CIFAR-100, and $0 . 8 2 \\%$ on ImageNet16-120; we also outperform the state-of-the-art on the DARTS search space Liu et al. (2019), for both CIFAR-10 and ImageNet, and match it on NAS-Bench-1Shot1 (Zela et al., 2020a).1 ", + "bbox": [ + 174, + 306, + 825, + 376 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Related Work. Most optimization analyses of NAS show monotonic improvement (Akimoto et al., 2019), asymptotic guarantees (Yao et al., 2020), or bounds on auxiliary quantities disconnected from any objective (Noy et al., 2019; Nayman et al., 2019; Carlucci et al., 2019). In contrast, we prove polynomial-time stationary-point convergence on a single-level objective for weight-sharing NAS, so far only studied empirically (Xie et al., 2019; Li et al., 2019). Our results draw upon the mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) and extend recent nonconvex convergence results Zhang & He (2018) to handle alternating descent. While there exist related results (Dang & Lan, 2015) the associated guarantees do not hold for the algorithms we propose. Finally, we note that a variant of GAEA that modifies first-order DARTS is related to XNAS (Nayman et al., 2019), whose update also involves exponentiated gradient; however, GAEA is simpler and easier to implement.2 Furthermore, the regret guarantees for XNAS do not relate to any meaningful performance measure for NAS such as speed or accuracy, whereas we guarantee convergence on the ERM objective. ", + "bbox": [ + 173, + 388, + 825, + 568 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 THE WEIGHT-SHARING OPTIMIZATION PROBLEM ", + "text_level": 1, + "bbox": [ + 174, + 582, + 619, + 598 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In supervised ML we have a dataset $T$ of labeled pairs $( x , y )$ drawn from a distribution $\\mathcal { D }$ over input/output spaces $X$ and $Y$ . The goal is to use $T$ to search a function class $H$ for $h _ { \\mathbf { w } } : X \\mapsto Y$ parameterized by $\\mathbf { w } \\in \\mathbb { R } ^ { d }$ that has low expected test loss $\\ell ( h _ { \\mathbf { w } } ( x ) , y )$ when using $x$ to predict the associated $y$ on unseen samples drawn from $D$ , as measured by some loss $\\ell : Y \\times Y \\mapsto [ 0 , \\infty )$ . A common way to do so is by approximate (regularized) empirical risk minimization (ERM), i.e. finding $\\mathbf { w } \\in \\mathbb { R } ^ { \\breve { d } }$ with the smallest average loss over $T$ , via some iterative method $\\mathtt { A l g }$ , e.g. SGD. ", + "bbox": [ + 174, + 609, + 826, + 695 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 THE BENEFITS AND CRITICISMS OF WEIGHT-SHARING FOR NAS ", + "text_level": 1, + "bbox": [ + 176, + 708, + 666, + 723 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "NAS is often viewed as hyperparameter optimization on top of $\\mathtt { A l g }$ , with each architecture $a \\in { \\mathcal { A } }$ corresponding to a function class $H _ { a } = \\{ h _ { \\mathbf { w } , a } : X \\mapsto Y , \\mathbf { w } \\in \\mathbb { R } ^ { d } \\}$ to be selected by using validation data $V \\subset X \\times Y$ to evaluate the predictor obtained by fixing $a$ and doing approximate ERM over $T$ : ", + "bbox": [ + 174, + 731, + 825, + 775 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/5ddbb5b436538d642f83308b412396a1deda8de0dc282e5f4c50d6ddfcf758dd.jpg", + "text": "$$\n\\operatorname* { m i n } _ { a \\in A } \\quad \\sum _ { ( x , y ) \\in V } \\ell ( h _ { \\mathbf { w } _ { a } , a } ( x ) , y ) \\qquad \\mathrm { s . t . } \\qquad \\mathbf { w } _ { a } = \\mathrm { A l } \\boldsymbol { \\mathrm { g } } ( T , a )\n$$", + "text_format": "latex", + "bbox": [ + 300, + 781, + 696, + 818 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Since training individual sets of weights for any sizeable number of architectures is prohibitive, weight-sharing methods instead use a single set of shared weights to obtain validation signal about many architectures at once. In its most simple form, RS-WS (Li & Talwalkar, 2019), these weights are trained to minimize a non-adaptive objective, $\\begin{array} { r } { \\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { d } } \\mathbb { E } _ { a } \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } _ { a } , a } ( x ) , y ) } \\end{array}$ , where the expectation is over a fixed distribution over architectures $\\mathcal { A }$ . The final architecture $a$ is then chosen to maximize the outer (validation) objective in (1) subject to ${ \\bf w } _ { a } = { \\bf w }$ . More frequently used is a bilevel objective over some continuous relaxation $\\Theta$ of the architecture space $\\mathcal { A }$ , after which a valid architecture is obtained via a discretization step $\\mathsf { M a p } : \\Theta \\mapsto \\mathcal { A }$ (Pham et al., 2018; Liu et al., 2019): ", + "bbox": [ + 176, + 818, + 828, + 859 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 175 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3cbedcd6e0d904cc850947134cd8a51bfa225464da3df144f93276cee63cb57a.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\quad \\sum _ { ( x , y ) \\in V } \\ell ( h _ { \\mathbf { w } , \\theta } ( x ) , y ) \\qquad \\mathrm { s . t . } \\qquad \\mathbf { w } \\in \\underset { \\mathbf { u } \\in \\mathbb { R } ^ { d } } { \\arg \\operatorname* { m i n } } \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { u } , \\theta } ( x ) , y )\n$$", + "text_format": "latex", + "bbox": [ + 246, + 176, + 748, + 212 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This objective is not significantly different from (2), since $\\mathbb { A } \\bot \\mathbb { g } ( T , a )$ approximately minimizes the empirical risk w.r.t. $T$ ; the difference is replacing discrete architectures with relaxed architecture parameters $\\theta \\in \\Theta$ , w.r.t. which we can take derivatives of the outer objective. This allows (2) to be approximated via alternating gradient updates w.r.t. w and $\\theta$ . Relaxations can be stochastic, so that $\\operatorname { M a p } ( \\theta )$ is a sample from a $\\theta$ -parameterized distribution (Pham et al., 2018; Dong & Yang, 2019), or a mixture, in which case $\\operatorname { M a p } ( \\theta )$ selects architectural decisions with the highest weight in a convex combination given by $\\theta$ (Liu et al., 2019). We overview this in more detail in Appendix A. ", + "bbox": [ + 173, + 213, + 825, + 310 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While weight-sharing significantly shortens search (Pham et al., 2018), it draws two main criticisms: ", + "bbox": [ + 174, + 316, + 823, + 332 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Rank disorder: this describes when the rank of an architecture $a$ according to the validation risk evaluated with fixed shared weights w is poorly correlated with the one using “standalone” weights $\\mathbf { w } _ { a } = \\mathbb { A } \\mathbb { 1 } \\mathtt { g } ( T , a )$ . This causes suboptimal architectures to be selected after shared weights search (Yu et al., 2020; Zela et al., 2020b; Zhang et al., 2020; Pourchot et al., 2020). ", + "bbox": [ + 174, + 335, + 821, + 390 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Poor performance: weight-sharing can converge to degenerate architectures (Zela et al., 2020a) and is outperformed by regular hyperparameter tuning on NAS-Bench-201 (Dong & Yang, 2020). ", + "bbox": [ + 178, + 391, + 825, + 417 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 SINGLE-LEVEL NAS AS A BASELINE OBJECT OF STUDY", + "text_level": 1, + "bbox": [ + 173, + 430, + 606, + 445 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Why are we able to apply weight-sharing to NAS? The key is that, unlike regular hyperparameters such as step-size, architectural hyperparameters directly affect the loss function without requiring a dependent change in the model weights w. Thus we can distinguish architectures without retraining simply by changing architectural decisions. Besides enabling weight-sharing, this point reveals that the goal of NAS is perhaps better viewed as a regular learning problem over an extended class ${ \\cal H } _ { \\cal A } = \\bigcup _ { a \\in { \\cal A } } { \\cal H } _ { a } = \\{ h _ { \\mathbf { w } , a } : \\hat { X } \\mapsto Y , \\mathbf { w } \\in \\mathbb { R } ^ { d } , a \\in \\mathcal { A } \\}$ that subsumes the architectural decisions as parameters of a larger model class, an unrelaxed “supernet.” The natural approach to solving this is by approximate empirical risk minimization, e.g. by approximating continuous objective below on the right using a gradient algorithm and passing the output $\\theta$ through Map to obtain a valid architecture: ", + "bbox": [ + 173, + 452, + 825, + 579 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d7bef258303cc1c0526ed0926466c7821a7f10fea5c9622dc4dcb756f19eb01d.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { d } , a \\in \\mathcal { A } } \\quad \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , a } ( x ) , y ) \\qquad \\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { d } , \\theta \\in \\Theta } \\quad \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , \\theta } ( x ) , y )\n$$", + "text_format": "latex", + "bbox": [ + 240, + 579, + 758, + 614 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Several works have optimized this single-level objective as an alternative to bilevel (2) (Xie et al., 2019; Li et al., 2019). We argue for its use as the baseline object of study in NAS for three reasons: ", + "bbox": [ + 173, + 635, + 825, + 662 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. As discussed above, it is the natural first approach to solving the statistical objective of NAS: finding a good predictor $h _ { \\mathbf { w } , a } \\in H _ { \\cal A }$ in the extended function class over architectures and weights. \n2. The common alternating gradient approach to the bilevel problem (2) is in practice very similar to alternating block approaches to ERM (3); as we will see, there are established ways of analyzing such methods for the latter objective, while for the former convergence is known only under very strong assumptions such as uniqueness of the inner minimum (Franceschi et al., 2018). \n3. While less frequently used in practice than bilevel, single-level optimization can be very effective: we use it to achieve new state-of-the-art results on NAS-Bench-201 (Dong & Yang, 2020). ", + "bbox": [ + 173, + 666, + 826, + 775 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Understanding NAS as single-level optimization—the usual deep learning setting—makes weightsharing a natural, not surprising, approach. Furthermore, for methods—both single-level and bilevel— that adapt architecture parameters during search, it suggests that we need not worry about rank disorder as long as we can use optimization to find a single feasible point that generalizes well; we explicitly do not need a ranking. Non-adaptive methods such as RS-WS still do require rank correlation to select good architectures after search, but they are explicitly not changing $\\theta$ and so have no variant solving (3). The single-level formulation thus reduces search method design to well-studied questions of how to best regularize and optimize ERM. While there are many techniques for regularizing weight-sharing—including partial channels (Xu et al., 2020) and validation Hessian penalization (Zela et al., 2020a)—we focus on the second question of optimization. ", + "bbox": [ + 173, + 785, + 826, + 925 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 GEOMETRY-AWARE GRADIENT ALGORITHMS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 584, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We seek to minimize the (possibly regularized) empirical risk $\\begin{array} { r } { f ( \\mathbf { w } , \\theta ) = \\frac { 1 } { | T | } \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , \\theta } ( x ) , y ) } \\end{array}$ over shared-weights $\\mathbf { w } \\in \\mathbb { R } ^ { d }$ and architecture parameters $\\theta \\in \\Theta$ . Assuming we have noisy gradients of $f$ w.r.t. w or $\\theta$ at any point $( \\mathbf { w } , \\theta ) \\in \\mathbb { R } ^ { d } \\times \\Theta .$ —i.e. $\\tilde { \\nabla } _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta )$ or $\\tilde { \\nabla } _ { \\boldsymbol { \\theta } } f ( \\mathbf { w } , \\boldsymbol { \\theta } )$ satisfying $\\mathbb { E } \\tilde { \\nabla } _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta ) = \\nabla _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta )$ or $\\mathbb { E } \\tilde { \\nabla } _ { \\theta } f ( \\mathbf { w } , \\theta ) = \\nabla _ { \\theta } f ( \\mathbf { w } , \\theta )$ , respectively—our goal is a point where $f$ , or at least its gradient, is small, while taking as few gradients as possible. Our main complication is that architecture parameters lie in a constrained, non-Euclidean domain $\\Theta$ . Most search spaces $\\mathcal { A }$ are product sets of categorical decisions—which operation $o \\in O$ to use at edge $e \\in E$ —so the natural relaxation is a product of $| E | \\mid O |$ -simplices. However, NAS methods often re-parameterize $\\Theta$ to be unconstrained using a softmax and then SGD or Adam (Kingma & Ba, 2015). Is there a better parameterization-algorithm co-design? We consider a geometry-aware approach that uses mirror descent to design NAS methods with better properties depending on the domain; a key desirable property is to return sparse architectural parameters to reduce loss from post-search discretization. ", + "bbox": [ + 173, + 127, + 825, + 304 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 BACKGROUND ON MIRROR DESCENT ", + "text_level": 1, + "bbox": [ + 176, + 313, + 473, + 327 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Mirror descent has many formulations (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003; ShalevShwartz, 2011); the proximal starts by noting that, in the unconstrained case, an SGD update at a point $\\theta \\in \\Theta = \\mathbb { R } ^ { k }$ given gradient estimate $\\tilde { \\nabla } \\bar { f } ( \\theta )$ with step-size $\\eta > 0$ is equivalent to ", + "bbox": [ + 174, + 335, + 825, + 380 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6767d31dd40b41e8ed16ad5ff5035e47dd14f49397f146195f569d72defaeb36.jpg", + "text": "$$\n\\theta - \\eta \\tilde { \\nabla } f ( \\theta ) \\quad = \\quad \\begin{array} { c c c c } { \\underset { \\mathbf { u } \\in \\mathbb { R } ^ { k } } { \\arg \\operatorname* { m i n } } } & { \\eta \\tilde { \\nabla } f ( \\theta ) \\cdot \\mathbf { u } } & { + } & { \\frac { 1 } { 2 } \\| \\mathbf { u } - \\theta \\| _ { 2 } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 380, + 705, + 411 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here the first term aligns the output with the gradient while the second (proximal) term regularizes for closeness to the previous point as measured by the Euclidean distance. While the SGD update has been found to work well for unconstrained high-dimensional optimization, e.g. deep nets, this choice of proximal regularization may be sub-optimal over a constrained space with sparse solutions. The canonical such setting is optimization over the unit simplex, i.e. when $\\Theta = \\{ \\theta \\in [ 0 , 1 ] ^ { k } : \\| \\theta \\| _ { 1 } = 1 \\}$ . Replacing the $\\ell _ { 2 }$ -regularizer in Equation 4 by the relative entropy $\\mathbf { u } \\cdot ( \\log \\mathbf { u } - \\log \\theta )$ , i.e. the KL-divergence, yields the exponentiated gradient (EG) update ( $\\odot$ denotes element-wise product): ", + "bbox": [ + 173, + 410, + 825, + 507 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/27dceb9822e865a594967885a988cbc1827ab302e85d87780a2f25d39c20c0b6.jpg", + "text": "$$\n\\begin{array} { r l r l r } { \\theta \\odot \\dot { \\exp ( - \\eta \\tilde { \\nabla } f ( \\theta ) ) } } & { { } \\propto } & { \\underset { \\mathbf { u } \\in \\Theta } { \\operatorname { a r g m i n } } } & { \\eta \\tilde { \\nabla } f ( \\theta ) \\cdot \\mathbf { u } } & { { } + } & { \\mathbf { u } \\cdot ( \\log \\mathbf { u } - \\log \\theta ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 506, + 754, + 532 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that the full EG update is obtained by $\\ell _ { 1 }$ -normalizing the l.h.s. It is well-known that EG over the $k$ -dimensional simplex requires only $\\mathcal { O } ( \\log k ) / \\varepsilon ^ { 2 }$ iterations to achieve a function value $\\varepsilon$ -away from optimal (Beck & Teboulle, 2003, Theorem 5.1), compared to the ${ \\cal O } ( k / \\varepsilon ^ { 2 } )$ guarantee of gradient descent. This nearly dimension-independent iteration complexity is achieved by choosing a regularizer—the KL divergence—well-suited to the underlying geometry—the simplex. More generally, mirror descent is specified by a distance-generating function (DGF) $\\phi$ that is stronglyconvex w.r.t. some norm. $\\phi$ induces a Bregman divergence $D _ { \\phi } ( { \\mathbf { u } } | | { \\mathbf { v } } ) = \\phi ( { \\mathbf { u } } ) - \\phi ( { \\mathbf { v } } ) - \\nabla \\phi ( { \\mathbf { v } } ) \\cdot ( { \\mathbf { u } } - { \\mathbf { v } } )$ (Bregman, 1967), a notion of distance on $\\Theta$ that acts as a regularizer in the mirror descent update: ", + "bbox": [ + 173, + 532, + 825, + 643 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c3b80053f1483c16c4567d44f6c48a49e1c82b3c5cf48203af0ccc6aacafcc47.jpg", + "text": "$$\n\\begin{array} { r l r } { \\underset { \\mathbf { u } \\in \\Theta } { \\mathrm { a r g m i n } } } & { { } \\eta \\tilde { \\nabla } f ( \\boldsymbol { \\theta } ) \\cdot \\mathbf { u } } & { + \\quad D _ { \\boldsymbol { \\phi } } ( \\mathbf { u } \\vert \\vert \\boldsymbol { \\theta } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 642, + 632, + 667 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For example, to recover SGD (4) we set $\\begin{array} { r } { \\phi ( \\mathbf { u } ) = \\frac 1 2 \\lVert \\mathbf { u } \\rVert _ { 2 } ^ { 2 } } \\end{array}$ , which is strongly-convex w.r.t. the Euclidean norm, while EG (5) is recovered by setting $\\phi ( \\mathbf { u } ) \\bar { \\mathbf { \\Psi } } = \\mathbf { u } \\cdot \\mathrm { l o g } \\mathbf { u }$ , strongly-convex w.r.t. the $\\ell _ { 1 }$ -norm. ", + "bbox": [ + 174, + 669, + 825, + 699 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 BLOCK-STOCHASTIC MIRROR DESCENT ", + "text_level": 1, + "bbox": [ + 174, + 707, + 495, + 722 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the previous section we saw how mirror descent can perform better over certain geometries such as the simplex. However, in weight-sharing we are interested in optimizing over a hybrid geometry containing both the shared weights in an unconstrained Euclidean space and the architecture parameters in a non-Euclidean domain. Thus we focus on optimization over two blocks: shared weights $\\mathbf { w } \\in \\mathbb { R } ^ { d }$ and architecture parameters $\\theta \\in \\Theta$ , the latter associated with a DGF $\\phi$ that is strongly-convex w.r.t. some norm $\\| \\cdot \\|$ . In NAS a common approach is to perform alternating gradient steps on each domain; for example, both ENAS (Pham et al., 2018) and first-order DARTS (Liu et al., 2019) alternate between SGD on the shared weights and Adam on architecture parameters. This approach is encapsulated in the block-stochastic algorithm described in Algorithm 1, which at each step chooses one block at random to update using mirror descent (recall that SGD is a variant) and after $T$ steps returns a random iterate. Algorithm 1 generalizes the single-level variant of both ENAS and first-order DARTS if SGD is used to update $\\theta$ instead of Adam, with some mild caveats: in practice blocks are picked cyclically and the algorithm returns the last iterate, not a a random one. To analyze the convergence of Algorithm 1 we first state some regularity assumptions on the function: ", + "bbox": [ + 174, + 729, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1: Block-stochastic mirror descent optimization of a function $f : \\mathbb { R } ^ { d } \\times \\Theta \\mapsto \\mathbb { R }$ . ", + "text_level": 1, + "bbox": [ + 173, + 108, + 776, + 123 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/779e9c9724a0201deb99d3afec3fcfe8261b00ad0593c912a84c8212b2218a2a.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
for iteration t=1,...,TdoInput: nitialization(w(1),(1)）∈Rd×,strongly-convex DGF𝜙:→R,numberofiterations T≥1,step-size n>0
sample bt~Unif{w,0} if block bt = w then//randomly select update block
w(t+1) ←w(t)-nwf(w(t),0(t))// SGD update to shared weights
θ(t+1)←0(t)// no update to architecture params
else w(t+1）←w(t)
// no update to shared weights
θ(t+1)← argminuεe nθf(w(t),θ(t))·u + D(ul|θ(t))// update architecture params
Output:(w(r),θ(r)) for r~ Unif{1,...,T}
", + "bbox": [ + 173, + 127, + 816, + 255 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 1. Suppose $\\phi$ is strongly-convex w.r.t. some norm $\\| \\cdot \\|$ on a convex set $\\Theta$ and the objective function $f : \\mathbb { R } ^ { d } \\times \\Theta \\mapsto [ 0 , \\infty )$ satisfies the following: ", + "bbox": [ + 174, + 286, + 820, + 314 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. $\\gamma$ -relatively-weak-convexity: $f ( \\mathbf { w } , \\theta ) + \\gamma \\phi ( \\theta )$ is convex on $\\mathbb { R } ^ { d } \\times \\Theta$ for some $\\gamma > 0$ . ", + "bbox": [ + 179, + 315, + 800, + 330 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2. gradient bound: $\\mathbb { E } \\| \\tilde { \\nabla } _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta ) \\| _ { 2 } ^ { 2 } \\leq G _ { \\mathbf { w } } ^ { 2 }$ and $\\mathbb { E } \\| \\tilde { \\nabla } _ { \\theta } f ( \\mathbf { w } , \\theta ) \\| _ { * } ^ { 2 } \\leq G _ { \\theta } ^ { 2 }$ for some $G _ { \\mathbf { w } } , G _ { \\theta } \\geq 0 .$ ", + "bbox": [ + 181, + 332, + 802, + 348 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The second assumption is a standard bound on the gradient norm while the first is a generalization of smoothness that allows all smooth and some non-smooth non-convex functions (Zhang & He, 2018). ", + "bbox": [ + 173, + 352, + 825, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our aim will be to show (first-order) $\\varepsilon$ -stationary-point convergence of Algorithm 1, a standard metric indicating that it has reached a point with no feasible descent direction, up to error $\\varepsilon$ ; for example, in the unconstrained Euclidean case an $\\varepsilon$ -stationary-point is simply one where the gradient has squared-norm $\\leq \\varepsilon$ . The number of steps required to obtain such a point thus measures how fast a first-order method terminates. Stationarity is also significant as a necessary condition for optimality. ", + "bbox": [ + 173, + 387, + 825, + 457 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In our case $\\Theta$ may be constrained and so the gradient may never be small, thus necessitating a measure other than gradient norm. We use Bregman stationarity (Zhang & He, 2018, Equation 2.11), which measures stationary at a point $( \\mathbf { w } , \\theta )$ using the Bregman divergence between the point and its proximal map $\\begin{array} { r } { \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\theta ) = \\arg \\operatorname* { m i n } _ { \\mathbf { u } \\in \\mathbb { R } ^ { d } \\times \\Theta } \\lambda f ( \\mathbf { u } ) + D _ { \\ell _ { 2 } , \\phi } ( \\mathbf { u } | | \\mathbf { w } , \\theta ) } \\end{array}$ for some $\\lambda > 0$ : ", + "bbox": [ + 174, + 463, + 826, + 521 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f517f9c017b14b0ee2b25d74cb53fa826727c14bb55a8d4870811d232aa2c391.jpg", + "text": "$$\n\\Delta _ { \\lambda } ( \\mathbf { w } , \\boldsymbol { \\theta } ) = \\frac { D _ { \\ell _ { 2 } , \\phi } ( \\mathbf { w } , \\boldsymbol { \\theta } | | \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\boldsymbol { \\theta } ) ) + D _ { \\ell _ { 2 } , \\phi } ( \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\boldsymbol { \\theta } ) | | \\mathbf { w } , \\boldsymbol { \\theta } ) } { \\lambda ^ { 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 269, + 525, + 727, + 556 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here $\\begin{array} { r } { \\lambda = \\frac { 1 } { 2 \\gamma } } \\end{array}$ and the Bregman divergence $D _ { \\ell _ { 2 } , \\phi }$ is that of the DGF $\\frac { 1 } { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } + \\phi ( { \\theta } )$ that encodes the geometry of the joint optimization domain over w $\\in \\mathbb { R } ^ { d }$ and $\\theta$ ; note that the dependence of the stationarity measure on $\\gamma$ is standard (Dang $\\&$ Lan, 2015; Zhang & He, 2018). ", + "bbox": [ + 174, + 559, + 825, + 607 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To understand why reaching a point $( \\mathbf { w } , \\theta )$ with small Bregman stationarity is a reasonable goal, note that the proximal operator $\\operatorname { p r o x } _ { \\lambda }$ has the property that its fixed points, i.e. those satisfying $( \\mathbf { w } , \\theta ) = \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\theta )$ , correspond to points where $f$ has no feasible descent direction. Thus measuring how close $( \\mathbf { w } , \\theta )$ is to being a fixed point of $\\operatorname { p r o x } _ { \\lambda }$ —as is done using the Bregman divergence in (7)—is a good measure of how far away the point is from being a stationary point of $f$ . Finally, note that if $f$ is smooth, $\\phi$ is Euclidean, and $\\Theta$ is unconstrained—i.e. if we are running SGD over architecture parameters as well—then $\\Delta _ { \\frac { 1 } { 2 \\gamma } } \\leq \\varepsilon$ implies an $O ( \\varepsilon )$ -bound on the squared gradient norm, recovering the standard definition of $\\varepsilon$ -stationarity. More intuition on proximal operators can be found in Parikh & Boyd (2013, Section 1.2), while further details on Bregman stationarity and how it relates to other notions of convergence can be found in Zhang & He (2018, Section 2.3). ", + "bbox": [ + 173, + 613, + 826, + 757 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The following result shows that Algorithm 1 needs polynomially many iterations to finds a point $( \\mathbf { w } , \\theta )$ with $\\varepsilon$ -small Bregman stationarity in-expectation: ", + "bbox": [ + 173, + 762, + 821, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1. Let $F = f ( \\mathbf { w } ^ { ( 1 ) } , \\theta ^ { ( 1 ) } )$ be the value of $f$ at initialization. Under Assumption 1, if we run Algorithm $I$ for $\\begin{array} { r } { T = \\frac { 1 6 \\gamma F } { \\varepsilon ^ { 2 } } ( G _ { \\mathbf { w } } ^ { 2 } + G _ { \\theta } ^ { 2 } ) } \\end{array}$ iterations with step-size $\\begin{array} { r } { \\eta = \\sqrt { \\frac { 4 F } { \\gamma ( G _ { \\mathbf { w } } ^ { 2 } + G _ { \\theta } ^ { 2 } ) T } } } \\end{array}$ then $\\begin{array} { r } { \\mathbb { E } \\Delta _ { \\frac { 1 } { 2 \\gamma } } \\big ( \\mathbf { w } ^ { ( r ) } , \\theta ^ { ( r ) } \\big ) \\leq \\varepsilon . } \\end{array}$ . Here the expectation is over the randomness of the algorithm and gradients. ", + "bbox": [ + 173, + 796, + 826, + 859 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The proof in the appendix follows from single-block analysis (Zhang & He, 2018, Theorem 3.1) and in fact holds for the general case of any number of blocks associated to any set of strongly-convex DGFs. Although there are prior results for the multi-block case (Dang & Lan, 2015), they do not hold for nonsmooth Bregman divergences such as the KL divergence needed for exponentiated gradient. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/d648ab607dd40f81f37634764c44572a01953b0b33302d329a29127ff5492113.jpg", + "image_caption": [ + "Figure 1: Sparsity: Evolution over search phase epochs of the average entropy of the operationweights for GAEA and approaches it modifies when run on the DARTS search space (left), NASBench-1Shot1 Search Space 1 (middle), and NASBench-201 on CIFAR-10 (right). GAEA reduces entropy much more quickly, allowing it to quickly obtain sparse architecture weights. This leads to both faster convergence to a single architecture and a lower loss when pruning at the end of search. " + ], + "image_footnote": [], + "bbox": [ + 179, + 103, + 815, + 209 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Thus Algorithm 1 returns an $\\varepsilon$ -stationary-point given $T = \\mathcal { O } ( G _ { \\mathbf { w } } ^ { 2 } + G _ { \\theta } ^ { 2 } ) / \\varepsilon ^ { 2 }$ iterations, where $G _ { \\mathbf { w } } ^ { 2 }$ bounds the squared $\\ell _ { 2 }$ -norm of the shared-weights gradient $\\tilde { \\nabla } _ { \\mathbf { w } }$ and $G _ { \\theta } ^ { 2 }$ bounds the squared magnitude of the architecture gradient $\\tilde { \\nabla } _ { \\theta }$ , as measured by the dual norm $\\| \\cdot \\| _ { * }$ of $\\| \\cdot \\|$ . Only the last term $G _ { \\theta }$ is affected by our choice of DGF $\\phi$ . The DGF of SGD is strongly-convex w.r.t. the $\\ell _ { 2 }$ -norm, which is its own dual, so $G _ { \\mathbf { w } } ^ { 2 }$ is defined via $\\ell _ { 2 }$ . However, for EG the DGF $\\phi ( \\mathbf { u } ) = \\mathbf { u } \\cdot \\log \\mathbf { u }$ is strongly-convex√ w.r.t. the $\\ell _ { 1 }$ -norm, whose dual is $\\ell _ { \\infty }$ . Since the $\\ell _ { 2 }$ -norm of a $k$ -dimensional vector can be $\\sqrt { k }$ times its $\\ell _ { \\infty }$ -norm, picking this DGF can lead to better bound on $G _ { \\theta }$ and thus on the number of iterations. ", + "bbox": [ + 173, + 306, + 825, + 414 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 GAEA: A GEOMETRY-AWARE EXPONENTIATED ALGORITHM", + "text_level": 1, + "bbox": [ + 174, + 422, + 642, + 438 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Equipped with these single-level guarantees, we turn to designing methods that can in-principle be applied to both the single-level and bilevel objectives, seeking parameterizations and algorithms that converge quickly and encourage favorable properties; in particular, we focus on returning architecture parameters that are sparse to reduce loss due to post-search discretization. EG is often considered to converge quickly to sparse solutions over the simplex (Bradley & Bagnell, 2008; Bubeck, 2019), which makes it a natural choice for the architecture update. We thus propose GAEA, a GeometryAware Exponentiated Algorithm in which operation weights on each edge are constrained to the simplex and trained using EG; as in DARTS, the shared weights w are trained using SGD. GAEA can be used as a simple, principled modification to the many NAS methods that treat architecture parameters $\\theta \\in \\Theta = \\mathbb { R } ^ { | E | \\times | O | }$ as real-valued “logits” to be passed through a softmax to obtain mixture weights or probabilities for simplices over the operations $O$ . Such methods include DARTS, PC-DARTS ( $\\mathrm { X u }$ et al., 2020), and GDAS (Dong & Yang, 2019). To apply GAEA, first re-parameterize $\\Theta$ to be the product set of $| E |$ simplices, each associated to an edge $( i , j ) \\in E$ ; thus $\\theta _ { i , j , o }$ corresponds directly to the weight or probability of operation $o \\in O$ for edge $( i , j )$ , not a logit. Then, given a stochastic gradient $\\tilde { \\nabla } _ { \\boldsymbol { \\theta } } f ( \\mathbf { w } ^ { ( t ) } , \\boldsymbol { \\theta } ^ { ( t ) } )$ and step-size $\\eta > 0$ , replace the architecture update by EG: ", + "bbox": [ + 173, + 445, + 826, + 659 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f4432e997e5095b873c6e7a17b3681374f73d397d2b0096c90ad6bde4ab9fd89.jpg", + "text": "$$\n\\begin{array} { r l r } & { \\tilde { \\theta } ^ { ( t + 1 ) } \\gets \\theta ^ { ( t ) } \\odot \\mathrm { e x p } \\left( - \\eta \\tilde { \\nabla } _ { \\theta } f \\big ( \\mathbf { w } ^ { ( t ) } , \\theta ^ { ( t ) } \\big ) \\right) } & { \\mathrm { ( m u l t i p l i c a t i v e ~ u p d a t e ) } } \\\\ & { \\theta _ { i , j , o } ^ { ( t + 1 ) } \\gets \\frac { \\tilde { \\theta } _ { i , j , o } ^ { ( t + 1 ) } } { \\sum _ { o ^ { \\prime } \\in O } \\tilde { \\theta } _ { i , j , o ^ { \\prime } } ^ { ( t + 1 ) } } \\ \\forall o \\in O , \\forall \\ ( i , j ) \\in E } & { \\mathrm { ( s i m p l e x ~ p r o j e c t i o n ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 664, + 741, + 739 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "These two simple modifications, $r e$ -parameterization and exponentiation, suffice to obtain state-ofthe-art results on several NAS benchmarks, as shown in Section 4. Note that to obtain a bilevel algorithm we simply replace the gradient w.r.t. $\\theta$ of the training loss with that of the validation loss. ", + "bbox": [ + 176, + 744, + 821, + 786 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "GAEA is equivalent to Algorithm 1 with $\\begin{array} { r } { \\phi ( \\theta ) = \\sum _ { ( i , j ) \\in E } \\sum _ { o \\in O } \\theta _ { i , j , o } \\log \\theta _ { i , j , o } } \\end{array}$ , which is stronglyconvex w.r.t. $\\| \\cdot \\| _ { 1 } / { \\sqrt { | E | } }$ over the product of $| E | \\mid O |$ -simplices. The dual is ${ \\sqrt { | E | } } \\| \\cdot \\| _ { \\infty }$ , so if $G _ { \\mathbf { w } }$ bounds the shared-weights gradient and we have an entry-wise bound on the architecture gradient then GAEA reach $\\varepsilon$ -stationarity in $\\mathcal { O } ( G _ { \\mathbf { w } } ^ { 2 } + | E | ) / \\varepsilon ^ { 2 }$ iterations. This can be up to a factor $| O |$ improvement over SGD, either over the simplex or the logit space. In addition, GAEA encourages sparsity in the architecture weights by using a multiplicative update over simplices and not an additive update over $\\mathbb { R } ^ { | E | \\times | O | }$ . Obtaining sparse architecture parameters is critical for good performance, both for the mixture relaxation, where it alleviates the effect of discretization on the validation loss, and for the stochastic relaxation, where it reduces noise when sampling architectures. ", + "bbox": [ + 173, + 791, + 825, + 925 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/d2b8f9aa236a5e72ec94b44075d51b3b0194ba075ddb46e0a7321909b464e9d1.jpg", + "table_caption": [ + "Table 1: DARTS: Comparison with SOTA NAS methods on the DARTS search space, plus three results on different search spaces with a similar number of parameters reported at the top for comparison. All evaluations and reported performances of models found on the DARTS search space use similar training routines; this includes auxiliary towers and cutout but no other modifications, e.g. label smoothing (Muller et al., 2019), AutoAugment (Cubuk et al., 2019), Swish (Ramachandran ¨ et al., 2017), Squeeze & Excite (Hu et al., 2018), etc. The specific training procedure we use is that of PC-DARTS, which differs slightly from the DARTS routine by a small change to the drop-path probability; PDARTS tunes both this and batch-size. Our results are averaged over 10 random seeds. Search cost is hardware-dependent; we used Tesla V100 GPUs. For more details see Tables 4 & 5. " + ], + "table_footnote": [ + "Search space/backbone differ from the DARTS setting; we show results for networks with a comparable number of parameters. For fair comparison to other work, we show the search cost for training the shared-weights network with a single initialization. " + ], + "table_body": "
Search Method (source)CIFAR-10 ErrorSearch Cost (GPU Days)ImageNet Error top-1Search Cost (GPU Days)method
BestAveragetop-5
NASNet-A* (Zoph et al.,2018)-2.65200026.08.41800RL
AmoebaNet-B* (Real et al.,2019)12.55± 0.05315024.37.63150evolution
ProxylessNAS*(Cai et al.,2019)2.08424.97.58.3gradient (WS)
ENAS (Pham et al., 2018)2.890.5--1RL (WS)
RS-Wst (Li& Talwalkar,2019)2.712.85±0.080.7-1-random (WS)
ASNG (Akimoto et al.,2019)12.83 ± 0.140.1--gradient (WS)
SNAS (Xie et al., 2019)=2.85±0.021.527.39.21.5gradient (WS)
DARTS (1st)† (Liu et al.,2019)=3.00 ±0.140.4---gradient (WS)
DARTS (2nd)† (Liu et al., 2019)2.76 ±0.09126.78.74.0gradient (WS)
PDARTS (Chen et al.,2019)2.50=0.324.47.40.3gradient (WS)
PC-DARTSt (Xu et al.,2020)-2.57 ±0.070.124.27.33.8gradient (WS)
GAEA PC-DARTS† (ours)2.392.50 ±0.060.124.07.33.8gradient (WS)
PC-DARTSt (Xu et al.,2020)(search on CIFAR-1O,train on ImageNet)25.17.80.1gradient (WS)
GAEA PC-DARTS† (ours)(search on CIFAR-10, train on ImageNet)24.37.30.1gradient (WS)
", + "bbox": [ + 173, + 231, + 826, + 421 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EMPIRICAL RESULTS USING GAEA ", + "text_level": 1, + "bbox": [ + 174, + 468, + 503, + 484 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate GAEA on three different computer vision benchmarks: the large and heavily studied search space from DARTS (Liu et al., 2019) and two smaller oracle evaluation benchmarks, NASBench-1Shot1 (Zela et al., 2020a), and NAS-Bench-201 (Dong & Yang, 2020). NAS-Bench-1Shot1 differs from the others by applying operations per node instead of per edge, while NAS-Bench-201 differs by not requiring edge-pruning. Since GAEA can modify a variety of methods, e.g. DARTS, PC-DARTS (Xu et al., 2020), and GDAS (Dong & Yang, 2019), on each benchmark we start by evaluating the GAEA variant of the current best method on that benchmark. We show that despite the diversity of search spaces, GAEA improves upon this state-of-the-art across all three. Note that we use the same step-size for GAEA variants of DARTS/PC-DARTS and do not require weight-decay on architecture parameters. We defer experimental details and hyperparameter settings to the appendix and release all code, hyperparameters, and random seeds for reproducibility. ", + "bbox": [ + 173, + 496, + 826, + 650 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 CONVERGENCE AND SPARSITY OF GAEA ", + "text_level": 1, + "bbox": [ + 174, + 660, + 504, + 674 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We first examine the impact of GAEA on convergence and sparsity. Figure 1 shows the entropy of the operation weights averaged across nodes for a GAEA-variant and its base method across the three benchmarks, demonstrating that it decreases much faster for GAEA-modified approaches. This validates our expectation that GAEA encourages sparse architecture parameters, which should alleviate the mismatch between the continuously relaxed architecture parameters and the discrete architecture returned. Indeed, we find that post-search discretization on the DARTS search space causes the validation accuracy of the PC-DARTS supernet to drop from $7 2 . 1 7 \\%$ to $1 5 . 2 7 \\%$ , while for GAEA PC-DARTS the drop is only $7 5 . 0 7 \\%$ to $3 3 . 2 3 \\%$ ; note that this is shared-weights accuracy, obtained without retraining the final network. The numbers demonstrate that GAEA both (1) achieves better supernet optimization of the weight-sharing objective and (2) suffers less due to discretization. ", + "bbox": [ + 174, + 681, + 825, + 821 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 GAEA ON THE DARTS SEARCH SPACE ", + "text_level": 1, + "bbox": [ + 174, + 830, + 490, + 845 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Here we evaluate GAEA on the task of designing CNN cells for CIFAR-10 (Krizhevksy, 2009) and ImageNet (Russakovsky et al., 2015) by using it to modify PC-DARTS (Xu et al., 2020), the current state-of-the-art method. We follow the same three stage process used by both DARTS and RS-WS for search and evaluation. Table 1 displays results on both datasets and demonstrates that GAEA’s parameterization and optimization scheme improves upon PC-DARTS. In fact, GAEA PC-DARTS outperforms all search methods except ProxylessNAS, which uses 1.5 times as many parameters on a different search space. Thus we improve the state-of-the-art on the DARTS search space. To meet a higher bar for reproducibility on CIFAR-10, in Appendix C we report “broad reproducibility” (Li & Talwalkar, 2019) by repeating our pipeline with new seeds. While GAEA PC-DARTS consistently finds good networks when selecting the best of four independent trials, multiple trials are required due to sensitivity to initialization, as is true for many approaches (Liu et al., 2019; Xu et al., 2020). ", + "bbox": [ + 174, + 853, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/256fe57190a403501ca5f7fd2ddac27d71516438ba9a18b29b2b46a4467b7320.jpg", + "image_caption": [ + "Figure 2: NAS-Bench-1Shot1: Online comparison of PC-DARTS and GAEA PC-DARTS in terms of the test regret at each epoch of shared-weights training, i.e. the difference between the ground truth test error of the proposed architecture and that of the best architecture in the search space. The dark lines indicate the mean of four random trials and the light colored bands $\\pm$ one standard deviation. The dashed line is the final regret of the best weight-sharing method according to Zela et al. (2020b); note that in our reproduction PC-DARTS performed better than their evaluation on spaces 1 and 3. " + ], + "image_footnote": [], + "bbox": [ + 171, + 98, + 825, + 212 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 333, + 825, + 417 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "On ImageNet, we follow $\\mathrm { X u }$ et al. (2020) by using subsamples containing $1 0 \\%$ and $2 . 5 \\%$ of the training images from ILSVRC-2012 (Russakovsky et al., 2015) as training and validation sets, respectively. We fix architecture parameters for the first 35 epochs, then run GAEA PC-DARTS with step-size 0.1. All other hyperparameters match those of $\\mathrm { X u }$ et al. (2020). Table 1 shows the final performance of both the architecture found by GAEA PC-DARTS on CIFAR-10 and the one found directly on ImageNet when trained from scratch for 250 epochs using the same settings as $\\mathrm { X u }$ et al. (2020). GAEA PC-DARTS achieves a top-1 test error of $2 4 . 0 \\%$ , which is state-of-the-art performance in the mobile setting when excluding additional training modifications, e.g. those in the caption. Additionally, the architecture found by GAEA PC-DARTS for CIFAR-10 and transferred achieves a test error of $2 4 . 2 \\%$ , comparable to the $2 4 . 2 \\%$ error of the one found by PC-DARTS directly on ImageNet. Top architectures found by GAEA PC-DARTS are depicted in Figure 3 in the appendix. ", + "bbox": [ + 174, + 424, + 826, + 577 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 GAEA ON NAS-BENCH-1SHOT1 ", + "text_level": 1, + "bbox": [ + 176, + 588, + 442, + 602 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "NAS-Bench-1Shot1 (Zela et al., 2020a) is a subset of NAS-Bench-101 (Ying et al., 2019) that allows benchmarking weight-sharing methods on three search spaces over CIFAR-10 that differ in the number of nodes considered and the number of input edges per node. Of the weight-sharing methods benchmarked by Zela et al. (2020a), we found that PC-DARTS achieves the best performance on 2 of 3 search spaces, so we again evaluate GAEA PC-DARTS here. Figure 2 shows that GAEA PC-DARTS consistently finds better architectures on average than PC-DARTS and thus exceeds the performance of the best method from Zela et al. (2020a) on 2 of 3 search spaces. We hypothesize that the benefits of GAEA are limited here due to the near-saturation of NAS methods. In particular, existing methods obtain within $1 \\%$ test error of the top network in each space, while the latters’ test errors when evaluated with different initializations are $0 . 3 7 \\%$ , $0 . 2 3 \\%$ and $0 . 1 9 \\%$ , respectively. ", + "bbox": [ + 174, + 611, + 825, + 750 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 GAEA ON NAS-BENCH-201 ", + "text_level": 1, + "bbox": [ + 176, + 762, + 415, + 776 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "NAS-Bench-201 has one search space on three datasets—CIFAR-10, CIFAR-100, and ImageNet16-120—that includes 4-node architectures with an operation from $O = \\{ \\mathrm { n o n e }$ , skip connect, 1x1 convolution, 3x3 convolution, $3 \\mathrm { x } 3 \\mathrm { a v g } \\mathrm { p o o l } \\}$ on each edge, yielding 15625 possible networks. Dong & Yang (2020) report results for several algorithms in the transfer NAS setting, where search is conducted on CIFAR-10 and the resulting networks are trained on a possibly different target dataset. Table 2 reports a subset of these results alongside evaluations of our implementation of several existing and GAEA-modified NAS methods in both the transfer and direct setting. Both the results from Dong & Yang (2020) and our reproductions show that GDAS is the best previous weight-sharing method; we evaluate GAEA GDAS and find that it achieves better results on CIFAR-100 and similar results on the other two datasets. ", + "bbox": [ + 173, + 785, + 825, + 922 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/3607b9f9fc2e9a5268b9b12dabc8c091084d63a8de4bb250c63408d425f01010.jpg", + "table_caption": [ + "Table 2: NAS-Bench-201: Results are separated into traditional hyperparameter optimization algorithms with search run on CIFAR-10 (top block), weight-sharing methods with search run on CIFAR-10 (middle block), and weight-sharing methods run directly on the dataset used for training (bottom block). The use of transfer NAS follows the evaluations conducted by Dong & Yang (2020); unless otherwise stated all non-GAEA results are from their paper. The best results in the transfer and direct settings on each dataset are bolded. " + ], + "table_footnote": [ + "Search cost reported for running the search algorithm on CIFAR-10. Search cost measured on NVIDIA P100 GPUs. ‡ Our reproduction or implementation of a non-GAEA method. " + ], + "table_body": "
Search*(seconds)CIFAR-10 (test)CIFAR-100 (test)ImageNet-16-120 (test)
RegularREAN/A93.92 ± 0.3071.84 ± 0.9945.54 ± 1.03
HO,RSN/A93.70±0.3671.04 ± 1.0844.57 ±1.25
search onREINFORCEN/A93.85 ± 0.3771.71 ± 1.0945.25 ± 1.18
CIFAR-10BOHBN/A93.61 ±0.5270.85 ±1.2844.42 ± 1.49
RSPS758787.66 ± 1.6958.33 ± 4.3431.14 ± 3.88
DARTS (bilevel)3578154.30 ±0.0015.61 ±0.0016.32 ± 0.00
WeightSETN3413987.64 ± 0.0059.05 ± 0.2432.52 ±0.21
sharing,GDAS3160993.61 ±0.0970.70 ± 0.3041.71 ± 0.98
search onDARTS‡ (bilevel)10683†54.30 ±0.0015.32 ± 0.0016.38 ± 0.00
CIFAR-10GAEA DARTS (bilevel)7930+91.63 ± 2.5768.39 ± 4.4741.59 ± 4.20
DARTS* (ERM)18112t84.39 ±3.8254.81 ± 7.0831.82 ± 4.78
GAEA DARTS (ERM)9061†94.10±0.2972.60± 0.8945.81 ± 0.51
WeightGDAS27923†93.52 ± 0.1567.52 ± 0.1540.91 ± 0.12
GAEA GDAS16754t93.55 ± 0.1370.47 ± 0.4740.91 ±0.12
sharing,DARTS‡ (bilevel)10683†54.30 ±0.0015.32 ±0.0028.96 ±10.22
directGAEA DARTS (bilevel)7930t91.63 ± 2.5771.87 ± 0.5745.69 ± 0.56
searchDARTS* (ERM)18112†84.39 ±3.8251.26 ±6.1431.35 ± 7.46
GAEA DARTS (ERM)9061†94.10±0.2973.43 ± 0.1346.36 ±0.00
ResNetN/A93.9770.8643.63
OptimalN/A94.3773.5147.31
", + "bbox": [ + 173, + 188, + 825, + 446 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Since we are interested in improving upon not only GAEA GDAS but also upon traditional hyperparameter optimization methods, we also investigate the performance of GAEA applied to first-order DARTS. We evaluate GAEA DARTS with both single-level (ERM) and bilevel optimization; recall that in the latter case we optimize architecture parameters w.r.t. the validation loss and the shared weights w.r.t. the training loss, whereas in ERM there is no data split. GAEA DARTS (ERM) achieves state-of-the-art performance on all three datasets in both the transfer and direct setting, exceeding the test accuracy of both weight-sharing and traditional hyperparameter tuning by a wide margin. GAEA DARTS (bilevel) performs worse but still exceeds all other methods on CIFAR-100 and ImageNet-16-120 in the direct search setting. The result thus also confirms the relevance of studying the single-level case to understand NAS; notably, the DARTS (ERM) baseline also improves substantially upon the DARTS (bilevel) baseline. ", + "bbox": [ + 174, + 523, + 826, + 676 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 715, + 318, + 732 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper we take an optimization-based view of NAS, arguing that the design of good NAS algorithms is largely a matter of successfully optimizing and regularizing the supernet. In support of this, we develop GAEA, a simple modification of gradient-based NAS that attains state-of-the-art performance on several computer vision benchmarks while enjoying favorable speed and sparsity properties. We believe that obtaining high-performance NAS algorithms for a wide variety of applications will continue to require a similar co-design of search space parameterizations and optimization methods, and that our geometry-aware framework can help accelerate this process. In particular, most modern NAS algorithms search over products of categorical decision spaces, to which our approach is directly applicable. More generally, as the field moves towards more ambitious search spaces, e.g. full-network topologies or generalizations of operations such as convolution or attention, these developments may result in new architecture domains for which our work can inform the design of appropriate, geometry-aware optimization methods. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 356, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Jeremy Cohen, Jeffrey Li, and Nicholas Roberts for helpful feedback. This work was supported in part by DARPA under cooperative agreements FA875017C0141 and HR0011202000, NSF grants CCF-1535967, CCF-1910321, IIS-1618714, IIS-1705121, IIS-1838017, IIS-1901403, and IIS-2046613, a Microsoft Research Faculty Fellowship, a Bloomberg Data Science research grant, an Amazon Research Award, an AWS Machine Learning Research Award, a Facebook Faculty Research Award, funding from Booz Allen Hamilton Inc., a Block Center Grant, a Carnegie Bosch Institute Research Award, and a Two Sigma Fellowship Award. 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", + "bbox": [ + 174, + 818, + 825, + 861 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A BACKGROUND ON NAS WITH WEIGHT-SHARING ", + "text_level": 1, + "bbox": [ + 174, + 102, + 617, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Here we review the NAS setup motivating our work. Weight-sharing methods almost exclusively use micro cell-based search spaces for their tractability and additional structure (Pham et al., 2018; Liu et al., 2019). These search spaces can be represented as directed acyclic graphs (DAGs) with a set of ordered nodes $N$ and edges $E$ . Each node $x ^ { ( i ) } \\in N$ is a feature representation and each edge $( i , j ) \\in E$ is associated with an operation on the feature of node $j$ passed to node $i$ and aggregated with other inputs to form $x ^ { ( j ) }$ , with the restriction that a given node $j$ can only receive edges from prior nodes as input. Hence, the feature at node $i$ is $\\begin{array} { r } { x ^ { ( i ) } = \\sum _ { j < i } \\overset { \\cdot } { o } ^ { ( i , j ) } ( x ^ { ( j ) } ) } \\end{array}$ . Search spaces are specified by the number of nodes, the number of edges per node, and the set of operations $O$ that can be applied at each edge. Thus for NAS, $\\mathcal { A } \\subset \\{ 0 , \\bar { 1 } \\} ^ { | E | \\times | O | }$ is the set of all valid architectures for encoded by edge and operation decisions. Treating both the shared weights $\\mathbf { w } \\in \\mathbb { R } ^ { d }$ and architecture decisions $a \\in { \\mathcal { A } }$ as parameters, weight-sharing methods train a single network subsuming all possible functions within the search space. ", + "bbox": [ + 173, + 133, + 825, + 308 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Gradient-based weight-sharing methods apply continuous relaxations to the architecture space $\\mathcal { A }$ in order to compute gradients in a continuous space $\\Theta$ . Methods like DARTS (Liu et al., 2019) and its variants (Chen et al., 2019; Laube & Zell, 2019; Hundt et al., 2019; Liang et al., 2019; Noy et al., 2019; Nayman et al., 2019) relax the search space by considering a mixture of operations per edge. For example, we will consider a relaxation where the architecture space $\\mathcal { A } = \\bar { \\{ 0 , 1 \\} } ^ { | E | \\times | O \\bar { 1 } }$ is relaxed into $\\Theta \\bar { = } [ 0 , 1 ] ^ { | E | \\times | O | }$ with the constraint that $\\textstyle \\sum _ { o \\in O } \\theta _ { i , j , o } = 1$ , i.e. the operation weights on each edge sum to 1. The feature at node $i$ is then $\\begin{array} { r } { x ^ { ( i ) } = \\sum _ { j < i } \\sum _ { o \\in O } \\theta _ { i , j , o } o ( x ^ { ( j ) } ) } \\end{array}$ . To get a valid architecture $a \\in { \\mathcal { A } }$ from a mixture $\\theta$ , rounding and pruning are typically employed after the search phase. ", + "bbox": [ + 173, + 314, + 825, + 446 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "An alternative, stochastic approach, such as that used by GDAS (Dong & Yang, 2019), instead uses $\\Theta$ -parameterized distributions $p _ { \\theta }$ over $\\mathcal { A }$ to sample architectures (Pham et al., 2018; Xie et al., 2019; Akimoto et al., 2019; Cai et al., 2019); unbiased gradients w.r.t. $\\theta \\in \\Theta$ can be computed using Monte Carlo sampling. The goal of all these relaxations is to use simple gradient-based approaches to approximately optimize (1) over $a \\in { \\mathcal { A } }$ by optimizing (2) over $\\theta \\in \\Theta$ instead. However, both the relaxation and the optimizer critically affect the convergence speed and solution quality. We next present a principled approach for understanding both mixture and stochastic methods. ", + "bbox": [ + 173, + 454, + 825, + 553 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 334, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This section contains proofs and generalizations of the non-convex optimization results in Section 3. Throughout this section, $\\mathbb { V }$ denotes a finite-dimensional real vector space with Euclidean inner product $\\langle \\cdot , \\cdot \\rangle$ , $\\mathbb { R } _ { + }$ denotes the set of nonnegative real numbers, and $\\overline { { \\mathbb { R } } }$ denotes the set of extended real numbers $\\mathbb { R } \\cup \\{ \\pm \\infty \\}$ . ", + "bbox": [ + 173, + 132, + 826, + 193 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.1 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 174, + 207, + 330, + 222 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Definition 1. Consider a closed and convex subset $\\mathcal { X } \\subset \\mathbb { V } .$ . For any $\\alpha > 0$ and norm $\\| \\cdot \\| : \\mathcal { X } \\mapsto \\mathbb { R } _ { + }$ an everywhere-subdifferentiable function $f : \\mathcal { X } \\mapsto \\overline { { \\mathbb { R } } }$ is called $\\alpha$ -strongly-convex w.r.t. $| | \\cdot | | i f$ $\\forall \\mathbf { x } , \\mathbf { y } \\in \\mathcal { X }$ we have ", + "bbox": [ + 173, + 233, + 828, + 277 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/86ead831e3b31f599c6e32c3d6cadafce7bd3683c713a1d2b025a48759db6ab3.jpg", + "text": "$$\nf ( \\mathbf { y } ) \\geq f ( \\mathbf { x } ) + \\langle \\nabla f ( \\mathbf { x } ) , \\mathbf { y } - \\mathbf { x } \\rangle + { \\frac { \\alpha } { 2 } } \\| \\mathbf { y } - \\mathbf { x } \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 341, + 275, + 655, + 303 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Definition 2. Consider a closed and convex subset $\\mathcal { X } \\subset \\mathbb { V } .$ . For any $\\beta > 0$ and norm $\\| \\cdot \\| : \\mathcal { X } \\mapsto \\mathbb { R } _ { + }$ an continuously-differentiable function $f : \\mathcal { X } \\mapsto \\mathbb { R }$ is called $\\beta$ -strongly-smooth w.r.t. $\\| \\cdot \\| i f$ $\\forall \\mathbf { x } , \\mathbf { y } \\in \\mathcal { X }$ we have ", + "bbox": [ + 174, + 305, + 823, + 347 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a01ee43e03c300852e5775c0ee37305433d7e3d0a326e3e6f77e3a1539c6bc70.jpg", + "text": "$$\nf ( \\mathbf { y } ) \\leq f ( \\mathbf { x } ) + \\langle \\nabla f ( \\mathbf { x } ) , \\mathbf { y } - \\mathbf { x } \\rangle + { \\frac { \\beta } { 2 } } \\| \\mathbf { y } - \\mathbf { x } \\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 341, + 344, + 655, + 376 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Definition 3. Let $\\mathcal { X }$ be a closed and convex subset of $\\mathbb { V }$ . The Bregman divergence induced by $a$ strictly convex, continuously-differentiable distance-generating function (DGF) $\\phi : \\mathcal { X } \\mapsto \\overline { { \\mathbb { R } } }$ is ", + "bbox": [ + 173, + 377, + 823, + 406 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7d82ff9bb672823a8976f51af52de832f28b1243fc84e26d6cd9d75488df8319.jpg", + "text": "$$\nD _ { \\phi } ( \\mathbf { x } | | \\mathbf { y } ) = \\phi ( \\mathbf { x } ) - \\phi ( \\mathbf { y } ) - \\left. \\nabla \\phi ( \\mathbf { y } ) , \\mathbf { x } - \\mathbf { y } \\right. \\forall \\mathbf { x } , \\mathbf { y } \\in \\mathcal { X }\n$$", + "text_format": "latex", + "bbox": [ + 308, + 412, + 689, + 431 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By definition, the Bregman divergence satisfies the following properties: ", + "bbox": [ + 173, + 436, + 647, + 452 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1e0423bab78b709363f2a9d4331d1341f98a7030174a480e5633b9831eccd068.jpg", + "text": "$$\nD _ { \\phi } ( \\mathbf { x } | | \\mathbf { y } ) \\geq 0 \\forall x , y \\in \\mathcal { X } a n d D _ { \\phi } ( \\mathbf { x } | | \\mathbf { y } ) = 0 \\iff x = y .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 462, + 622, + 479 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2. If $\\phi$ is $\\alpha$ -strongly-convex w.r.t. norm $\\| \\cdot \\|$ then so is $D _ { \\phi } ( \\cdot | | \\mathbf { y } ) \\ \\forall \\ \\mathbf { y } \\in \\mathcal { X }$ . Furthermore, $\\begin{array} { r } { D _ { \\phi } ( \\mathbf { x } | | \\mathbf { y } ) \\geq \\frac { \\alpha } { 2 } \\| \\mathbf { x } - \\mathbf { y } \\| ^ { 2 } \\forall \\mathbf { x } , \\mathbf { y } \\in \\mathcal { X } . } \\end{array}$ . ", + "bbox": [ + 210, + 486, + 825, + 518 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "3. If $\\phi$ is $\\beta$ -strongly-smooth w.r.t. norm $\\| \\cdot \\|$ then so is $D _ { \\phi } ( \\cdot | | \\mathbf { y } ) \\ \\forall \\ \\mathbf { y } \\in \\mathcal { X }$ . Furthermore, $\\begin{array} { r } { D _ { \\phi } ( \\mathbf { x } | | \\mathbf { y } ) \\leq \\frac { \\beta } { 2 } \\| \\mathbf { x } - \\mathbf { y } \\| ^ { 2 } \\forall \\mathbf { x } , \\mathbf { y } \\in \\mathcal { X } } \\end{array}$ . ", + "bbox": [ + 209, + 525, + 825, + 559 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Definition 4. (Zhang & He, 2018, Definition 2.1) Consider a closed and convex subset $\\mathcal { X } \\subset \\mathbb { V } .$ . For any $\\gamma > 0$ and ${ \\overset { \\vartriangle } { \\phi } } : { \\boldsymbol { \\mathcal { X } } } \\mapsto { \\overline { { \\mathbb { R } } } }$ an everywhere-subdifferentiable function $f : \\mathcal { X } \\mapsto \\mathbb { R }$ is called $\\gamma$ -relatively-weakly-convex ( $\\langle \\gamma$ -RWC) w.r.t. $\\phi$ if $f ( \\cdot ) + \\gamma \\phi ( \\cdot )$ is convex on $\\mathcal { X }$ . ", + "bbox": [ + 176, + 560, + 826, + 604 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Definition 5. (Zhang & He, 2018, Definition 2.3) Consider a closed and convex subset $\\mathcal { X } \\subset \\mathbb { V } .$ . For any $\\lambda > 0$ , function $f : \\mathcal { X } \\mapsto \\mathbb { R } ,$ , and D $G F \\phi : \\mathcal { X } \\mapsto \\overline { { \\mathbb { R } } }$ the Bregman proximal operator of $f$ is ", + "bbox": [ + 173, + 607, + 823, + 637 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/11eabf6c7f7fd3272f38e01fb1cff92065710b6a5f4b423d0919d512eb668d44.jpg", + "text": "$$\n\\operatorname { p r o x } _ { \\lambda } ( \\mathbf { x } ) = \\underset { \\mathbf { u } \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\lambda f ( \\mathbf { u } ) + D _ { \\phi } ( \\mathbf { u } | | \\mathbf { x } )\n$$", + "text_format": "latex", + "bbox": [ + 362, + 643, + 633, + 669 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Definition 6. (Zhang & He, 2018, Equation 2.11) Consider a closed and convex subset $\\mathcal { X } \\subset \\mathbb { V } .$ . For any $\\lambda > 0$ , function $f : \\mathcal { X } \\mapsto \\mathbb { R }$ , and DGF ${ \\vec { \\mathbf { \\tau } } } \\phi : \\mathcal { X } \\mapsto \\overline { { \\mathbb { R } } }$ the Bregman stationarity of $f$ at any point $\\mathbf { x } \\in \\mathcal { X }$ is ", + "bbox": [ + 173, + 674, + 826, + 715 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/63b694794ee3994e7a1b6f9187536510250eafa66f36a03c300264488fba4272.jpg", + "text": "$$\n\\Delta _ { \\lambda } ( { \\bf x } ) = \\frac { D _ { \\phi } ( { \\bf x } | | \\mathrm { p r o x } _ { \\lambda } ( { \\bf x } ) ) + D _ { \\phi } ( \\mathrm { p r o x } _ { \\lambda } ( { \\bf x } ) | | { \\bf x } ) } { \\lambda ^ { 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 333, + 713, + 663, + 744 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.2 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 282, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Throughout this subsection let $\\mathbb { V } = \\times _ { i = 1 } ^ { b } \\mathbb { V } _ { i }$ be a product space of $b$ finite-dimensional real vector spaces $\\mathbb { V } _ { i }$ , each with an associated norm $\\| \\cdot \\| _ { i } : \\mathbb { V } _ { i } \\mapsto \\mathbb { R } _ { + }$ , and $\\chi = \\times _ { i = 1 } ^ { b } \\chi _ { i }$ be a product set of $b$ subsets $\\mathcal { X } _ { i } \\subset \\mathbb { V } _ { i }$ , each with an associated 1-strongly-convex DGF $\\phi _ { i } : \\mathcal { X } _ { i } \\mapsto \\overline { { \\mathbb { R } } }$ w.r.t. $\\| \\cdot \\| _ { i }$ . For each $i \\in [ b ]$ will use $\\| \\cdot \\| _ { i , * }$ to denote the dual norm of $\\| \\cdot \\| _ { i }$ and for any element $\\mathbf { x } \\in \\mathcal { X }$ we will use $\\mathbf { x } _ { i }$ to denote its component in block $i$ and $\\mathbf { x } _ { - i }$ to denote the component across all blocks other than $i$ . Define the functions $\\| \\cdot \\| : \\mathbb { V } \\mapsto \\mathbb { R } _ { + }$ and $\\| \\cdot \\| _ { * } \\mathbb { V } \\mapsto \\mathbb { R } _ { + }$ for any $\\mathbf { x } \\in \\mathbb { V }$ by $\\begin{array} { r } { \\| \\mathbf { x } \\| ^ { 2 } = \\sum _ { i = 1 } ^ { b } \\| \\mathbf { x } _ { i } \\| _ { i } ^ { 2 } } \\end{array}$ and $\\begin{array} { r } { \\| \\mathbf { x } \\| _ { * } ^ { 2 } = \\sum _ { i = 1 } ^ { b } \\| \\mathbf { x } _ { i } \\| _ { i , * } ^ { 2 } } \\end{array}$ , respectively, and the function $\\phi : \\mathcal { X } \\mapsto \\overline { { \\mathbb { R } } }$ for any $\\mathbf { x } \\in \\mathcal { X }$ by $\\begin{array} { r } { \\phi ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { b } \\phi _ { i } ( \\mathbf { x } ) } \\end{array}$ . Finally, for any $n \\in \\mathbb { N }$ we will use $[ n ]$ to denote the set $\\{ 1 , \\ldots , n \\}$ . Setting 1. For some fixed constants $\\gamma _ { i } , L _ { i } > 0$ for each $i \\in [ b ]$ we have the following: ", + "bbox": [ + 173, + 126, + 826, + 277 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1. $f : \\mathcal { X } \\mapsto \\mathbb { R }$ is everywhere-subdifferentiable with minimum $f ^ { * } > - \\infty$ and for all $\\mathbf { x } \\in \\mathcal { X }$ and each $i \\in [ b ]$ the restriction $f ( \\cdot , \\mathbf { x } _ { - i } )$ is $\\gamma _ { i }$ -RWC w.r.t. $\\phi _ { i }$ . \n2. For each $i \\in [ b ]$ there exists a stochastic oracle $G _ { i }$ that for input $\\mathbf { x } \\in \\mathcal { X }$ outputs a random vector $G _ { i } ( \\mathbf { x } , \\boldsymbol { \\xi } )$ s.t. $\\mathbb { E } _ { \\xi } G _ { i } ( \\mathbf { x } , \\xi ) \\in \\partial _ { i } f ( \\mathbf { x } )$ , where $\\partial _ { i } f ( \\mathbf { x } )$ is the subdifferential set of the restriction $f ( \\cdot , \\mathbf { x } _ { - i } )$ at $\\mathbf { x } _ { i }$ . Moreover, $\\mathbb { E } _ { \\xi } \\| G _ { i } ( \\mathbf { x } , \\xi ) \\| _ { i , * } ^ { 2 } \\le L _ { i } ^ { 2 }$ . ", + "bbox": [ + 210, + 287, + 825, + 372 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Define $\\gamma = \\operatorname* { m a x } _ { i \\in [ b ] } \\gamma _ { i }$ and $\\begin{array} { r } { L ^ { 2 } = \\sum _ { i = 1 } ^ { b } L _ { i } ^ { 2 } } \\end{array}$ ", + "bbox": [ + 174, + 383, + 464, + 404 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Claim 1. $\\| \\cdot \\|$ is a norm on $\\mathbb { V }$ . ", + "bbox": [ + 174, + 406, + 377, + 421 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. Positivity and homogeneity are trivial. For the triangle inequality, note that for any $\\lambda \\in [ 0 , 1 ]$ and any $\\mathbf { x } , \\mathbf { y } \\in { \\mathcal { X } }$ we have that ", + "bbox": [ + 173, + 436, + 823, + 465 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c04092cdca1e993a5e1cb4a8c08e17abaef5411926ce19a6a0a4e14a1819569f.jpg", + "text": "$$\n\\begin{array} { r l } { \\| \\lambda \\mathbf { x } + ( 1 - \\lambda ) \\mathbf { y } \\| = \\displaystyle \\sqrt { \\displaystyle \\sum _ { i = 1 } ^ { b } \\| \\lambda \\mathbf { x } _ { i } + ( 1 - \\lambda ) \\mathbf { y } _ { i } \\| _ { i } ^ { 2 } } } & { } \\\\ { \\leq \\sqrt { \\displaystyle \\sum _ { i = 1 } ^ { b } ( \\lambda \\| \\mathbf { x } _ { i } \\| _ { i } + ( 1 - \\lambda ) \\| \\mathbf { y } _ { i } \\| _ { i } ) ^ { 2 } } } & { } \\\\ { \\leq \\lambda \\sqrt { \\displaystyle \\sum _ { i = 1 } ^ { b } \\| \\mathbf { x } _ { i } \\| _ { i } ^ { 2 } } + ( 1 - \\lambda ) \\sqrt { \\displaystyle \\sum _ { i = 1 } ^ { b } \\| \\mathbf { y } _ { i } \\| _ { i } ^ { 2 } } } & { } \\\\ { = \\lambda \\| \\mathbf { x } \\| + ( 1 - \\lambda ) \\| \\mathbf { y } \\| } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 473, + 696, + 643 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the first inequality follows by convexity of the norms $\\| \\cdot \\| _ { i } \\forall i \\in [ b ]$ and the fact that the Euclidean norm on $\\mathbb { R } ^ { b }$ is nondecreasing in each argument, while the second inequality follows by convexity of the Euclidean norm on $\\mathbb { R } ^ { \\bar { b } }$ . Setting $\\begin{array} { r } { \\lambda = \\frac { 1 } { 2 } } \\end{array}$ and multiplying both sides by 2 yields the triangle inequality. □ ", + "bbox": [ + 173, + 647, + 826, + 705 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Algorithm 2: Block-stochastic mirror descent over $\\mathcal { X } = \\times _ { i = 1 } ^ { b } \\mathcal { X } _ { i }$ given associated DGFs $\\phi _ { i } : \\mathcal { X } _ { i } \\mapsto \\overline { { \\mathbb { R } } }$ . ", + "bbox": [ + 171, + 108, + 808, + 143 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Input: initialization $\\mathbf { x } ^ { ( 1 ) } \\in \\mathcal { X }$ , number of steps $T \\geq 1$ , step-size sequence $\\{ \\eta _ { t } \\} _ { t = 1 } ^ { T }$ ", + "bbox": [ + 173, + 147, + 715, + 164 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "r iteration $t \\in [ T ]$ do \nsample $i \\sim \\mathrm { U n i f } [ b ]$ \nset x−i $\\mathbf { x } _ { - i } ^ { ( t + 1 ) } = \\mathbf { x } _ { - i } ^ { ( t ) }$ \nget $\\mathbf { g } = G _ { i } ( \\mathbf { x } ^ { ( t ) } , \\xi _ { t } )$ \nset xi $\\begin{array} { r } { \\mathbf { x } _ { i } ^ { ( t + 1 ) } = \\arg \\operatorname* { m i n } _ { \\mathbf { u } \\in \\mathcal { X } _ { i } } \\eta _ { t } \\langle \\mathbf { g } , \\mathbf { u } \\rangle + D _ { \\phi _ { i } } ( \\mathbf { u } | | \\mathbf { x } _ { i } ^ { ( t ) } ) } \\end{array}$ ", + "bbox": [ + 194, + 165, + 534, + 246 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Claim 2. $\\frac { 1 } { 2 } \\| \\cdot \\| _ { * } ^ { 2 }$ is the convex conjugate of ${ \\frac { 1 } { 2 } } \\| \\cdot \\| ^ { 2 }$ . ", + "bbox": [ + 174, + 300, + 514, + 318 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Consider any $\\mathbf { u } \\in \\mathbb { V }$ . To upper-bound the convex conjugate note that ", + "bbox": [ + 174, + 330, + 673, + 347 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/83c25aff1a1ea644c8624de2bf7e3cccdf092178422f69d475d76fdc2a7558ac.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { s u p } _ { \\mathbf { x } \\in \\mathbb { V } } \\langle \\mathbf { u } , \\mathbf { x } \\rangle - \\frac { \\| \\mathbf { x } \\| ^ { 2 } } { 2 } = \\displaystyle \\operatorname* { s u p } _ { \\mathbf { x } \\in \\mathbb { V } } \\displaystyle \\sum _ { i = 1 } ^ { b } \\langle \\mathbf { u } _ { i } , \\mathbf { x } _ { i } \\rangle - \\frac { \\| \\mathbf { x } _ { i } \\| _ { i } ^ { 2 } } { 2 } } \\\\ { \\displaystyle \\quad \\leq \\operatorname* { s u p } _ { \\mathbf { x } \\in \\mathbb { V } } \\displaystyle \\sum _ { i = 1 } ^ { b } \\| \\mathbf { u } _ { i } \\| _ { i , * } \\| \\mathbf { x } _ { i } \\| _ { i } - \\frac { \\| \\mathbf { x } _ { i } \\| _ { i } ^ { 2 } } { 2 } } \\\\ { \\displaystyle \\quad = \\frac { 1 } { 2 } \\displaystyle \\sum _ { i = 1 } ^ { b } \\| \\mathbf { u } _ { i } \\| _ { i , * } ^ { 2 } } \\\\ { \\displaystyle \\quad = \\frac { \\| \\mathbf { u } \\| _ { * } ^ { 2 } } { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 353, + 673, + 520 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the first inequality follows by definition of a dual norm and the second by maximizing each term w.r.t. $\\| \\mathbf { x } _ { i } \\| _ { i }$ . For the lower bound, pick $\\mathbf { x } \\in \\mathbb { V }$ s.t. $\\langle \\mathbf { u } _ { i } , \\mathbf { x } _ { i } \\rangle = \\| \\mathbf { u } _ { i } \\| _ { i , * } \\| \\mathbf { x } _ { i } \\| _ { i }$ and $\\| \\mathbf { x } _ { i } \\| _ { i } =$ $\\| \\mathbf { u } _ { i } \\| _ { i , * } \\forall i \\in [ b ]$ , which must exist by the definition of a dual norm. Then ", + "bbox": [ + 174, + 523, + 825, + 566 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/804672c22ad35d5d95c79cb195d177ccd21c22c0fc2f7c88ab6ab460614b3178.jpg", + "text": "$$\n\\langle \\mathbf { u } , \\mathbf { x } \\rangle - \\frac { \\| \\mathbf { x } \\| ^ { 2 } } { 2 } = \\sum _ { i = 1 } ^ { b } \\langle \\mathbf { u } _ { i } , \\mathbf { x } _ { i } \\rangle - \\frac { \\| \\mathbf { x } _ { i } \\| _ { i } ^ { 2 } } { 2 } = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { b } \\frac { \\| \\mathbf { u } _ { i } \\| _ { i , * } ^ { 2 } } { 2 } = \\frac { \\| \\mathbf { u } \\| _ { * } ^ { 2 } } { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 281, + 573, + 715, + 617 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "so $\\begin{array} { r } { \\operatorname* { s u p } _ { \\mathbf { x } \\in \\mathbb { V } } \\langle \\mathbf { u } , \\mathbf { x } \\rangle - \\frac { 1 } { 2 } \\| \\mathbf { x } \\| ^ { 2 } \\geq \\frac { 1 } { 2 } \\| \\mathbf { u } \\| _ { * } ^ { 2 } } \\end{array}$ , completing the proof. ", + "bbox": [ + 173, + 622, + 563, + 642 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Theorem 2. Let $\\hat { x }$ be the output of Algorithm 2 after $T$ iterations with non-increasing step-size sequence $\\{ \\eta _ { t } \\} _ { t = 1 } ^ { T }$ . Then under Setting $^ { l }$ , for any $\\hat { \\gamma } > \\gamma$ we have that ", + "bbox": [ + 169, + 102, + 825, + 133 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/3e2c408691e5a4606956e526aece1a38a1f2222cf17c374d9831ef484fbb4a08.jpg", + "text": "$$\n\\mathbb { E } \\Delta _ { \\frac { 1 } { \\hat { \\mathbf { \\xi } } } } ( \\hat { \\mathbf { x } } ) \\leq \\frac { \\hat { \\gamma } b } { \\hat { \\gamma } - \\gamma } \\frac { \\operatorname* { m i n } _ { \\mathbf { u } \\in \\mathcal { X } } f ( \\mathbf { u } ) + \\hat { \\gamma } D _ { \\phi } ( \\mathbf { u } | | \\mathbf { x } ^ { ( 1 ) } ) - f ^ { * } + \\frac { \\hat { \\gamma } L ^ { 2 } } { 2 b } \\sum _ { t = 1 } ^ { T } \\eta _ { t } ^ { 2 } } { \\sum _ { t = 1 } ^ { T } \\eta _ { t } }\n$$", + "text_format": "latex", + "bbox": [ + 258, + 138, + 738, + 183 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the expectation is w.r.t. $\\xi _ { t }$ and the randomness of the algorithm. ", + "bbox": [ + 176, + 186, + 633, + 202 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. Define transforms be a stochastic oracle tha $\\mathbf { U } _ { i } , i \\in \\ [ b ]$ $\\mathbf { U } _ { i } ^ { T } \\mathbf { x } = \\mathbf { x } _ { i }$ $\\begin{array} { r } { \\mathbf { x } = \\sum _ { i = 1 } ^ { b } \\mathbf { U } _ { i } \\mathbf { x } _ { i } \\ \\forall \\textbf { x } \\in \\mathcal { X } } \\end{array}$ . Let impli $G$ $\\textbf { x } \\in { \\mathcal { X } }$ $G ( { \\bf x } , i , \\xi ) = b { \\bf U } _ { i } G _ { i } ( { \\bf x } , \\xi )$ $\\begin{array} { r } { \\mathbb { E } _ { i , \\xi } G ( \\mathbf { x } , i , \\xi ) = \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } b \\mathbf { U } _ { i } \\mathbb { E } _ { \\xi } G _ { i } ( \\mathbf { x } , \\xi ) \\in \\sum _ { i = 1 } ^ { b } \\mathbf { U } _ { i } \\partial _ { i } f ( \\mathbf { x } ) = \\partial f ( \\mathbf { x } ) } \\end{array}$ and $\\mathbb { E } _ { i , \\xi } \\Vert G ( \\mathbf { x } , i , \\xi ) \\Vert _ { * } ^ { 2 } =$ $\\begin{array} { r } { \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } b ^ { 2 } \\mathbb { E } _ { \\xi } \\| \\mathbf { U } _ { i } G _ { i } ( \\mathbf { x } , \\boldsymbol { \\xi } ) \\| _ { i , * } ^ { 2 } \\leq b \\sum _ { i = 1 } ^ { b } L _ { i } ^ { 2 } = b L ^ { 2 } . \\mathrm { \\bf { T } } } \\end{array}$ hen ", + "bbox": [ + 174, + 217, + 825, + 285 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/070aa552d5e2831a317e3305ef3b56dfc0943f49d02863fcb016a81d0d2c9e57.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { x } ^ { ( t + 1 ) } = \\mathbf { U } _ { i } \\left[ \\underset { \\mathbf { u } \\in \\mathcal { X } _ { i } } { \\arg \\operatorname* { m i n } } \\eta _ { t } \\langle g , u \\rangle + D _ { \\phi _ { i } } ( \\mathbf { u } | | \\mathbf { x } _ { i } ^ { ( t ) } ) \\right] } \\\\ & { \\quad \\quad \\quad = \\mathbf { U } _ { i } \\mathbf { U } _ { i } ^ { T } \\left[ \\underset { \\mathbf { u } \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\eta _ { t } \\langle \\mathbf { U } _ { i } G _ { i } ( \\mathbf { x } ^ { ( t ) } , \\xi _ { t } ) , \\mathbf { u } \\rangle + \\underset { i = 1 } { \\overset { b } { \\sum } } D _ { \\phi _ { i } } ( \\mathbf { u } _ { i } | | \\mathbf { x } _ { i } ^ { ( t ) } ) \\right] } \\\\ & { \\quad \\quad \\quad = \\underset { \\mathbf { u } \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\frac { \\eta _ { t } } { b } \\langle G ( \\mathbf { x } , i , \\xi _ { t } ) , \\mathbf { u } \\rangle + D _ { \\phi } ( \\mathbf { u } | | \\mathbf { x } ^ { ( t ) } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 291, + 732, + 404 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Thus Algorithm 2 is equivalent to Zhang & He (2018, Algorithm 1) with stochastic oracle $G ( \\mathbf { x } , i , \\xi )$ , step-size sequence $\\{ \\eta _ { t } / b \\} _ { t = 1 } ^ { T }$ , and no regularizer. Note that $\\phi$ is 1-strongly-convex w.r.t. $\\| \\cdot \\|$ and $f$ is $\\gamma$ -RWC w.r.t. $\\phi$ , so in light of Claims 1 and 2 our setup satisfies Assumption 3.1 of Zhang & He (2018). The result then follows from Theorem 3.1 of the same. □ ", + "bbox": [ + 174, + 409, + 825, + 467 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Corollary 1. Under Setting $^ { l }$ let ˆx be the output of Algorithm 2 with constant step-size $\\eta _ { t } ~ =$ q 2b(f(1)−f∗)γL2T ∀ t ∈ [T ], where f (1) = f (x(1)). Then we have ", + "bbox": [ + 173, + 473, + 826, + 511 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/76c89a24f42e69932dc7143e58b2ac512fa359d39b4fef87e712ac1268659bfd.jpg", + "text": "$$\n\\mathbb { E } \\Delta _ { \\frac { 1 } { 2 \\gamma } } \\big ( \\hat { \\mathbf { x } } \\big ) \\leq 2 L \\sqrt { \\frac { 2 b \\gamma \\big ( f ^ { ( 1 ) } - f ^ { * } \\big ) } { T } }\n$$", + "text_format": "latex", + "bbox": [ + 382, + 520, + 614, + 554 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the expectation is w.r.t. $\\xi _ { t }$ and the randomness of the algorithm. Equivalently, we can reach a point ˆx satisfying $\\begin{array} { r } { \\mathbb { E } \\Delta _ { \\frac { 1 } { 2 \\gamma } } ( \\hat { \\mathbf { x } } ) \\leq \\varepsilon i n \\frac { 8 \\gamma b L ^ { 2 } ( f ^ { ( 1 ) } - f ^ { * } ) } { \\varepsilon ^ { 2 } } } \\end{array}$ stochastic oracle calls. ", + "bbox": [ + 173, + 560, + 825, + 597 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.3 A SINGLE-LEVEL ANALYSIS OF ENAS AND DARTS", + "bbox": [ + 173, + 103, + 583, + 117 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In this section we apply our analysis to understanding two existing NAS algorithms, ENAS (Pham et al., 2018) and DARTS (Liu et al., 2019). For simplicity, we assume objectives induced by architectures in the relaxed search space are $\\gamma$ -smooth, which excludes components such as ReLU. However, such cases can be smoothed via Gaussian convolution, i.e. adding noise to every gradient; thus given the noisiness of SGD training we believe the following analysis is still informative (Kleinberg et al., 2018). ", + "bbox": [ + 174, + 128, + 826, + 213 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ENAS continuously relaxes $\\mathcal { A }$ via a neural controller that samples architectures $a \\in { \\mathcal { A } }$ , so $\\Theta =$ $\\mathbb { R } ^ { \\mathcal { O } ( h ^ { 2 } ) }$ , where $h$ is the number of hidden units. The controller is trained with Monte Carlo gradients. On the other hand, first-order DARTS uses a mixture relaxation similar to the one in Section A but using a softmax instead of constraining parameters to the simplex. Thus $\\Theta = \\mathbb { R } ^ { | E | \\times | O | }$ for $E$ the set of learnable edges and $O$ the set of possible operations. If we assume that both algorithms use SGD for the architecture parameters then to compare them we are interested in their respective values of $G _ { \\theta }$ , which we will refer to as $G _ { \\mathrm { E N A S } }$ and $G _ { \\mathrm { D A R T S } }$ . Before proceeding, we note again that our theory holds only for the single-level objective and when using SGD as the architecture optimizer, whereas both algorithms specify the bilevel objective and Adam (Kingma & Ba, 2015), respectively. ", + "bbox": [ + 173, + 219, + 825, + 348 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "At a very high level, the Monte Carlo gradients used by ENAS are known to be high-variance, so $G _ { \\mathrm { E N A S } }$ may be much larger than $G _ { \\mathrm { D A R T S } }$ , yielding faster convergence for DARTS, which is reflected in practice (Liu et al., 2019). We can also do a simple low-level analysis under the assumption that all architecture gradients are bounded entry-wise, i.e. in $\\ell _ { \\infty }$ -norm, by some constant; then since the squared $\\ell _ { 2 }$ -norm is bounded by the product of the dimension and the squared $\\ell _ { \\infty }$ -norm we have $G _ { \\mathrm { E N A S } } ^ { 2 } = \\mathcal { O } ( h ^ { 2 } )$ while $G _ { \\mathrm { D A R T S } } ^ { 2 } = \\mathcal { O } ( | \\bar { E } | | { \\cal O } | )$ . Since ENAS uses a hidden state size of $h = 1 0 0$ and the DARTS search space has $| E | = 1 4$ edges and $| O | = 7$ operations, this also points to DARTS needing fewer iterations to converge. ", + "bbox": [ + 173, + 354, + 825, + 467 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 419, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We provide additional detail on the experimental setup and hyperparameter settings used for each benchmark studied in Section 4. We also provide a more detailed discussion of how XNAS differs from GAEA, along with empirical results for XNAS on the NAS-Bench-201 benchmark. ", + "bbox": [ + 174, + 133, + 825, + 175 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C.1 DARTS SEARCH SPACE ", + "text_level": 1, + "bbox": [ + 174, + 191, + 385, + 207 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We consider the same search space as DARTS (Liu et al., 2019), which has become one of the standard search spaces for CNN cell search (Xie et al., 2019; Nayman et al., 2019; Chen et al., 2019; Noy et al., 2019; Liang et al., 2019). Following the evaluation procedure used in Liu et al. (2019) and Xu et al. (2020) , our evaluation of GAEA PC-DARTS consists of three stages: ", + "bbox": [ + 174, + 218, + 825, + 273 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Stage 1: In the search phase, we run GAEA PC-DARTS with 5 random seeds to reduce variance from different initialization of the shared-weights network.3 \n• Stage 2: We evaluate the best architecture identified by each search run by training from scratch. \n• Stage 3: We perform a more thorough evaluation of the best architecture from stage 2 by training with ten different random seed initializations. ", + "bbox": [ + 174, + 285, + 825, + 361 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For completeness, we describe the convolutional neural network search space considered. A cell consists of 2 input nodes and 4 intermediate nodes for a total of 6 nodes. The nodes are ordered and subsequent nodes can receive the output of prior nodes as input so for a given node $k$ , there are $k - 1$ possible input edges to node $k$ . Therefore, there are a total of $2 + 3 + 4 + 5 = 1 4$ edges in the weight-sharing network. ", + "bbox": [ + 173, + 372, + 825, + 441 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "An architecture is defined by selecting 2 input edges per intermediate node and also selecting a single operation per edge from the following 8 operations: (1) $3 \\times 3$ separable convolution, (2) $5 \\times 5$ separable convolution, (3) $3 \\times 3$ dilated convolution, (4) $5 \\times 5$ dilated convolution, (5) max pooling, (6) average pooling, (7) identity (8) zero. We use the same search space to design a “normal” cell and a “reduction” cell; the normal cells have stride 1 operations that do not change the dimension of the input, while the reduction cells have stride 2 operations that half the length and width dimensions of the input. In the experiments, for both cell types, , after which the output of all intermediate nodes are concatenated to form the output of the cell. ", + "bbox": [ + 173, + 448, + 825, + 559 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C.1.1 STAGE 1: ARCHITECTURE SEARCH ", + "text_level": 1, + "bbox": [ + 176, + 574, + 473, + 589 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For stage 1, as is done by DARTS and PC-DARTS, we use GAEA PC-DARTS to update architecture parameters for a smaller shared-weights network in the search phase with 8 layers and 16 initial channels. All hyperparameters for training the weight-sharing network are the same as that used by PC-DARTS: ", + "bbox": [ + 173, + 598, + 825, + 654 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "train: scheduler: cosine lr_anneal_cycles: 1 smooth_cross_entropy: false batch_size: 256 learning_rate: 0.1 learning_rate_min: 0.0 momentum: 0.9 weight_decay: 0.0003 init_channels: 16 layers: 8 autoaugment: false cutout: false auxiliary: false drop_path_prob: 0 grad_clip: 5 ", + "bbox": [ + 176, + 667, + 480, + 893 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For GAEA PC-DARTS, we initialize the architecture parameters with equal weight across all options (equal weight across all operations per edge and equal weight across all input edges per node). Then, we train the shared-weights network for 10 epochs without performing any architecture updates to warmup the weights. Then, we use a learning rate of 0.1 for the exponentiated gradient update for GAEA PC-DARTS. ", + "bbox": [ + 173, + 103, + 825, + 172 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.1.2 STAGE 2 AND 3: ARCHITECTURE EVALUATION ", + "text_level": 1, + "bbox": [ + 174, + 189, + 555, + 204 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For stages 2 and 3, we train each architecture for 600 epochs with the same hyperparameter settings as PC-DARTS: ", + "bbox": [ + 176, + 213, + 823, + 241 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "train: scheduler: cosine lr_anneal_cycles: 1 smooth_cross_entropy: false batch_size: 128 learning_rate: 0.025 learning_rate_min: 0.0 momentum: 0.9 weight_decay: 0.0003 init_channels: 36 layers: 20 autoaugment: false cutout: true cutout_length: 16 auxiliary: true auxiliary_weight: 0.4 drop_path_prob: 0.3 grad_clip: 5 ", + "bbox": [ + 176, + 256, + 480, + 503 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.1.3 BROAD REPRODUCIBILITY ", + "text_level": 1, + "bbox": [ + 176, + 518, + 415, + 532 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Our ‘broad reproducibility’ results in Table 3 show the final stage 3 evaluation performance of GAEA PC-DARTS for 2 additional sets of random seeds from stage 1 search. The performance of GAEA PC-DARTS for one set is similar to that reported in Table 1, while the other is on par with the performance reported for PC-DARTS in Xu et al. (2020). We do observe non-negligible variance in the performance of the architecture found by different random seed initializations of the shared-weights network, necessitating running multiple searches before selecting an architecture. We also found that it was possible to identify and eliminate poor performing architectures in just 20 epochs of training during stage 2 intermediate evaluation, thereby reducing the total training cost by over $7 5 \\%$ (we only trained 3 out of 10 architectures for the entire 600 epochs). ", + "bbox": [ + 173, + 542, + 826, + 667 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/793f07b1b6970a1cc0cd4a37b8b093cbbbe52f440c45bf771c47941791617714.jpg", + "table_caption": [ + "Stage 3 Evaluation ", + "Table 3: GAEA PC-DARTS Stage 3 Evaluation for 3 sets of random seeds. " + ], + "table_footnote": [], + "table_body": "
Set 1 (Reported)Set2Set3
2.50 ± 0.072.50±0.092.60±0.09
", + "bbox": [ + 334, + 695, + 663, + 726 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We depict the top architectures found by GAEA PC-DARTS for CIFAR-10 and ImageNet in Figure 3 and detailed results in Tables 4 and 5. ", + "bbox": [ + 174, + 765, + 823, + 795 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/ce6d48674ef443b3731e351bae3f145e958b8559063ac5fc11eb15040c8f7056.jpg", + "table_caption": [], + "table_footnote": [ + "We show results for networks with a comparable number of parameters. For fair comparison to other work, we show the search cost for training the shared-weights network with a single initialization. ‡ Search space and backbone architecture (PyramidNet) differ from the DARTS setting. PDARTS results not reported for multiple seeds. Additionally, PDARTS uses deeper weight-sharing networks during search, on which PC-DARTS has also been shown to improve performance (Xu et al., 2020), so we GAEA PC-DARTS to further improve if modified similarly. " + ], + "table_body": "
ArchitectureTest Error AverageParams (M)Search Cost (GPU Days)Comparable Search SpaceSearch Method
Shake-Shake(DeVries&Taylor,2017)Best N/A2.5626.2--manual
PyramidNet (Yamada et al.,2018)2.31N/A261-manual
NASNet-A* (Zoph et al.,2018)N/A2.653.32000NRL
AmoebaNet-B*(Real et al.,2019)N/A2.55± 0.052.83150Nevolution
ProxylessNAS* (Cai et al.,2019)2.08N/A5.74Ngradient
ENAS (Pham et al.,2018)2.89N/A4.60.5YRL
Random search Wst (Li& Talwalkar,2019)2.712.85 ±0.083.80.7Yrandom
ASNG-NAS (Akimoto et al.,2019)N/A2.83 ± 0.143.90.1Ygradient
SNAS (Xie et al.,2019)N/A2.85 ±0.022.81.5Ygradient
DARTS (lst-order)†(Liu et al.,2019)N/A3.00 ±0.143.30.4Ygradient
DARTS (2nd-order)†(Liu et al.,2019)N/A2.76±0.093.31Ygradient
PDARTS# (Chen et al., 2019)2.50N/A3.40.3Ygradient
PC-DARTS† (Xu et al.,2020)N/A2.57 ±0.073.60.1Ygradient
GAEA PC-DARTS† (Ours)2.392.50±0.063.70.1Ygradient
", + "bbox": [ + 179, + 101, + 818, + 273 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 4: DARTS (CIFAR-10): Comparison with manually designed networks and those found by SOTA NAS methods, mainly on the DARTS search space (Liu et al., 2019). Results grouped by the type of search method: manually designed, full-evaluation NAS, and weight-sharing NAS. All test errors are for models trained with auxiliary towers and cutout (parameter counts exclude auxiliary weights). Test errors we report are averaged over 10 seeds. “-” indicates that the field does not apply while “N/A” indicates unknown. Note that search cost is hardware-dependent; our results used Tesla V100 GPUs. ", + "bbox": [ + 179, + 340, + 818, + 438 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/399412b5470967178407c4e9ade3226f37522cb29c885d543e2e8928b5febc3f.jpg", + "table_caption": [], + "table_footnote": [ + "∗,† ,‡ ,# See notes in Table 4. " + ], + "table_body": "
Test ErrorParams (M)Search Cost (GPU Days)Search Method
ArchitectureSourcetop-1top-5
MobileNet(Howard et al.,2017)29.410.54.2-manual
ShuffleNet V2 2x(Ma et al.,2018)25.1N/A~5-manual
NASNet-A*(Zoph et al.,2018)26.08.45.31800RL
AmoebaNet-C*(Real et al.,2019)24.37.66.43150evolution
DARTS(Liu et al.,2019)26.78.74.74.0gradient
SNAS(Xie et al.,2019)27.39.24.31.5gradient
ProxylessNAS‡(Cai et al.,2019)24.97.57.18.3gradient
PDARTS#(Chen et al., 2019)24.47.44.90.3gradient
PC-DARTS (CIFAR-10)+(Xu et al.,2020)25.17.85.30.1gradient
PC-DARTS (ImageNet)t(Xu et al.,2020)24.27.35.33.8gradient
GAEA PC-DARTS (CIFAR-10)†Ours24.37.35.30.1gradient
GAEA PC-DARTS (ImageNet)†Ours24.07.35.63.8gradient
", + "bbox": [ + 202, + 527, + 794, + 680 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 5: DARTS (ImageNet): Comparison with manually designed networks and those found by SOTA NAS methods, mainly on the DARTS search space (Liu et al., 2019). Results are grouped by the type of search method: manually designed, full-evaluation NAS, and weight-sharing NAS. All test errors are for models trained with auxiliary towers and cutout but no other modifications, e.g. label smoothing (Muller et al., 2019), AutoAugment (Cubuk ¨ et al., 2019), Swish (Ramachandran et al., 2017), squeeze and excite modules (Hu et al., 2018), etc. “-” indicates that the field does not apply while “N/A” indicates unknown. Note that search cost is hardware-dependent; our results used Tesla V100 GPUs. ", + "bbox": [ + 204, + 695, + 794, + 808 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/63ae622cf29eb0f9ab780252106da6d0cf794f22106cbaf1aa1ef10d1b7f747b.jpg", + "image_caption": [ + "(a) CIFAR-10: Normal Cell " + ], + "image_footnote": [], + "bbox": [ + 187, + 121, + 810, + 251 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/3569ea92af070f0ff5181fc6a9eb008f2cb5fae43d928684d613531cc3d4a3c7.jpg", + "image_caption": [ + "(b) CIFAR-10: Reduction Cell " + ], + "image_footnote": [], + "bbox": [ + 186, + 309, + 812, + 383 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/148a34e424faf3bd2389b857129d4478f7e86f99095c5089b79873d477d6b55d.jpg", + "image_caption": [ + "(c) ImageNet: Normal Cell " + ], + "image_footnote": [], + "bbox": [ + 187, + 435, + 810, + 551 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/97ad583bc5680581ac1635958f01f41737c5cabe214a5c5fd5f49240ac4f5fba.jpg", + "image_caption": [ + "Figure 3: The best normal and reduction cells found by GAEA PC-DARTS on CIFAR-10 (top) and ImageNet (bottom). " + ], + "image_footnote": [], + "bbox": [ + 187, + 609, + 810, + 780 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/a9695b181215ef2198739bbcc3c51586579237fa3521e3b6eb9418b8493325f8.jpg", + "image_caption": [ + "Figure 4: NAS-Bench-201: Learning Curves. Evolution over search phase epochs of the best architecture according to the NAS method. DARTS (first-order) converges to nearly all skip connections while GAEA is able to suppress overfitting to the mixture relaxation by encouraging sparsity in operation weights. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 812, + 492 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.2 NAS-BENCH-1SHOT1 ", + "text_level": 1, + "bbox": [ + 176, + 599, + 369, + 614 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "The NAS-Bench-1Shot1 benchmark (Zela et al., 2020a) contains 3 different search spaces that are subsets of the NAS-Bench-101 search space. The search spaces differ in the number of nodes and the number of input edges selected per node. We refer the reader to (Zela et al., 2020a) for details about each individual search space. ", + "bbox": [ + 174, + 631, + 825, + 686 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Of the NAS methods evaluated in Zela et al. (2020a), PC-DARTS had the most robust performance across the three search spaces and converged to the best architecture in search spaces 1 and 3. GDAS, a probabilistic gradient NAS method, achieved the best performance on search space 2. Hence, we focused on applying a geometry-aware approach to PC-DARTS. We implemented GAEA PCDARTS within the repository provided by the authors of Zela et al. (2020a) available at https: //github.com/automl/nasbench-1shot1. We used the same hyperparameter settings for training the weight-sharing network as that used by Zela et al. (2020a) for PC-DARTS. Similar to the previous benchmark, we initialize architecture parameters to allocate equal weight to all options. For the architecture updates, the only hyperparameter for GAEA PC-DARTS is the learning rate for exponentiated gradient, which we set to 0.1. ", + "bbox": [ + 173, + 694, + 826, + 833 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "As mentioned in Section 4, the search spaces considered in this benchmark differ in that operations are applied after aggregating all edge inputs to a node instead of per edge input as in the DARTS and NAS-Bench-201 search spaces. This structure inherently limits the size of the weight-sharing network to scale with the number of nodes instead of the number of edges $( \\mathcal { O } ( | \\mathrm { n o d e s } | ^ { \\bar { 2 } } ) ;$ , thereby limiting the degree of overparameterization. Understanding the impact of overparameterization on the performance of weight-sharing NAS methods is a direction for future study. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.3 NAS-BENCH-201 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 339, + 117 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "The NAS-Bench-201 benchmark (Dong & Yang, 2020) evaluates a single search space across 3 datasets: CIFAR-10, CIFAR-100, and a miniature version of ImageNet (ImageNet-16-120). ImageNet-16-120 is a downsampled version of ImageNet with $1 6 \\times 1 6$ images and 120 classes for a total of $1 5 1 . 7 \\mathrm { k }$ training images, 3k validation images, and 3k test images. The authors of Dong & Yang (2020) evaluated the architecture search performance of multiple weight-sharing methods and traditional hyperparameter optimization methods on all three datasets. According to the results from Dong & Yang (2020), GDAS outperformed other weight-sharing methods by a large margin. Hence, we first evaluated the performance of GAEA GDAS on each of the three datasets. Our implementation of GAEA GDAS uses an architecture learning rate of 0.1, which matches the learning rate used for GAEA approaches in the previous two benchmarks. Additionally, we run GAEA GDAS for 150 epochs instead of 250 epochs used for GDAS in the original benchmarked results; this is why the search cost is lower for GAEA GDAS. All other hyperparameter settings are the same. Our results for GAEA GDAS is comparable to the reported results for GDAS on CIFAR-10 and CIFAR-100 but slightly lower on ImageNet-16-120. Compared to our reproduced results for GDAS, GAEA GDAS outperforms GDAS on CIFAR-100 and matches it on CIFAR-10 and ImageNet-16-120. ", + "bbox": [ + 173, + 130, + 825, + 337 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Next, to see if we can use GAEA to further improve the performance of weight-sharing methods, we evaluated GAEA DARTS (first order) applied to both the single-level (ERM) and bi-level optimization problems. Again, we used a learning rate of 0.1 and trained GAEA DARTS for 25 epochs on each dataset. The one additional modification we made was to exclude the zero operation, which limits GAEA DARTS to a subset of the search space. To isolate the impact of this modification, we also evaluated first-order DARTS with this modification. Similar to (Dong & Yang, 2020), we observe DARTS with this modification to also converge to architectures with nearly all skip connections, resulting in similar performance as that reported in Dong & Yang (2020). We present the learning curves of the oracle architecture recommended by DARTS and GAEA DARTS (when excluding zero operation) over the training horizon for 4 different runs in Figure 4. For GAEA GDAS and GAEA DARTS, we train the weight-sharing network with the following hyperparameters: ", + "bbox": [ + 174, + 344, + 825, + 498 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "train: scheduler: cosine lr_anneal_cycles: 1 smooth_cross_entropy: false batch_size: 64 learning_rate: 0.025 learning_rate_min: 0.001 momentum: 0.9 weight_decay: 0.0003 init_channels: 16 layers: 5 autoaugment: false cutout: false auxiliary: false auxiliary_weight: 0.4 drop_path_prob: 0 grad_clip: 5 ", + "bbox": [ + 176, + 511, + 480, + 746 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Surprisingly, we observe single-level optimization to yield better performance than solving the bilevel problem with GAEA DARTS on this search space. In fact, the performance of GAEA DARTS (ERM) not only exceeds that of GDAS, but also outperforms traditional hyperparameter optimization approaches on all three datasets, nearly reaching the optimal accuracy on all three datasets. In contrast, GAEA DARTS (bi-level) outperforms GDAS on CIFAR-100 and ImageNet-16-120 but underperforms slightly on CIFAR-10. The single-level results on this benchmark provides concrete support for our convergence analysis, which only applies to the ERM problem. As noted in Section 4, the search space considered in this benchmark differs from the prior two in that there is no subsequent edge pruning. Additionally, the search space is fairly small with only 3 nodes for which architecture decisions must be made. The success of GAEA DARTS (ERM) on this benchmark indicate the need for a better understanding of when single-level optimization should be used in favor of the default bi-level optimization problem and how the search space impacts the decision. ", + "bbox": [ + 173, + 757, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/80d0caa75d9919dc7f9a33b41a5523e5c17f1bbf9e12492fe2d2427fdc69679c.jpg", + "image_caption": [ + "Figure 5: NAS-Bench-201: XNAS Learning Curves. Evolution over search phase epochs of the best architecture according 4 runs of XNAS. XNAS exhibits the same behavior as DARTS and converges to nearly all skip connections. " + ], + "image_footnote": [], + "bbox": [ + 178, + 103, + 812, + 222 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "C.4 COMPARISON TO XNAS ", + "text_level": 1, + "bbox": [ + 174, + 300, + 387, + 315 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "As discussed in Section 1, XNAS is similar to GAEA in that it uses an exponentiated gradient update but is motivated from a regret minimization perspective. Nayman et al. (2019) provides regret bounds for XNAS relative to the observed sequence of validation losses, however, this is not equivalent to the regret relative to the best architecture in the search space, which would have generated a different sequence of validation losses. ", + "bbox": [ + 173, + 327, + 825, + 396 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "XNAS also differs in its implementation in two ways: (1) a wipeout routine is used to zero out operations that cannot recover to exceed the current best operation within the remaining number of iterations and (2) architecture gradient clipping is applied per data point before aggregating to form the update. These differences are motivated from the regret analysis and meaningfully increase the complexity of the algorithm. Unfortunately, the authors do not provide the code for architecture search in their code release at https://github.com/NivNayman/XNAS. Nonetheless, we implemented XNAS for the NAS-Bench-201 search space to provide a point of comparison to GAEA. ", + "bbox": [ + 173, + 402, + 825, + 501 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Our results shown in Figure 5 demonstrate that XNAS exhibits much of the same behavior as DARTS in that the operations all converge to skip connections. We hypothesize that this is due to the gradient clipping, which obscures the signal kept by GAEA in favor of convolutional operations. 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Methods", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "such as DARTS (Liu et al., 2019), GDAS (Dong & Yang, 2019), and many others (Pham et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "2018; Zheng et al., 2019; Yang et al., 2020; Xie et al., 2019; Liu et al., 2018; Laube & Zell, 2019;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Cai et al., 2019; Akimoto et al., 2019; Xu et al., 2020) combine weight-sharing with a continuous", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "relaxation of the discrete search space to allow cheap gradient updates, enabling the use of popular", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "optimizers. 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While many regularization", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "approaches have been implicitly proposed in recent NAS efforts, we focus instead on the question", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "of optimizing architecture parameters, which may not be amenable to standard procedures such as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "SGD that work well for standard neural network weights. 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Theoretically, we draw", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "upon the mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) to give", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "convergence guarantees when using any of a broad class of such geometry-aware gradient methods", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "to optimize the weight-sharing objective; empirically, we show that our solution leads to strong", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 453, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 453, + 733 + ], + "score": 1.0, + "content": "improvements on several NAS benchmarks. 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We argue for the study of single-level empirical risk minimization to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 256, + 470, + 269 + ], + "spans": [ + { + "bbox": [ + 142, + 256, + 470, + 269 + ], + "score": 1.0, + "content": "understand NAS with weight-sharing, reducing the design of NAS methods to de-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 267, + 470, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 280 + ], + "score": 1.0, + "content": "vising optimizers and regularizers that can quickly obtain high-quality solutions to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 278, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 292 + ], + "score": 1.0, + "content": "this problem. 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Notably, we exceed the best published results", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "score": 1.0, + "content": "for both CIFAR and ImageNet on both the DARTS search space and NAS-Bench-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 344, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 470, + 356 + ], + "score": 1.0, + "content": "201; on the latter we achieve near-oracle-optimal performance on CIFAR-10 and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 354, + 469, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 369 + ], + "score": 1.0, + "content": "CIFAR-100. Together, our theory and experiments demonstrate a principled way", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 365, + 471, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 471, + 379 + ], + "score": 1.0, + "content": "to co-design optimizers and continuous relaxations of discrete NAS search spaces.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 213, + 471, + 379 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 397, + 206, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 208, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 208, + 413 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "Neural architecture search has become an important tool for automating machine learning (ML) but", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "can require hundreds of thousands of GPU-hours to train. Recently, weight-sharing approaches have", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "achieved state-of-the-art performance while drastically reducing the computational cost of NAS to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "just that of training a single shared-weights network (Pham et al., 2018; Liu et al., 2019). Methods", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "such as DARTS (Liu et al., 2019), GDAS (Dong & Yang, 2019), and many others (Pham et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "2018; Zheng et al., 2019; Yang et al., 2020; Xie et al., 2019; Liu et al., 2018; Laube & Zell, 2019;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Cai et al., 2019; Akimoto et al., 2019; Xu et al., 2020) combine weight-sharing with a continuous", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "relaxation of the discrete search space to allow cheap gradient updates, enabling the use of popular", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "optimizers. However, despite some empirical success, weight-sharing remains poorly understood", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "and has received criticism due to (1) rank-disorder (Yu et al., 2020; Zela et al., 2020b; Zhang et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "2020; Pourchot et al., 2020), where the shared-weights performance is a poor surrogate of standalone", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 539, + 499, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 499, + 552 + ], + "score": 1.0, + "content": "performance, and (2) poor results on recent benchmarks (Dong & Yang, 2020; Zela et al., 2020a).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 418, + 506, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "Motivated by the challenge of developing simple and efficient methods that achieve state-of-the-art", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "score": 1.0, + "content": "performance, we study how to best handle the goals and optimization objectives of NAS. We start", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 592 + ], + "score": 1.0, + "content": "by observing that weight-sharing subsumes architecture hyperparameters as another set of learned", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 507, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 507, + 602 + ], + "score": 1.0, + "content": "parameters of the shared-weights network, in effect extending the class of functions being learned.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "This suggests that a reasonable approach towards obtaining high-quality NAS solutions is to study", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "how to regularize and optimize the empirical risk over this extended class. While many regularization", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "approaches have been implicitly proposed in recent NAS efforts, we focus instead on the question", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "of optimizing architecture parameters, which may not be amenable to standard procedures such as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "SGD that work well for standard neural network weights. In particular, to better-satisfy desirable", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "properties such as generalization and sparsity of architectural decisions, we propose to constrain", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "architecture parameters to the simplex and update them using exponentiated gradient, which has", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "favorable convergence properties due to the underlying problem structure. Theoretically, we draw", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "upon the mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) to give", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "convergence guarantees when using any of a broad class of such geometry-aware gradient methods", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "to optimize the weight-sharing objective; empirically, we show that our solution leads to strong", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 453, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 453, + 733 + ], + "score": 1.0, + "content": "improvements on several NAS benchmarks. We summarize these contributions below:", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 556, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "1. We argue for studying NAS with weight-sharing as a single-level objective over a structured", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 118, + 93, + 507, + 106 + ], + "spans": [ + { + "bbox": [ + 118, + 93, + 507, + 106 + ], + "score": 1.0, + "content": "function class in which architectural decisions are treated as learned parameters rather than hyper-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 117, + 105, + 507, + 117 + ], + "spans": [ + { + "bbox": [ + 117, + 105, + 507, + 117 + ], + "score": 1.0, + "content": "parameters. Our setup clarifies recent concerns about rank disorder and makes clear that proper", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 118, + 116, + 495, + 128 + ], + "spans": [ + { + "bbox": [ + 118, + 116, + 495, + 128 + ], + "score": 1.0, + "content": "regularization and optimization of this objective is critical to obtaining high-quality solutions.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "score": 1.0, + "content": "2. Focusing on optimization, we propose to improve existing NAS algorithms by re-parameterizing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 118, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 118, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "architecture parameters over the simplex and updating them using exponentiated gradient, a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 117, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 117, + 149, + 505, + 164 + ], + "score": 1.0, + "content": "variant of mirror descent that converges quickly over this domain and enjoys favorable sparsity", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 117, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 117, + 161, + 506, + 174 + ], + "score": 1.0, + "content": "properties. This simple modification—which we call the Geometry-Aware Exponentiated", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 117, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 117, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "Algorithm (GAEA)—is easily applicable to numerous methods, including first-order DARTS", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 117, + 183, + 444, + 196 + ], + "spans": [ + { + "bbox": [ + 117, + 183, + 444, + 196 + ], + "score": 1.0, + "content": "Liu et al. (2019), GDAS Dong & Yang (2019), and PC-DARTS (Xu et al., 2020).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "3. To show correctness and efficiency of our scheme, we prove polynomial-time stationary-point", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 118, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 118, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "convergence of block-stochastic mirror descent—a family of geometry-aware gradient algorithms", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 117, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 117, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "that includes GAEA—over a continuous relaxation of the single-level NAS objective. To the best", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 118, + 229, + 504, + 242 + ], + "spans": [ + { + "bbox": [ + 118, + 229, + 504, + 242 + ], + "score": 1.0, + "content": "of our knowledge these are the first finite-time convergence guarantees for gradient-based NAS.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "4. We demonstrate that GAEA improves upon state-of-the-art methods on three of the latest NAS", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 118, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "benchmarks for computer vision. Specifically, we beat the current best results on NAS-Bench-201", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 118, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 118, + 264, + 221, + 277 + ], + "score": 1.0, + "content": "(Dong & Yang, 2020) by", + "type": "text" + }, + { + "bbox": [ + 221, + 265, + 249, + 276 + ], + "score": 0.87, + "content": "0 . 1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 264, + 310, + 277 + ], + "score": 1.0, + "content": "on CIFAR-10,", + "type": "text" + }, + { + "bbox": [ + 310, + 265, + 337, + 276 + ], + "score": 0.87, + "content": "1 . 5 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 264, + 420, + 277 + ], + "score": 1.0, + "content": "on CIFAR-100, and", + "type": "text" + }, + { + "bbox": [ + 420, + 265, + 447, + 275 + ], + "score": 0.87, + "content": "0 . 8 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "on ImageNet-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 118, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 118, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "16-120; we also outperform the state-of-the-art on the DARTS search space Liu et al. (2019), for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 118, + 286, + 475, + 299 + ], + "spans": [ + { + "bbox": [ + 118, + 286, + 475, + 299 + ], + "score": 1.0, + "content": "both CIFAR-10 and ImageNet, and match it on NAS-Bench-1Shot1 (Zela et al., 2020a).1", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "Related Work. Most optimization analyses of NAS show monotonic improvement (Akimoto et al.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "2019), asymptotic guarantees (Yao et al., 2020), or bounds on auxiliary quantities disconnected", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "from any objective (Noy et al., 2019; Nayman et al., 2019; Carlucci et al., 2019). In contrast, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "prove polynomial-time stationary-point convergence on a single-level objective for weight-sharing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "NAS, so far only studied empirically (Xie et al., 2019; Li et al., 2019). Our results draw upon the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) and extend", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "recent nonconvex convergence results Zhang & He (2018) to handle alternating descent. While there", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "exist related results (Dang & Lan, 2015) the associated guarantees do not hold for the algorithms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "we propose. 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Most optimization analyses of NAS show monotonic improvement (Akimoto et al.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "2019), asymptotic guarantees (Yao et al., 2020), or bounds on auxiliary quantities disconnected", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "from any objective (Noy et al., 2019; Nayman et al., 2019; Carlucci et al., 2019). In contrast, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "prove polynomial-time stationary-point convergence on a single-level objective for weight-sharing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "NAS, so far only studied empirically (Xie et al., 2019; Li et al., 2019). Our results draw upon the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "mirror descent meta-algorithm (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003) and extend", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "recent nonconvex convergence results Zhang & He (2018) to handle alternating descent. While there", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "exist related results (Dang & Lan, 2015) the associated guarantees do not hold for the algorithms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "we propose. 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We overview this in more detail in Appendix A.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 504, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "While weight-sharing significantly shortens search (Pham et al., 2018), it draws two main criticisms:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 503, + 309 + ], + "lines": [ + { + "bbox": [ + 107, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 107, + 266, + 369, + 278 + ], + "score": 1.0, + "content": "• Rank disorder: this describes when the rank of an architecture", + "type": "text" + }, + { + "bbox": [ + 370, + 268, + 376, + 276 + ], + "score": 0.63, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "according to the validation risk", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 116, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "evaluated with fixed shared weights w is poorly correlated with the one using “standalone” weights", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 116, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 116, + 288, + 185, + 300 + ], + "score": 0.92, + "content": "\\mathbf { w } _ { a } = \\mathbb { A } \\mathbb { 1 } \\mathtt { g } ( T , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 288, + 505, + 300 + ], + "score": 1.0, + "content": ". 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The key is that, unlike regular hyperparameters", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 384 + ], + "score": 1.0, + "content": "such as step-size, architectural hyperparameters directly affect the loss function without requiring a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 396 + ], + "score": 1.0, + "content": "dependent change in the model weights w. Thus we can distinguish architectures without retraining", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "simply by changing architectural decisions. Besides enabling weight-sharing, this point reveals", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "that the goal of NAS is perhaps better viewed as a regular learning problem over an extended class", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 412, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 324, + 427 + ], + "score": 0.87, + "content": "{ \\cal H } _ { \\cal A } = \\bigcup _ { a \\in { \\cal A } } { \\cal H } _ { a } = \\{ h _ { \\mathbf { w } , a } : \\hat { X } \\mapsto Y , \\mathbf { w } \\in \\mathbb { R } ^ { d } , a \\in \\mathcal { A } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 412, + 507, + 428 + ], + "score": 1.0, + "content": "that subsumes the architectural decisions as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "parameters of a larger model class, an unrelaxed “supernet.” The natural approach to solving this is by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "approximate empirical risk minimization, e.g. by approximating continuous objective below on the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 502, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 324, + 460 + ], + "score": 1.0, + "content": "right using a gradient algorithm and passing the output", + "type": "text" + }, + { + "bbox": [ + 324, + 448, + 330, + 457 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 447, + 502, + 460 + ], + "score": 1.0, + "content": "through Map to obtain a valid architecture:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 459, + 464, + 487 + ], + "lines": [ + { + "bbox": [ + 147, + 459, + 464, + 487 + ], + "spans": [ + { + "bbox": [ + 147, + 459, + 464, + 487 + ], + "score": 0.79, + "content": "\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { d } , a \\in \\mathcal { A } } \\quad \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , a } ( x ) , y ) \\qquad \\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { d } , \\theta \\in \\Theta } \\quad \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , \\theta } ( x ) , y )", + "type": "interline_equation", + "image_path": "d7bef258303cc1c0526ed0926466c7821a7f10fea5c9622dc4dcb756f19eb01d.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 147, + 459, + 464, + 468.3333333333333 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 147, + 468.3333333333333, + 464, + 477.66666666666663 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 147, + 477.66666666666663, + 464, + 486.99999999999994 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "Several works have optimized this single-level objective as an alternative to bilevel (2) (Xie et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "2019; Li et al., 2019). We argue for its use as the baseline object of study in NAS for three reasons:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "score": 1.0, + "content": "1. As discussed above, it is the natural first approach to solving the statistical objective of NAS:", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 118, + 537, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 118, + 537, + 215, + 552 + ], + "score": 1.0, + "content": "finding a good predictor", + "type": "text" + }, + { + "bbox": [ + 216, + 539, + 263, + 551 + ], + "score": 0.92, + "content": "h _ { \\mathbf { w } , a } \\in H _ { \\cal A }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 537, + 507, + 552 + ], + "score": 1.0, + "content": "in the extended function class over architectures and weights.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 547, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 507, + 563 + ], + "score": 1.0, + "content": "2. The common alternating gradient approach to the bilevel problem (2) is in practice very similar to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 117, + 558, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 117, + 558, + 506, + 574 + ], + "score": 1.0, + "content": "alternating block approaches to ERM (3); as we will see, there are established ways of analyzing", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 117, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 117, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "such methods for the latter objective, while for the former convergence is known only under very", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 117, + 582, + 467, + 594 + ], + "spans": [ + { + "bbox": [ + 117, + 582, + 467, + 594 + ], + "score": 1.0, + "content": "strong assumptions such as uniqueness of the inner minimum (Franceschi et al., 2018).", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 117, + 602 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 118, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "While less frequently used in practice than bilevel, single-level optimization can be very effective:", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 118, + 602, + 481, + 616 + ], + "spans": [ + { + "bbox": [ + 118, + 602, + 481, + 616 + ], + "score": 1.0, + "content": "we use it to achieve new state-of-the-art results on NAS-Bench-201 (Dong & Yang, 2020).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 634 + ], + "score": 1.0, + "content": "Understanding NAS as single-level optimization—the usual deep learning setting—makes weight-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "sharing a natural, not surprising, approach. Furthermore, for methods—both single-level and bilevel—", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "that adapt architecture parameters during search, it suggests that we need not worry about rank", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "disorder as long as we can use optimization to find a single feasible point that generalizes well;", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "we explicitly do not need a ranking. Non-adaptive methods such as RS-WS still do require rank", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 468, + 690 + ], + "score": 1.0, + "content": "correlation to select good architectures after search, but they are explicitly not changing", + "type": "text" + }, + { + "bbox": [ + 468, + 678, + 475, + 688 + ], + "score": 0.68, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "and so", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "have no variant solving (3). The single-level formulation thus reduces search method design to", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "well-studied questions of how to best regularize and optimize ERM. While there are many techniques", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "for regularizing weight-sharing—including partial channels (Xu et al., 2020) and validation Hessian", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 721, + 442, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 721, + 442, + 734 + ], + "score": 1.0, + "content": "penalization (Zela et al., 2020a)—we focus on the second question of optimization.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 49.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 139 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 104, + 81, + 507, + 140 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 140, + 458, + 168 + ], + "lines": [ + { + "bbox": [ + 151, + 140, + 458, + 168 + ], + "spans": [ + { + "bbox": [ + 151, + 140, + 458, + 168 + ], + "score": 0.9, + "content": "\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\quad \\sum _ { ( x , y ) \\in V } \\ell ( h _ { \\mathbf { w } , \\theta } ( x ) , y ) \\qquad \\mathrm { s . t . } \\qquad \\mathbf { w } \\in \\underset { \\mathbf { u } \\in \\mathbb { R } ^ { d } } { \\arg \\operatorname* { m i n } } \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { u } , \\theta } ( x ) , y )", + "type": "interline_equation", + "image_path": "3cbedcd6e0d904cc850947134cd8a51bfa225464da3df144f93276cee63cb57a.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 151, + 140, + 458, + 149.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 151, + 149.33333333333334, + 458, + 158.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 151, + 158.66666666666669, + 458, + 168.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 169, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 340, + 182 + ], + "score": 1.0, + "content": "This objective is not significantly different from (2), since", + "type": "text" + }, + { + "bbox": [ + 341, + 169, + 384, + 181 + ], + "score": 0.93, + "content": "\\mathbb { A } \\bot \\mathbb { g } ( T , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 168, + 505, + 182 + ], + "score": 1.0, + "content": "approximately minimizes the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 189, + 192 + ], + "score": 1.0, + "content": "empirical risk w.r.t.", + "type": "text" + }, + { + "bbox": [ + 189, + 181, + 198, + 190 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "; the difference is replacing discrete architectures with relaxed architecture", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 154, + 203 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 155, + 191, + 180, + 201 + ], + "score": 0.9, + "content": "\\theta \\in \\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 191, + 505, + 203 + ], + "score": 1.0, + "content": ", w.r.t. which we can take derivatives of the outer objective. This allows (2) to be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 343, + 214 + ], + "score": 1.0, + "content": "approximated via alternating gradient updates w.r.t. w and", + "type": "text" + }, + { + "bbox": [ + 343, + 202, + 349, + 212 + ], + "score": 0.73, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 201, + 506, + 214 + ], + "score": 1.0, + "content": ". Relaxations can be stochastic, so that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 138, + 225 + ], + "score": 0.89, + "content": "\\operatorname { M a p } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 212, + 214, + 226 + ], + "score": 1.0, + "content": "is a sample from a", + "type": "text" + }, + { + "bbox": [ + 215, + 213, + 221, + 223 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 212, + 506, + 226 + ], + "score": 1.0, + "content": "-parameterized distribution (Pham et al., 2018; Dong & Yang, 2019), or", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 223, + 205, + 237 + ], + "score": 1.0, + "content": "a mixture, in which case", + "type": "text" + }, + { + "bbox": [ + 206, + 224, + 237, + 235 + ], + "score": 0.91, + "content": "\\operatorname { M a p } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "selects architectural decisions with the highest weight in a convex", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 235, + 469, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 195, + 247 + ], + "score": 1.0, + "content": "combination given by", + "type": "text" + }, + { + "bbox": [ + 196, + 235, + 202, + 245 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 235, + 469, + 247 + ], + "score": 1.0, + "content": "(Liu et al., 2019). We overview this in more detail in Appendix A.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 168, + 506, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 504, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "While weight-sharing significantly shortens search (Pham et al., 2018), it draws two main criticisms:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 250, + 506, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 503, + 309 + ], + "lines": [ + { + "bbox": [ + 107, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 107, + 266, + 369, + 278 + ], + "score": 1.0, + "content": "• Rank disorder: this describes when the rank of an architecture", + "type": "text" + }, + { + "bbox": [ + 370, + 268, + 376, + 276 + ], + "score": 0.63, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "according to the validation risk", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 116, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "evaluated with fixed shared weights w is poorly correlated with the one using “standalone” weights", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 116, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 116, + 288, + 185, + 300 + ], + "score": 0.92, + "content": "\\mathbf { w } _ { a } = \\mathbb { A } \\mathbb { 1 } \\mathtt { g } ( T , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 288, + 505, + 300 + ], + "score": 1.0, + "content": ". This causes suboptimal architectures to be selected after shared weights search", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 116, + 299, + 425, + 311 + ], + "spans": [ + { + "bbox": [ + 116, + 299, + 425, + 311 + ], + "score": 1.0, + "content": "(Yu et al., 2020; Zela et al., 2020b; Zhang et al., 2020; Pourchot et al., 2020).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 107, + 266, + 505, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 310, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 506, + 322 + ], + "score": 1.0, + "content": "• Poor performance: weight-sharing can converge to degenerate architectures (Zela et al., 2020a)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 115, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 115, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "and is outperformed by regular hyperparameter tuning on NAS-Bench-201 (Dong & Yang, 2020).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 307, + 506, + 333 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 341, + 371, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 372, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 372, + 354 + ], + "score": 1.0, + "content": "2.2 SINGLE-LEVEL NAS AS A BASELINE OBJECT OF STUDY", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "Why are we able to apply weight-sharing to NAS? The key is that, unlike regular hyperparameters", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 384 + ], + "score": 1.0, + "content": "such as step-size, architectural hyperparameters directly affect the loss function without requiring a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 396 + ], + "score": 1.0, + "content": "dependent change in the model weights w. Thus we can distinguish architectures without retraining", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "simply by changing architectural decisions. Besides enabling weight-sharing, this point reveals", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "that the goal of NAS is perhaps better viewed as a regular learning problem over an extended class", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 412, + 507, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 324, + 427 + ], + "score": 0.87, + "content": "{ \\cal H } _ { \\cal A } = \\bigcup _ { a \\in { \\cal A } } { \\cal H } _ { a } = \\{ h _ { \\mathbf { w } , a } : \\hat { X } \\mapsto Y , \\mathbf { w } \\in \\mathbb { R } ^ { d } , a \\in \\mathcal { A } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 412, + 507, + 428 + ], + "score": 1.0, + "content": "that subsumes the architectural decisions as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "parameters of a larger model class, an unrelaxed “supernet.” The natural approach to solving this is by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "approximate empirical risk minimization, e.g. by approximating continuous objective below on the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 502, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 324, + 460 + ], + "score": 1.0, + "content": "right using a gradient algorithm and passing the output", + "type": "text" + }, + { + "bbox": [ + 324, + 448, + 330, + 457 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 447, + 502, + 460 + ], + "score": 1.0, + "content": "through Map to obtain a valid architecture:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 359, + 507, + 460 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 459, + 464, + 487 + ], + "lines": [ + { + "bbox": [ + 147, + 459, + 464, + 487 + ], + "spans": [ + { + "bbox": [ + 147, + 459, + 464, + 487 + ], + "score": 0.79, + "content": "\\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { d } , a \\in \\mathcal { A } } \\quad \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , a } ( x ) , y ) \\qquad \\operatorname* { m i n } _ { \\mathbf { w } \\in \\mathbb { R } ^ { d } , \\theta \\in \\Theta } \\quad \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , \\theta } ( x ) , y )", + "type": "interline_equation", + "image_path": "d7bef258303cc1c0526ed0926466c7821a7f10fea5c9622dc4dcb756f19eb01d.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 147, + 459, + 464, + 468.3333333333333 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 147, + 468.3333333333333, + 464, + 477.66666666666663 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 147, + 477.66666666666663, + 464, + 486.99999999999994 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "Several works have optimized this single-level objective as an alternative to bilevel (2) (Xie et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "2019; Li et al., 2019). We argue for its use as the baseline object of study in NAS for three reasons:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 502, + 506, + 527 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 528, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 507, + 541 + ], + "score": 1.0, + "content": "1. As discussed above, it is the natural first approach to solving the statistical objective of NAS:", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 537, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 118, + 537, + 215, + 552 + ], + "score": 1.0, + "content": "finding a good predictor", + "type": "text" + }, + { + "bbox": [ + 216, + 539, + 263, + 551 + ], + "score": 0.92, + "content": "h _ { \\mathbf { w } , a } \\in H _ { \\cal A }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 537, + 507, + 552 + ], + "score": 1.0, + "content": "in the extended function class over architectures and weights.", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 547, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 507, + 563 + ], + "score": 1.0, + "content": "2. The common alternating gradient approach to the bilevel problem (2) is in practice very similar to", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 117, + 558, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 117, + 558, + 506, + 574 + ], + "score": 1.0, + "content": "alternating block approaches to ERM (3); as we will see, there are established ways of analyzing", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 117, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 117, + 569, + 505, + 585 + ], + "score": 1.0, + "content": "such methods for the latter objective, while for the former convergence is known only under very", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 117, + 582, + 467, + 594 + ], + "spans": [ + { + "bbox": [ + 117, + 582, + 467, + 594 + ], + "score": 1.0, + "content": "strong assumptions such as uniqueness of the inner minimum (Franceschi et al., 2018).", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 117, + 602 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 118, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "While less frequently used in practice than bilevel, single-level optimization can be very effective:", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 602, + 481, + 616 + ], + "spans": [ + { + "bbox": [ + 118, + 602, + 481, + 616 + ], + "score": 1.0, + "content": "we use it to achieve new state-of-the-art results on NAS-Bench-201 (Dong & Yang, 2020).", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 528, + 507, + 616 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 634 + ], + "score": 1.0, + "content": "Understanding NAS as single-level optimization—the usual deep learning setting—makes weight-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "sharing a natural, not surprising, approach. Furthermore, for methods—both single-level and bilevel—", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "that adapt architecture parameters during search, it suggests that we need not worry about rank", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "disorder as long as we can use optimization to find a single feasible point that generalizes well;", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "we explicitly do not need a ranking. Non-adaptive methods such as RS-WS still do require rank", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 468, + 690 + ], + "score": 1.0, + "content": "correlation to select good architectures after search, but they are explicitly not changing", + "type": "text" + }, + { + "bbox": [ + 468, + 678, + 475, + 688 + ], + "score": 0.68, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "and so", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "have no variant solving (3). The single-level formulation thus reduces search method design to", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "well-studied questions of how to best regularize and optimize ERM. While there are many techniques", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "for regularizing weight-sharing—including partial channels (Xu et al., 2020) and validation Hessian", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 721, + 442, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 721, + 442, + 734 + ], + "score": 1.0, + "content": "penalization (Zela et al., 2020a)—we focus on the second question of optimization.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 49.5, + "bbox_fs": [ + 104, + 623, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 358, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 360, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 360, + 96 + ], + "score": 1.0, + "content": "3 GEOMETRY-AWARE GRADIENT ALGORITHMS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 101, + 505, + 241 + ], + "lines": [ + { + "bbox": [ + 104, + 100, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 100, + 349, + 119 + ], + "score": 1.0, + "content": "We seek to minimize the (possibly regularized) empirical risk", + "type": "text" + }, + { + "bbox": [ + 349, + 101, + 505, + 118 + ], + "score": 0.94, + "content": "\\begin{array} { r } { f ( \\mathbf { w } , \\theta ) = \\frac { 1 } { | T | } \\sum _ { ( x , y ) \\in T } \\ell ( h _ { \\mathbf { w } , \\theta } ( x ) , y ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 188, + 130 + ], + "score": 1.0, + "content": "over shared-weights", + "type": "text" + }, + { + "bbox": [ + 189, + 116, + 222, + 128 + ], + "score": 0.9, + "content": "\\mathbf { w } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 117, + 334, + 130 + ], + "score": 1.0, + "content": "and architecture parameters", + "type": "text" + }, + { + "bbox": [ + 335, + 118, + 361, + 128 + ], + "score": 0.89, + "content": "\\theta \\in \\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 117, + 505, + 130 + ], + "score": 1.0, + "content": ". Assuming we have noisy gradients", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 127, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 104, + 127, + 119, + 144 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 119, + 130, + 127, + 141 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 127, + 181, + 144 + ], + "score": 1.0, + "content": "w.r.t. w or", + "type": "text" + }, + { + "bbox": [ + 181, + 130, + 187, + 140 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 127, + 245, + 144 + ], + "score": 1.0, + "content": "at any point", + "type": "text" + }, + { + "bbox": [ + 245, + 128, + 323, + 141 + ], + "score": 0.9, + "content": "( \\mathbf { w } , \\theta ) \\in \\mathbb { R } ^ { d } \\times \\Theta .", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 127, + 351, + 144 + ], + "score": 1.0, + "content": "—i.e.", + "type": "text" + }, + { + "bbox": [ + 352, + 128, + 399, + 142 + ], + "score": 0.93, + "content": "\\tilde { \\nabla } _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 127, + 414, + 144 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 415, + 128, + 460, + 142 + ], + "score": 0.93, + "content": "\\tilde { \\nabla } _ { \\boldsymbol { \\theta } } f ( \\mathbf { w } , \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 127, + 506, + 144 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 141, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 221, + 154 + ], + "score": 0.91, + "content": "\\mathbb { E } \\tilde { \\nabla } _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta ) = \\nabla _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 141, + 233, + 155 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 234, + 142, + 343, + 154 + ], + "score": 0.9, + "content": "\\mathbb { E } \\tilde { \\nabla } _ { \\theta } f ( \\mathbf { w } , \\theta ) = \\nabla _ { \\theta } f ( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 141, + 505, + 155 + ], + "score": 1.0, + "content": ", respectively—our goal is a point where", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 153, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 107, + 154, + 114, + 165 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 153, + 505, + 166 + ], + "score": 1.0, + "content": ", or at least its gradient, is small, while taking as few gradients as possible. Our main complication", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 410, + 177 + ], + "score": 1.0, + "content": "is that architecture parameters lie in a constrained, non-Euclidean domain", + "type": "text" + }, + { + "bbox": [ + 410, + 165, + 419, + 174 + ], + "score": 0.76, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 164, + 505, + 177 + ], + "score": 1.0, + "content": ". Most search spaces", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 107, + 175, + 116, + 185 + ], + "score": 0.77, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 174, + 357, + 188 + ], + "score": 1.0, + "content": "are product sets of categorical decisions—which operation", + "type": "text" + }, + { + "bbox": [ + 358, + 176, + 384, + 185 + ], + "score": 0.91, + "content": "o \\in O", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 174, + 443, + 188 + ], + "score": 1.0, + "content": "to use at edge", + "type": "text" + }, + { + "bbox": [ + 443, + 175, + 469, + 186 + ], + "score": 0.86, + "content": "e \\in E", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "—so the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 237, + 199 + ], + "score": 1.0, + "content": "natural relaxation is a product of", + "type": "text" + }, + { + "bbox": [ + 237, + 186, + 267, + 198 + ], + "score": 0.64, + "content": "| E | \\mid O |", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 186, + 495, + 199 + ], + "score": 1.0, + "content": "-simplices. However, NAS methods often re-parameterize", + "type": "text" + }, + { + "bbox": [ + 495, + 186, + 504, + 196 + ], + "score": 0.75, + "content": "\\Theta", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "score": 1.0, + "content": "to be unconstrained using a softmax and then SGD or Adam (Kingma & Ba, 2015). Is there a better", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 208, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 506, + 221 + ], + "score": 1.0, + "content": "parameterization-algorithm co-design? We consider a geometry-aware approach that uses mirror", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "descent to design NAS methods with better properties depending on the domain; a key desirable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 230, + 502, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 502, + 243 + ], + "score": 1.0, + "content": "property is to return sparse architectural parameters to reduce loss from post-search discretization.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 248, + 290, + 259 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 291, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 291, + 261 + ], + "score": 1.0, + "content": "3.1 BACKGROUND ON MIRROR DESCENT", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "Mirror descent has many formulations (Nemirovski & Yudin, 1983; Beck & Teboulle, 2003; Shalev-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "Shwartz, 2011); the proximal starts by noting that, in the unconstrained case, an SGD update at a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 455, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 129, + 303 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 130, + 289, + 181, + 300 + ], + "score": 0.92, + "content": "\\theta \\in \\Theta = \\mathbb { R } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 288, + 277, + 303 + ], + "score": 1.0, + "content": "given gradient estimate", + "type": "text" + }, + { + "bbox": [ + 277, + 288, + 305, + 302 + ], + "score": 0.92, + "content": "\\tilde { \\nabla } \\bar { f } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 288, + 364, + 303 + ], + "score": 1.0, + "content": "with step-size", + "type": "text" + }, + { + "bbox": [ + 364, + 290, + 389, + 301 + ], + "score": 0.89, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 288, + 455, + 303 + ], + "score": 1.0, + "content": "is equivalent to", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 301, + 432, + 326 + ], + "lines": [ + { + "bbox": [ + 178, + 301, + 432, + 326 + ], + "spans": [ + { + "bbox": [ + 178, + 301, + 432, + 326 + ], + "score": 0.9, + "content": "\\theta - \\eta \\tilde { \\nabla } f ( \\theta ) \\quad = \\quad \\begin{array} { c c c c } { \\underset { \\mathbf { u } \\in \\mathbb { R } ^ { k } } { \\arg \\operatorname* { m i n } } } & { \\eta \\tilde { \\nabla } f ( \\theta ) \\cdot \\mathbf { u } } & { + } & { \\frac { 1 } { 2 } \\| \\mathbf { u } - \\theta \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "6767d31dd40b41e8ed16ad5ff5035e47dd14f49397f146195f569d72defaeb36.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 178, + 301, + 432, + 326 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "Here the first term aligns the output with the gradient while the second (proximal) term regularizes", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "for closeness to the previous point as measured by the Euclidean distance. 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This nearly dimension-independent iteration complexity is achieved by choosing", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "a regularizer—the KL divergence—well-suited to the underlying geometry—the simplex. 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\\phi ( { \\mathbf { v } } ) - \\nabla \\phi ( { \\mathbf { v } } ) \\cdot ( { \\mathbf { u } } - { \\mathbf { v } } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 499, + 498, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 272, + 512 + ], + "score": 1.0, + "content": "(Bregman, 1967), a notion of distance on", + "type": "text" + }, + { + "bbox": [ + 272, + 499, + 281, + 509 + ], + "score": 0.8, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 499, + 498, + 512 + ], + "score": 1.0, + "content": "that acts as a regularizer in the mirror descent update:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 509, + 387, + 529 + ], + "lines": [ + { + "bbox": [ + 225, + 509, + 387, + 529 + ], + "spans": [ + { + "bbox": [ + 225, + 509, + 387, + 529 + ], + "score": 0.8, + "content": "\\begin{array} { r l r } { \\underset { \\mathbf { u } \\in \\Theta } { \\mathrm { a r g m i n } } } & { { } \\eta \\tilde { \\nabla } f ( \\boldsymbol { \\theta } ) \\cdot \\mathbf { u } } & { + \\quad D _ { \\boldsymbol { \\phi } } ( \\mathbf { u } \\vert \\vert \\boldsymbol { \\theta } ) } \\end{array}", + "type": "interline_equation", + "image_path": "c3b80053f1483c16c4567d44f6c48a49e1c82b3c5cf48203af0ccc6aacafcc47.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 225, + 509, + 387, + 529 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 530, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 263, + 545 + ], + "score": 1.0, + "content": "For example, to recover SGD (4) we set", + "type": "text" + }, + { + "bbox": [ + 263, + 530, + 324, + 543 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\phi ( \\mathbf { u } ) = \\frac 1 2 \\lVert \\mathbf { u } \\rVert _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 529, + 506, + 545 + ], + "score": 1.0, + "content": ", which is strongly-convex w.r.t. the Euclidean", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 541, + 494, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 279, + 555 + ], + "score": 1.0, + "content": "norm, while EG (5) is recovered by setting", + "type": "text" + }, + { + "bbox": [ + 279, + 543, + 349, + 554 + ], + "score": 0.89, + "content": "\\phi ( \\mathbf { u } ) \\bar { \\mathbf { \\Psi } } = \\mathbf { u } \\cdot \\mathrm { l o g } \\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 541, + 456, + 555 + ], + "score": 1.0, + "content": ", strongly-convex w.r.t. the", + "type": "text" + }, + { + "bbox": [ + 456, + 542, + 466, + 553 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 541, + 494, + 555 + ], + "score": 1.0, + "content": "-norm.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 107, + 560, + 303, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 304, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 304, + 573 + ], + "score": 1.0, + "content": "3.2 BLOCK-STOCHASTIC MIRROR DESCENT", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "In the previous section we saw how mirror descent can perform better over certain geometries", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "such as the simplex. However, in weight-sharing we are interested in optimizing over a hybrid", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "geometry containing both the shared weights in an unconstrained Euclidean space and the architecture", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "parameters in a non-Euclidean domain. 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This", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "approach is encapsulated in the block-stochastic algorithm described in Algorithm 1, which at each", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "step chooses one block at random to update using mirror descent (recall that SGD is a variant) and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 127, + 700 + ], + "score": 1.0, + "content": "after", + "type": "text" + }, + { + "bbox": [ + 127, + 688, + 136, + 698 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "steps returns a random iterate. Algorithm 1 generalizes the single-level variant of both ENAS", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 308, + 711 + ], + "score": 1.0, + "content": "and first-order DARTS if SGD is used to update", + "type": "text" + }, + { + "bbox": [ + 309, + 700, + 315, + 709 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "instead of Adam, with some mild caveats: in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "practice blocks are picked cyclically and the algorithm returns the last iterate, not a a random one. 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\\phi ( { \\mathbf { v } } ) - \\nabla \\phi ( { \\mathbf { v } } ) \\cdot ( { \\mathbf { u } } - { \\mathbf { v } } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 499, + 498, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 272, + 512 + ], + "score": 1.0, + "content": "(Bregman, 1967), a notion of distance on", + "type": "text" + }, + { + "bbox": [ + 272, + 499, + 281, + 509 + ], + "score": 0.8, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 499, + 498, + 512 + ], + "score": 1.0, + "content": "that acts as a regularizer in the mirror descent update:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 421, + 506, + 512 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 509, + 387, + 529 + ], + "lines": [ + { + "bbox": [ + 225, + 509, + 387, + 529 + ], + "spans": [ + { + "bbox": [ + 225, + 509, + 387, + 529 + ], + "score": 0.8, + "content": "\\begin{array} { r l r } { \\underset { \\mathbf { u } \\in \\Theta } { \\mathrm { a r g m i n } } } & { { } \\eta \\tilde { \\nabla } f ( \\boldsymbol { \\theta } ) \\cdot \\mathbf { u } } & { + \\quad D _ { \\boldsymbol { \\phi } } ( \\mathbf { u } \\vert \\vert \\boldsymbol { \\theta } ) } \\end{array}", + "type": "interline_equation", + "image_path": "c3b80053f1483c16c4567d44f6c48a49e1c82b3c5cf48203af0ccc6aacafcc47.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 225, + 509, + 387, + 529 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 530, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 263, + 545 + ], + "score": 1.0, + "content": "For example, to recover SGD (4) we set", + "type": "text" + }, + { + "bbox": [ + 263, + 530, + 324, + 543 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\phi ( \\mathbf { u } ) = \\frac 1 2 \\lVert \\mathbf { u } \\rVert _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 529, + 506, + 545 + ], + "score": 1.0, + "content": ", which is strongly-convex w.r.t. the Euclidean", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 541, + 494, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 279, + 555 + ], + "score": 1.0, + "content": "norm, while EG (5) is recovered by setting", + "type": "text" + }, + { + "bbox": [ + 279, + 543, + 349, + 554 + ], + "score": 0.89, + "content": "\\phi ( \\mathbf { u } ) \\bar { \\mathbf { \\Psi } } = \\mathbf { u } \\cdot \\mathrm { l o g } \\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 541, + 456, + 555 + ], + "score": 1.0, + "content": ", strongly-convex w.r.t. the", + "type": "text" + }, + { + "bbox": [ + 456, + 542, + 466, + 553 + ], + "score": 0.86, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 541, + 494, + 555 + ], + "score": 1.0, + "content": "-norm.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 529, + 506, + 555 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 560, + 303, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 304, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 304, + 573 + ], + "score": 1.0, + "content": "3.2 BLOCK-STOCHASTIC MIRROR DESCENT", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "In the previous section we saw how mirror descent can perform better over certain geometries", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "such as the simplex. However, in weight-sharing we are interested in optimizing over a hybrid", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "geometry containing both the shared weights in an unconstrained Euclidean space and the architecture", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "parameters in a non-Euclidean domain. Thus we focus on optimization over two blocks: shared", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 141, + 634 + ], + "score": 1.0, + "content": "weights", + "type": "text" + }, + { + "bbox": [ + 141, + 621, + 178, + 632 + ], + "score": 0.87, + "content": "\\mathbf { w } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 622, + 297, + 634 + ], + "score": 1.0, + "content": "and architecture parameters", + "type": "text" + }, + { + "bbox": [ + 297, + 622, + 326, + 632 + ], + "score": 0.91, + "content": "\\theta \\in \\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 622, + 468, + 634 + ], + "score": 1.0, + "content": ", the latter associated with a DGF", + "type": "text" + }, + { + "bbox": [ + 468, + 622, + 475, + 633 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "that is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 240, + 646 + ], + "score": 1.0, + "content": "strongly-convex w.r.t. some norm", + "type": "text" + }, + { + "bbox": [ + 240, + 633, + 256, + 645 + ], + "score": 0.89, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 633, + 506, + 646 + ], + "score": 1.0, + "content": ". In NAS a common approach is to perform alternating gradient", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "steps on each domain; for example, both ENAS (Pham et al., 2018) and first-order DARTS (Liu et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "2019) alternate between SGD on the shared weights and Adam on architecture parameters. This", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "approach is encapsulated in the block-stochastic algorithm described in Algorithm 1, which at each", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "step chooses one block at random to update using mirror descent (recall that SGD is a variant) and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 127, + 700 + ], + "score": 1.0, + "content": "after", + "type": "text" + }, + { + "bbox": [ + 127, + 688, + 136, + 698 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "steps returns a random iterate. Algorithm 1 generalizes the single-level variant of both ENAS", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 308, + 711 + ], + "score": 1.0, + "content": "and first-order DARTS if SGD is used to update", + "type": "text" + }, + { + "bbox": [ + 309, + 700, + 315, + 709 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "instead of Adam, with some mild caveats: in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "practice blocks are picked cyclically and the algorithm returns the last iterate, not a a random one. To", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "score": 1.0, + "content": "analyze the convergence of Algorithm 1 we first state some regularity assumptions on the function:", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 578, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 86, + 475, + 98 + ], + "lines": [ + { + "bbox": [ + 105, + 86, + 476, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 86, + 402, + 99 + ], + "score": 1.0, + "content": "Algorithm 1: Block-stochastic mirror descent optimization of a function", + "type": "text" + }, + { + "bbox": [ + 402, + 86, + 473, + 99 + ], + "score": 0.88, + "content": "f : \\mathbb { R } ^ { d } \\times \\Theta \\mapsto \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 86, + 476, + 99 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table", + "bbox": [ + 106, + 101, + 500, + 202 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 101, + 500, + 202 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 101, + 500, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 500, + 202 + ], + "score": 0.396, + "html": "
for iteration t=1,...,TdoInput: nitialization(w(1),(1)）∈Rd×,strongly-convex DGF𝜙:→R,numberofiterations T≥1,step-size n>0
sample bt~Unif{w,0} if block bt = w then//randomly select update block
w(t+1) ←w(t)-nwf(w(t),0(t))// SGD update to shared weights
θ(t+1)←0(t)// no update to architecture params
else w(t+1）←w(t)
// no update to shared weights
θ(t+1)← argminuεe nθf(w(t),θ(t))·u + D(ul|θ(t))// update architecture params
Output:(w(r),θ(r)) for r~ Unif{1,...,T}
", + "type": "table", + "image_path": "779e9c9724a0201deb99d3afec3fcfe8261b00ad0593c912a84c8212b2218a2a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 106, + 101, + 500, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 134.66666666666666, + 500, + 168.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 106, + 168.33333333333331, + 500, + 201.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 502, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 210, + 240 + ], + "score": 1.0, + "content": "Assumption 1. 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The number of steps required to obtain such a point thus measures how fast a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "first-order method terminates. Stationarity is also significant as a necessary condition for optimality.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 506, + 413 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 156, + 381 + ], + "score": 1.0, + "content": "In our case", + "type": "text" + }, + { + "bbox": [ + 156, + 368, + 165, + 378 + ], + "score": 0.78, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "may be constrained and so the gradient may never be small, thus necessitating a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "measure other than gradient norm. 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note that the dependence of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 469, + 422, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 201, + 482 + ], + "score": 1.0, + "content": "stationarity measure on", + "type": "text" + }, + { + "bbox": [ + 202, + 471, + 209, + 481 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 469, + 282, + 482 + ], + "score": 1.0, + "content": "is standard (Dang", + "type": "text" + }, + { + "bbox": [ + 283, + 470, + 292, + 480 + ], + "score": 0.27, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 469, + 422, + 482 + ], + "score": 1.0, + "content": "Lan, 2015; Zhang & He, 2018).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 506, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 257, + 500 + ], + "score": 1.0, + "content": "To understand why reaching a point", + "type": "text" + }, + { + "bbox": [ + 258, + 487, + 283, + 498 + ], + "score": 0.92, + "content": "( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 485, + 506, + 500 + ], + "score": 1.0, + "content": "with small Bregman stationarity is a reasonable goal,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 496, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 238, + 512 + ], + "score": 1.0, + "content": "note that the proximal operator", + "type": "text" + }, + { + "bbox": [ + 238, + 499, + 264, + 510 + ], + "score": 0.83, + "content": "\\operatorname { p r o x } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 496, + 506, + 512 + ], + "score": 1.0, + "content": "has the property that its fixed points, i.e. those satisfying", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 201, + 520 + ], + "score": 0.91, + "content": "( \\mathbf { w } , \\theta ) = \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 508, + 324, + 522 + ], + "score": 1.0, + "content": ", correspond to points where", + "type": "text" + }, + { + "bbox": [ + 325, + 509, + 332, + 520 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "has no feasible descent direction. Thus", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 198, + 533 + ], + "score": 1.0, + "content": "measuring how close", + "type": "text" + }, + { + "bbox": [ + 199, + 520, + 225, + 532 + ], + "score": 0.92, + "content": "( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 520, + 343, + 533 + ], + "score": 1.0, + "content": "is to being a fixed point of", + "type": "text" + }, + { + "bbox": [ + 343, + 521, + 369, + 531 + ], + "score": 0.71, + "content": "\\operatorname { p r o x } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "—as is done using the Bregman", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 496, + 543 + ], + "score": 1.0, + "content": "divergence in (7)—is a good measure of how far away the point is from being a stationary point of", + "type": "text" + }, + { + "bbox": [ + 496, + 531, + 503, + 542 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 531, + 506, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 183, + 554 + ], + "score": 1.0, + "content": "Finally, note that if", + "type": "text" + }, + { + "bbox": [ + 184, + 542, + 191, + 553 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 541, + 235, + 554 + ], + "score": 1.0, + "content": "is smooth,", + "type": "text" + }, + { + "bbox": [ + 235, + 542, + 243, + 553 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 541, + 313, + 554 + ], + "score": 1.0, + "content": "is Euclidean, and", + "type": "text" + }, + { + "bbox": [ + 314, + 542, + 323, + 551 + ], + "score": 0.84, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "is unconstrained—i.e. if we are running SGD", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 277, + 565 + ], + "score": 1.0, + "content": "over architecture parameters as well—then", + "type": "text" + }, + { + "bbox": [ + 278, + 552, + 316, + 568 + ], + "score": 0.93, + "content": "\\Delta _ { \\frac { 1 } { 2 \\gamma } } \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 551, + 360, + 565 + ], + "score": 1.0, + "content": "implies an", + "type": "text" + }, + { + "bbox": [ + 360, + 552, + 381, + 564 + ], + "score": 0.92, + "content": "O ( \\varepsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "-bound on the squared gradient", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 280, + 579 + ], + "score": 1.0, + "content": "norm, recovering the standard definition of", + "type": "text" + }, + { + "bbox": [ + 280, + 568, + 286, + 576 + ], + "score": 0.78, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "-stationarity. More intuition on proximal operators can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "be found in Parikh & Boyd (2013, Section 1.2), while further details on Bregman stationarity and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 587, + 488, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 488, + 600 + ], + "score": 1.0, + "content": "how it relates to other notions of convergence can be found in Zhang & He (2018, Section 2.3).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 503, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "The following result shows that Algorithm 1 needs polynomially many iterations to finds a point", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 615, + 335, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 132, + 628 + ], + "score": 0.9, + "content": "( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 615, + 153, + 629 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 154, + 617, + 159, + 626 + ], + "score": 0.75, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 615, + 335, + 629 + ], + "score": 1.0, + "content": "-small Bregman stationarity in-expectation:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 506, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 508, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 177, + 646 + ], + "score": 1.0, + "content": "Theorem 1. Let", + "type": "text" + }, + { + "bbox": [ + 178, + 631, + 256, + 644 + ], + "score": 0.92, + "content": "F = f ( \\mathbf { w } ^ { ( 1 ) } , \\theta ^ { ( 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 629, + 324, + 646 + ], + "score": 1.0, + "content": "be the value of", + "type": "text" + }, + { + "bbox": [ + 325, + 633, + 332, + 644 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 629, + 508, + 646 + ], + "score": 1.0, + "content": "at initialization. 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Here the expectation is over the randomness of the algorithm and gradients.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "The proof in the appendix follows from single-block analysis (Zhang & He, 2018, Theorem 3.1) and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "in fact holds for the general case of any number of blocks associated to any set of strongly-convex", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "DGFs. 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for iteration t=1,...,TdoInput: nitialization(w(1),(1)）∈Rd×,strongly-convex DGF𝜙:→R,numberofiterations T≥1,step-size n>0
sample bt~Unif{w,0} if block bt = w then//randomly select update block
w(t+1) ←w(t)-nwf(w(t),0(t))// SGD update to shared weights
θ(t+1)←0(t)// no update to architecture params
else w(t+1）←w(t)
// no update to shared weights
θ(t+1)← argminuεe nθf(w(t),θ(t))·u + D(ul|θ(t))// update architecture params
Output:(w(r),θ(r)) for r~ Unif{1,...,T}
", + "type": "table", + "image_path": "779e9c9724a0201deb99d3afec3fcfe8261b00ad0593c912a84c8212b2218a2a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 106, + 101, + 500, + 134.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 134.66666666666666, + 500, + 168.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 106, + 168.33333333333331, + 500, + 201.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 502, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 210, + 240 + ], + "score": 1.0, + "content": "Assumption 1. Suppose", + "type": "text" + }, + { + "bbox": [ + 210, + 228, + 218, + 239 + ], + "score": 0.81, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 226, + 371, + 240 + ], + "score": 1.0, + "content": "is strongly-convex w.r.t. some norm", + "type": "text" + }, + { + "bbox": [ + 371, + 227, + 391, + 239 + ], + "score": 0.9, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 226, + 460, + 240 + ], + "score": 1.0, + "content": "on a convex set", + "type": "text" + }, + { + "bbox": [ + 461, + 227, + 470, + 237 + ], + "score": 0.76, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 236, + 363, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 236, + 180, + 252 + ], + "score": 1.0, + "content": "objective function", + "type": "text" + }, + { + "bbox": [ + 181, + 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( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 248, + 364, + 265 + ], + "score": 1.0, + "content": "is convex on", + "type": "text" + }, + { + "bbox": [ + 364, + 250, + 397, + 262 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { d } \\times \\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 248, + 435, + 265 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 435, + 251, + 461, + 262 + ], + "score": 0.89, + "content": "\\gamma > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 248, + 464, + 265 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 108, + 248, + 464, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 263, + 491, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 261, + 489, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 191, + 279 + ], + "score": 1.0, + "content": "2. gradient bound:", + "type": "text" + }, + { + "bbox": [ + 191, + 262, + 288, + 276 + ], + "score": 0.94, + "content": "\\mathbb { E } \\| \\tilde { \\nabla } _ { \\mathbf { w } } f ( \\mathbf { w } , \\theta ) \\| _ { 2 } ^ { 2 } \\leq G _ { \\mathbf { w } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 261, + 307, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 262, + 399, + 276 + ], + "score": 0.9, + "content": "\\mathbb { E } \\| \\tilde { \\nabla } _ { \\theta } f ( \\mathbf { w } , \\theta ) \\| _ { * } ^ { 2 } \\leq G _ { \\theta } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 261, + 438, + 279 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 438, + 263, + 489, + 275 + ], + "score": 0.9, + "content": "G _ { \\mathbf { w } } , G _ { \\theta } \\geq 0 .", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 261, + 489, + 279 + ] + }, + { + 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case an", + "type": "text" + }, + { + "bbox": [ + 301, + 331, + 307, + 339 + ], + "score": 0.76, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "-stationary-point is simply one where the gradient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 178, + 352 + ], + "score": 1.0, + "content": "has squared-norm", + "type": "text" + }, + { + "bbox": [ + 179, + 340, + 195, + 351 + ], + "score": 0.87, + "content": "\\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 340, + 506, + 352 + ], + "score": 1.0, + "content": ". The number of steps required to obtain such a point thus measures how fast a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 364 + ], + "score": 1.0, + "content": "first-order method terminates. Stationarity is also significant as a necessary condition for optimality.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 307, + 506, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 506, + 413 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 156, + 381 + ], + "score": 1.0, + "content": "In our case", + "type": "text" + }, + { + "bbox": [ + 156, + 368, + 165, + 378 + ], + "score": 0.78, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 367, + 506, + 381 + ], + "score": 1.0, + "content": "may be constrained and so the gradient may never be small, thus necessitating a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "measure other than gradient norm. We use Bregman stationarity (Zhang & He, 2018, Equation 2.11),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 253, + 402 + ], + "score": 1.0, + "content": "which measures stationary at a point", + "type": "text" + }, + { + "bbox": [ + 254, + 390, + 279, + 401 + ], + "score": 0.9, + "content": "( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "using the Bregman divergence between the point and its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 464, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 164, + 414 + ], + "score": 1.0, + "content": "proximal map", + "type": "text" + }, + { + "bbox": [ + 164, + 401, + 395, + 414 + ], + "score": 0.82, + "content": "\\begin{array} { r } { \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\theta ) = \\arg \\operatorname* { m i n } _ { \\mathbf { u } \\in \\mathbb { R } ^ { d } \\times \\Theta } \\lambda f ( \\mathbf { u } ) + D _ { \\ell _ { 2 } , \\phi } ( \\mathbf { u } | | \\mathbf { w } , \\theta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 401, + 433, + 414 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 433, + 401, + 459, + 411 + ], + "score": 0.9, + "content": "\\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 401, + 464, + 414 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 367, + 506, + 414 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 416, + 445, + 441 + ], + "lines": [ + { + "bbox": [ + 165, + 416, + 445, + 441 + ], + "spans": [ + { + "bbox": [ + 165, + 416, + 445, + 441 + ], + "score": 0.9, + "content": "\\Delta _ { \\lambda } ( \\mathbf { w } , \\boldsymbol { \\theta } ) = \\frac { D _ { \\ell _ { 2 } , \\phi } ( \\mathbf { w } , \\boldsymbol { \\theta } | | \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\boldsymbol { \\theta } ) ) + D _ { \\ell _ { 2 } , \\phi } ( \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\boldsymbol { \\theta } ) | | \\mathbf { w } , \\boldsymbol { \\theta } ) } { \\lambda ^ { 2 } }", + "type": "interline_equation", + "image_path": "f517f9c017b14b0ee2b25d74cb53fa826727c14bb55a8d4870811d232aa2c391.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 165, + 416, + 445, + 441 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 103, + 440, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 103, + 440, + 129, + 462 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 443, + 162, + 459 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\lambda = \\frac { 1 } { 2 \\gamma } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 440, + 284, + 462 + ], + "score": 1.0, + "content": "and the Bregman divergence", + "type": "text" + }, + { + "bbox": [ + 284, + 444, + 308, + 457 + ], + "score": 0.93, + "content": "D _ { \\ell _ { 2 } , \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 440, + 389, + 462 + ], + "score": 1.0, + "content": "is that of the DGF", + "type": "text" + }, + { + "bbox": [ + 389, + 443, + 450, + 457 + ], + "score": 0.91, + "content": "\\frac { 1 } { 2 } \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } + \\phi ( { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 440, + 507, + 462 + ], + "score": 1.0, + "content": "that encodes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 325, + 471 + ], + "score": 1.0, + "content": "the geometry of the joint optimization domain over w", + "type": "text" + }, + { + "bbox": [ + 326, + 458, + 349, + 469 + ], + "score": 0.82, + "content": "\\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 457, + 366, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 367, + 459, + 373, + 469 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "; note that the dependence of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 469, + 422, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 201, + 482 + ], + "score": 1.0, + "content": "stationarity measure on", + "type": "text" + }, + { + "bbox": [ + 202, + 471, + 209, + 481 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 469, + 282, + 482 + ], + "score": 1.0, + "content": "is standard (Dang", + "type": "text" + }, + { + "bbox": [ + 283, + 470, + 292, + 480 + ], + "score": 0.27, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 469, + 422, + 482 + ], + "score": 1.0, + "content": "Lan, 2015; Zhang & He, 2018).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 103, + 440, + 507, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 506, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 257, + 500 + ], + "score": 1.0, + "content": "To understand why reaching a point", + "type": "text" + }, + { + "bbox": [ + 258, + 487, + 283, + 498 + ], + "score": 0.92, + "content": "( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 485, + 506, + 500 + ], + "score": 1.0, + "content": "with small Bregman stationarity is a reasonable goal,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 496, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 238, + 512 + ], + "score": 1.0, + "content": "note that the proximal operator", + "type": "text" + }, + { + "bbox": [ + 238, + 499, + 264, + 510 + ], + "score": 0.83, + "content": "\\operatorname { p r o x } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 496, + 506, + 512 + ], + "score": 1.0, + "content": "has the property that its fixed points, i.e. those satisfying", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 201, + 520 + ], + "score": 0.91, + "content": "( \\mathbf { w } , \\theta ) = \\mathrm { p r o x } _ { \\lambda } ( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 508, + 324, + 522 + ], + "score": 1.0, + "content": ", correspond to points where", + "type": "text" + }, + { + "bbox": [ + 325, + 509, + 332, + 520 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "has no feasible descent direction. Thus", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 198, + 533 + ], + "score": 1.0, + "content": "measuring how close", + "type": "text" + }, + { + "bbox": [ + 199, + 520, + 225, + 532 + ], + "score": 0.92, + "content": "( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 520, + 343, + 533 + ], + "score": 1.0, + "content": "is to being a fixed point of", + "type": "text" + }, + { + "bbox": [ + 343, + 521, + 369, + 531 + ], + "score": 0.71, + "content": "\\operatorname { p r o x } _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "—as is done using the Bregman", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 496, + 543 + ], + "score": 1.0, + "content": "divergence in (7)—is a good measure of how far away the point is from being a stationary point of", + "type": "text" + }, + { + "bbox": [ + 496, + 531, + 503, + 542 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 531, + 506, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 183, + 554 + ], + "score": 1.0, + "content": "Finally, note that if", + "type": "text" + }, + { + "bbox": [ + 184, + 542, + 191, + 553 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 541, + 235, + 554 + ], + "score": 1.0, + "content": "is smooth,", + "type": "text" + }, + { + "bbox": [ + 235, + 542, + 243, + 553 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 541, + 313, + 554 + ], + "score": 1.0, + "content": "is Euclidean, and", + "type": "text" + }, + { + "bbox": [ + 314, + 542, + 323, + 551 + ], + "score": 0.84, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "is unconstrained—i.e. if we are running SGD", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 277, + 565 + ], + "score": 1.0, + "content": "over architecture parameters as well—then", + "type": "text" + }, + { + "bbox": [ + 278, + 552, + 316, + 568 + ], + "score": 0.93, + "content": "\\Delta _ { \\frac { 1 } { 2 \\gamma } } \\leq \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 551, + 360, + 565 + ], + "score": 1.0, + "content": "implies an", + "type": "text" + }, + { + "bbox": [ + 360, + 552, + 381, + 564 + ], + "score": 0.92, + "content": "O ( \\varepsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "-bound on the squared gradient", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 280, + 579 + ], + "score": 1.0, + "content": "norm, recovering the standard definition of", + "type": "text" + }, + { + "bbox": [ + 280, + 568, + 286, + 576 + ], + "score": 0.78, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "-stationarity. More intuition on proximal operators can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "be found in Parikh & Boyd (2013, Section 1.2), while further details on Bregman stationarity and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 587, + 488, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 488, + 600 + ], + "score": 1.0, + "content": "how it relates to other notions of convergence can be found in Zhang & He (2018, Section 2.3).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 485, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 503, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "The following result shows that Algorithm 1 needs polynomially many iterations to finds a point", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 615, + 335, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 132, + 628 + ], + "score": 0.9, + "content": "( \\mathbf { w } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 615, + 153, + 629 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 154, + 617, + 159, + 626 + ], + "score": 0.75, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 615, + 335, + 629 + ], + "score": 1.0, + "content": "-small Bregman stationarity in-expectation:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 604, + 505, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 506, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 508, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 177, + 646 + ], + "score": 1.0, + "content": "Theorem 1. Let", + "type": "text" + }, + { + "bbox": [ + 178, + 631, + 256, + 644 + ], + "score": 0.92, + "content": "F = f ( \\mathbf { w } ^ { ( 1 ) } , \\theta ^ { ( 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 629, + 324, + 646 + ], + "score": 1.0, + "content": "be the value of", + "type": "text" + }, + { + "bbox": [ + 325, + 633, + 332, + 644 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 629, + 508, + 646 + ], + "score": 1.0, + "content": "at initialization. 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EG is often considered", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "to converge quickly to sparse solutions over the simplex (Bradley & Bagnell, 2008; Bubeck, 2019),", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 407, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 422 + ], + "score": 1.0, + "content": "which makes it a natural choice for the architecture update. We thus propose GAEA, a Geometry-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Aware Exponentiated Algorithm in which operation weights on each edge are constrained to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "simplex and trained using EG; as in DARTS, the shared weights w are trained using SGD. GAEA", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "can be used as a simple, principled modification to the many NAS methods that treat architecture", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 451, + 507, + 468 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 154, + 468 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 155, + 452, + 235, + 464 + ], + "score": 0.92, + "content": "\\theta \\in \\Theta = \\mathbb { R } ^ { | E | \\times | O | }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 451, + 507, + 468 + ], + "score": 1.0, + "content": "as real-valued “logits” to be passed through a softmax to obtain", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 367, + 477 + ], + "score": 1.0, + "content": "mixture weights or probabilities for simplices over the operations", + "type": "text" + }, + { + "bbox": [ + 367, + 465, + 376, + 474 + ], + "score": 0.71, + "content": "O", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 464, + 506, + 477 + ], + "score": 1.0, + "content": ". Such methods include DARTS,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 158, + 489 + ], + "score": 1.0, + "content": "PC-DARTS (", + "type": "text" + }, + { + "bbox": [ + 158, + 476, + 172, + 486 + ], + "score": 0.32, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 473, + 506, + 489 + ], + "score": 1.0, + "content": "et al., 2020), and GDAS (Dong & Yang, 2019). 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We thus propose GAEA, a Geometry-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Aware Exponentiated Algorithm in which operation weights on each edge are constrained to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "simplex and trained using EG; as in DARTS, the shared weights w are trained using SGD. 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Then, given a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 509, + 488, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 183, + 524 + ], + "score": 1.0, + "content": "stochastic gradient", + "type": "text" + }, + { + "bbox": [ + 184, + 509, + 248, + 522 + ], + "score": 0.94, + "content": "\\tilde { \\nabla } _ { \\boldsymbol { \\theta } } f ( \\mathbf { w } ^ { ( t ) } , \\boldsymbol { \\theta } ^ { ( t ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 509, + 304, + 524 + ], + "score": 1.0, + "content": "and step-size", + "type": "text" + }, + { + "bbox": [ + 304, + 511, + 329, + 522 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 509, + 488, + 524 + ], + "score": 1.0, + "content": ", replace the architecture update by EG:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 354, + 507, + 524 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 526, + 454, + 586 + ], + "lines": [ + { + "bbox": [ + 155, + 526, + 454, + 586 + ], + "spans": [ + { + "bbox": [ + 155, + 526, + 454, + 586 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } & { \\tilde { \\theta } ^ { ( t + 1 ) } \\gets \\theta ^ { ( t ) } \\odot \\mathrm { e x p } \\left( - \\eta \\tilde { \\nabla } _ { \\theta } f \\big ( \\mathbf { w } ^ { ( t ) } , \\theta ^ { ( t ) } \\big ) \\right) } & { \\mathrm { ( m u l t i p l i c a t i v e ~ u p d a t e ) } } \\\\ & { \\theta _ { i , j , o } ^ { ( t + 1 ) } \\gets \\frac { \\tilde { \\theta } _ { i , j , o } ^ { ( t + 1 ) } } { \\sum _ { o ^ { \\prime } \\in O } \\tilde { \\theta } _ { i , j , o ^ { \\prime } } ^ { ( t + 1 ) } } \\ \\forall o \\in O , \\forall \\ ( i , j ) \\in E } & { \\mathrm { ( s i m p l e x ~ p r o j e c t i o n ) } } \\end{array}", + "type": "interline_equation", + "image_path": "f4432e997e5095b873c6e7a17b3681374f73d397d2b0096c90ad6bde4ab9fd89.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 155, + 526, + 454, + 546.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 155, + 546.0, + 454, + 566.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 155, + 566.0, + 454, + 586.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 590, + 503, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 239, + 603 + ], + "score": 1.0, + "content": "These two simple modifications,", + "type": "text" + }, + { + "bbox": [ + 239, + 592, + 248, + 600 + ], + "score": 0.57, + "content": "r e", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "-parameterization and exponentiation, suffice to obtain state-of-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "the-art results on several NAS benchmarks, as shown in Section 4. Note that to obtain a bilevel", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 107, + 612, + 295, + 624 + ], + "score": 1.0, + "content": "algorithm we simply replace the gradient w.r.t.", + "type": "text" + }, + { + "bbox": [ + 295, + 612, + 302, + 621 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "of the training loss with that of the validation loss.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 589, + 505, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 271, + 644 + ], + "score": 1.0, + "content": "GAEA is equivalent to Algorithm 1 with", + "type": "text" + }, + { + "bbox": [ + 272, + 628, + 429, + 642 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\phi ( \\theta ) = \\sum _ { ( i , j ) \\in E } \\sum _ { o \\in O } \\theta _ { i , j , o } \\log \\theta _ { i , j , o } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 627, + 506, + 644 + ], + "score": 1.0, + "content": ", which is strongly-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 642, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 160, + 657 + ], + "score": 1.0, + "content": "convex w.r.t.", + "type": "text" + }, + { + "bbox": [ + 160, + 642, + 211, + 656 + ], + "score": 0.93, + "content": "\\| \\cdot \\| _ { 1 } / { \\sqrt { | E | } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 642, + 291, + 657 + ], + "score": 1.0, + "content": "over the product of", + "type": "text" + }, + { + "bbox": [ + 291, + 643, + 321, + 655 + ], + "score": 0.76, + "content": "| E | \\mid O |", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 642, + 414, + 657 + ], + "score": 1.0, + "content": "-simplices. The dual is", + "type": "text" + }, + { + "bbox": [ + 415, + 642, + 464, + 656 + ], + "score": 0.9, + "content": "{ \\sqrt { | E | } } \\| \\cdot \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 642, + 488, + 657 + ], + "score": 1.0, + "content": ", so if", + "type": "text" + }, + { + "bbox": [ + 488, + 643, + 504, + 655 + ], + "score": 0.84, + "content": "G _ { \\mathbf { w } }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "bounds the shared-weights gradient and we have an entry-wise bound on the architecture gradient then", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 159, + 678 + ], + "score": 1.0, + "content": "GAEA reach", + "type": "text" + }, + { + "bbox": [ + 159, + 667, + 165, + 675 + ], + "score": 0.74, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 665, + 223, + 678 + ], + "score": 1.0, + "content": "-stationarity in", + "type": "text" + }, + { + "bbox": [ + 224, + 665, + 292, + 677 + ], + "score": 0.93, + "content": "\\mathcal { O } ( G _ { \\mathbf { w } } ^ { 2 } + | E | ) / \\varepsilon ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 665, + 435, + 678 + ], + "score": 1.0, + "content": "iterations. This can be up to a factor", + "type": "text" + }, + { + "bbox": [ + 436, + 666, + 450, + 677 + ], + "score": 0.9, + "content": "| O |", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "improvement", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "over SGD, either over the simplex or the logit space. 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Obtaining sparse architecture parameters is critical for good performance, both for the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 504, + 722 + ], + "score": 1.0, + "content": "mixture relaxation, where it alleviates the effect of discretization on the validation loss, and for the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 403, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 403, + 734 + ], + "score": 1.0, + "content": "stochastic relaxation, where it reduces noise when sampling architectures.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 627, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 183, + 506, + 334 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 79, + 505, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 1: DARTS: Comparison with SOTA NAS methods on the DARTS search space, plus three", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 506, + 104 + ], + "score": 1.0, + "content": "results on different search spaces with a similar number of parameters reported at the top for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 101, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 104, + 101, + 506, + 116 + ], + "score": 1.0, + "content": "comparison. All evaluations and reported performances of models found on the DARTS search space", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 126 + ], + "score": 1.0, + "content": "use similar training routines; this includes auxiliary towers and cutout but no other modifications,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 123, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 104, + 123, + 507, + 139 + ], + "score": 1.0, + "content": "e.g. label smoothing (Muller et al., 2019), AutoAugment (Cubuk et al., 2019), Swish (Ramachandran ¨", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "score": 1.0, + "content": "et al., 2017), Squeeze & Excite (Hu et al., 2018), etc. The specific training procedure we use is that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 145, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 506, + 159 + ], + "score": 1.0, + "content": "of PC-DARTS, which differs slightly from the DARTS routine by a small change to the drop-path", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "score": 1.0, + "content": "probability; PDARTS tunes both this and batch-size. Our results are averaged over 10 random seeds.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 167, + 502, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 502, + 180 + ], + "score": 1.0, + "content": "Search cost is hardware-dependent; we used Tesla V100 GPUs. For more details see Tables 4 & 5.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "table_body", + "bbox": [ + 106, + 183, + 506, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 183, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 334 + ], + "score": 0.984, + "html": "
Search Method (source)CIFAR-10 ErrorSearch Cost (GPU Days)ImageNet Error top-1Search Cost (GPU Days)method
BestAveragetop-5
NASNet-A* (Zoph et al.,2018)-2.65200026.08.41800RL
AmoebaNet-B* (Real et al.,2019)12.55± 0.05315024.37.63150evolution
ProxylessNAS*(Cai et al.,2019)2.08424.97.58.3gradient (WS)
ENAS (Pham et al., 2018)2.890.5--1RL (WS)
RS-Wst (Li& Talwalkar,2019)2.712.85±0.080.7-1-random (WS)
ASNG (Akimoto et al.,2019)12.83 ± 0.140.1--gradient (WS)
SNAS (Xie et al., 2019)=2.85±0.021.527.39.21.5gradient (WS)
DARTS (1st)† (Liu et al.,2019)=3.00 ±0.140.4---gradient (WS)
DARTS (2nd)† (Liu et al., 2019)2.76 ±0.09126.78.74.0gradient (WS)
PDARTS (Chen et al.,2019)2.50=0.324.47.40.3gradient (WS)
PC-DARTSt (Xu et al.,2020)-2.57 ±0.070.124.27.33.8gradient (WS)
GAEA PC-DARTS† (ours)2.392.50 ±0.060.124.07.33.8gradient (WS)
PC-DARTSt (Xu et al.,2020)(search on CIFAR-1O,train on ImageNet)25.17.80.1gradient (WS)
GAEA PC-DARTS† (ours)(search on CIFAR-10, train on ImageNet)24.37.30.1gradient (WS)
", + "type": "table", + "image_path": "d2b8f9aa236a5e72ec94b44075d51b3b0194ba075ddb46e0a7321909b464e9d1.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 106, + 183, + 506, + 233.33333333333334 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 106, + 233.33333333333334, + 506, + 283.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 106, + 283.6666666666667, + 506, + 334.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 111, + 336, + 474, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 334, + 475, + 346 + ], + "spans": [ + { + "bbox": [ + 116, + 334, + 475, + 346 + ], + "score": 1.0, + "content": "Search space/backbone differ from the DARTS setting; we show results for networks with a comparable number of parameters.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 117, + 343, + 474, + 353 + ], + "spans": [ + { + "bbox": [ + 117, + 343, + 474, + 353 + ], + "score": 1.0, + "content": "For fair comparison to other work, we show the search cost for training the shared-weights network with a single initialization.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 371, + 308, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 308, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 308, + 386 + ], + "score": 1.0, + "content": "4 EMPIRICAL RESULTS USING GAEA", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 506, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "We evaluate GAEA on three different computer vision benchmarks: the large and heavily studied", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "search space from DARTS (Liu et al., 2019) and two smaller oracle evaluation benchmarks, NAS-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "Bench-1Shot1 (Zela et al., 2020a), and NAS-Bench-201 (Dong & Yang, 2020). NAS-Bench-1Shot1", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 426, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 440 + ], + "score": 1.0, + "content": "differs from the others by applying operations per node instead of per edge, while NAS-Bench-201", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "differs by not requiring edge-pruning. Since GAEA can modify a variety of methods, e.g. DARTS,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "PC-DARTS (Xu et al., 2020), and GDAS (Dong & Yang, 2019), on each benchmark we start by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "evaluating the GAEA variant of the current best method on that benchmark. We show that despite the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "diversity of search spaces, GAEA improves upon this state-of-the-art across all three. Note that we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "use the same step-size for GAEA variants of DARTS/PC-DARTS and do not require weight-decay on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "architecture parameters. We defer experimental details and hyperparameter settings to the appendix", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 504, + 412, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 412, + 516 + ], + "score": 1.0, + "content": "and release all code, hyperparameters, and random seeds for reproducibility.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 523, + 309, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 310, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 310, + 535 + ], + "score": 1.0, + "content": "4.1 CONVERGENCE AND SPARSITY OF GAEA", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 555 + ], + "score": 1.0, + "content": "We first examine the impact of GAEA on convergence and sparsity. Figure 1 shows the entropy", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "of the operation weights averaged across nodes for a GAEA-variant and its base method across", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "the three benchmarks, demonstrating that it decreases much faster for GAEA-modified approaches.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "This validates our expectation that GAEA encourages sparse architecture parameters, which should", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "alleviate the mismatch between the continuously relaxed architecture parameters and the discrete", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "architecture returned. Indeed, we find that post-search discretization on the DARTS search space", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 389, + 619 + ], + "score": 1.0, + "content": "causes the validation accuracy of the PC-DARTS supernet to drop from", + "type": "text" + }, + { + "bbox": [ + 389, + 607, + 421, + 617 + ], + "score": 0.89, + "content": "7 2 . 1 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 606, + 432, + 619 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 432, + 607, + 463, + 617 + ], + "score": 0.88, + "content": "1 5 . 2 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 606, + 506, + 619 + ], + "score": 1.0, + "content": ", while for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 507, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 254, + 631 + ], + "score": 1.0, + "content": "GAEA PC-DARTS the drop is only", + "type": "text" + }, + { + "bbox": [ + 255, + 618, + 288, + 628 + ], + "score": 0.88, + "content": "7 5 . 0 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 616, + 299, + 631 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 300, + 618, + 332, + 628 + ], + "score": 0.88, + "content": "3 3 . 2 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 616, + 507, + 631 + ], + "score": 1.0, + "content": "; note that this is shared-weights accuracy,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "obtained without retraining the final network. The numbers demonstrate that GAEA both (1) achieves", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "better supernet optimization of the weight-sharing objective and (2) suffers less due to discretization.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 658, + 300, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 302, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 302, + 672 + ], + "score": 1.0, + "content": "4.2 GAEA ON THE DARTS SEARCH SPACE", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "Here we evaluate GAEA on the task of designing CNN cells for CIFAR-10 (Krizhevksy, 2009) and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "ImageNet (Russakovsky et al., 2015) by using it to modify PC-DARTS (Xu et al., 2020), the current", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "state-of-the-art method. We follow the same three stage process used by both DARTS and RS-WS", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "for search and evaluation. Table 1 displays results on both datasets and demonstrates that GAEA’s", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "parameterization and optimization scheme improves upon PC-DARTS. In fact, GAEA PC-DARTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 183, + 506, + 334 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 79, + 505, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 1: DARTS: Comparison with SOTA NAS methods on the DARTS search space, plus three", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 506, + 104 + ], + "score": 1.0, + "content": "results on different search spaces with a similar number of parameters reported at the top for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 101, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 104, + 101, + 506, + 116 + ], + "score": 1.0, + "content": "comparison. All evaluations and reported performances of models found on the DARTS search space", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 126 + ], + "score": 1.0, + "content": "use similar training routines; this includes auxiliary towers and cutout but no other modifications,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 123, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 104, + 123, + 507, + 139 + ], + "score": 1.0, + "content": "e.g. label smoothing (Muller et al., 2019), AutoAugment (Cubuk et al., 2019), Swish (Ramachandran ¨", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 148 + ], + "score": 1.0, + "content": "et al., 2017), Squeeze & Excite (Hu et al., 2018), etc. The specific training procedure we use is that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 145, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 506, + 159 + ], + "score": 1.0, + "content": "of PC-DARTS, which differs slightly from the DARTS routine by a small change to the drop-path", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "score": 1.0, + "content": "probability; PDARTS tunes both this and batch-size. Our results are averaged over 10 random seeds.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 167, + 502, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 502, + 180 + ], + "score": 1.0, + "content": "Search cost is hardware-dependent; we used Tesla V100 GPUs. For more details see Tables 4 & 5.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "table_body", + "bbox": [ + 106, + 183, + 506, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 183, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 334 + ], + "score": 0.984, + "html": "
Search Method (source)CIFAR-10 ErrorSearch Cost (GPU Days)ImageNet Error top-1Search Cost (GPU Days)method
BestAveragetop-5
NASNet-A* (Zoph et al.,2018)-2.65200026.08.41800RL
AmoebaNet-B* (Real et al.,2019)12.55± 0.05315024.37.63150evolution
ProxylessNAS*(Cai et al.,2019)2.08424.97.58.3gradient (WS)
ENAS (Pham et al., 2018)2.890.5--1RL (WS)
RS-Wst (Li& Talwalkar,2019)2.712.85±0.080.7-1-random (WS)
ASNG (Akimoto et al.,2019)12.83 ± 0.140.1--gradient (WS)
SNAS (Xie et al., 2019)=2.85±0.021.527.39.21.5gradient (WS)
DARTS (1st)† (Liu et al.,2019)=3.00 ±0.140.4---gradient (WS)
DARTS (2nd)† (Liu et al., 2019)2.76 ±0.09126.78.74.0gradient (WS)
PDARTS (Chen et al.,2019)2.50=0.324.47.40.3gradient (WS)
PC-DARTSt (Xu et al.,2020)-2.57 ±0.070.124.27.33.8gradient (WS)
GAEA PC-DARTS† (ours)2.392.50 ±0.060.124.07.33.8gradient (WS)
PC-DARTSt (Xu et al.,2020)(search on CIFAR-1O,train on ImageNet)25.17.80.1gradient (WS)
GAEA PC-DARTS† (ours)(search on CIFAR-10, train on ImageNet)24.37.30.1gradient (WS)
", + "type": "table", + "image_path": "d2b8f9aa236a5e72ec94b44075d51b3b0194ba075ddb46e0a7321909b464e9d1.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 106, + 183, + 506, + 233.33333333333334 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 106, + 233.33333333333334, + 506, + 283.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 106, + 283.6666666666667, + 506, + 334.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 111, + 336, + 474, + 353 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 334, + 475, + 346 + ], + "spans": [ + { + "bbox": [ + 116, + 334, + 475, + 346 + ], + "score": 1.0, + "content": "Search space/backbone differ from the DARTS setting; we show results for networks with a comparable number of parameters.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 117, + 343, + 474, + 353 + ], + "spans": [ + { + "bbox": [ + 117, + 343, + 474, + 353 + ], + "score": 1.0, + "content": "For fair comparison to other work, we show the search cost for training the shared-weights network with a single initialization.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 371, + 308, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 308, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 308, + 386 + ], + "score": 1.0, + "content": "4 EMPIRICAL RESULTS USING GAEA", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 506, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "We evaluate GAEA on three different computer vision benchmarks: the large and heavily studied", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "search space from DARTS (Liu et al., 2019) and two smaller oracle evaluation benchmarks, NAS-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "Bench-1Shot1 (Zela et al., 2020a), and NAS-Bench-201 (Dong & Yang, 2020). NAS-Bench-1Shot1", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 426, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 440 + ], + "score": 1.0, + "content": "differs from the others by applying operations per node instead of per edge, while NAS-Bench-201", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "differs by not requiring edge-pruning. Since GAEA can modify a variety of methods, e.g. DARTS,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "PC-DARTS (Xu et al., 2020), and GDAS (Dong & Yang, 2019), on each benchmark we start by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "evaluating the GAEA variant of the current best method on that benchmark. We show that despite the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "diversity of search spaces, GAEA improves upon this state-of-the-art across all three. Note that we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "use the same step-size for GAEA variants of DARTS/PC-DARTS and do not require weight-decay on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "architecture parameters. We defer experimental details and hyperparameter settings to the appendix", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 504, + 412, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 412, + 516 + ], + "score": 1.0, + "content": "and release all code, hyperparameters, and random seeds for reproducibility.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 394, + 506, + 516 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 523, + 309, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 310, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 310, + 535 + ], + "score": 1.0, + "content": "4.1 CONVERGENCE AND SPARSITY OF GAEA", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 555 + ], + "score": 1.0, + "content": "We first examine the impact of GAEA on convergence and sparsity. Figure 1 shows the entropy", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "of the operation weights averaged across nodes for a GAEA-variant and its base method across", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "the three benchmarks, demonstrating that it decreases much faster for GAEA-modified approaches.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "This validates our expectation that GAEA encourages sparse architecture parameters, which should", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "alleviate the mismatch between the continuously relaxed architecture parameters and the discrete", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "architecture returned. Indeed, we find that post-search discretization on the DARTS search space", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 389, + 619 + ], + "score": 1.0, + "content": "causes the validation accuracy of the PC-DARTS supernet to drop from", + "type": "text" + }, + { + "bbox": [ + 389, + 607, + 421, + 617 + ], + "score": 0.89, + "content": "7 2 . 1 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 606, + 432, + 619 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 432, + 607, + 463, + 617 + ], + "score": 0.88, + "content": "1 5 . 2 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 606, + 506, + 619 + ], + "score": 1.0, + "content": ", while for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 507, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 254, + 631 + ], + "score": 1.0, + "content": "GAEA PC-DARTS the drop is only", + "type": "text" + }, + { + "bbox": [ + 255, + 618, + 288, + 628 + ], + "score": 0.88, + "content": "7 5 . 0 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 616, + 299, + 631 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 300, + 618, + 332, + 628 + ], + "score": 0.88, + "content": "3 3 . 2 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 616, + 507, + 631 + ], + "score": 1.0, + "content": "; note that this is shared-weights accuracy,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "obtained without retraining the final network. The numbers demonstrate that GAEA both (1) achieves", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "better supernet optimization of the weight-sharing objective and (2) suffers less due to discretization.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 539, + 507, + 652 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 658, + 300, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 302, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 302, + 672 + ], + "score": 1.0, + "content": "4.2 GAEA ON THE DARTS SEARCH SPACE", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "Here we evaluate GAEA on the task of designing CNN cells for CIFAR-10 (Krizhevksy, 2009) and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "ImageNet (Russakovsky et al., 2015) by using it to modify PC-DARTS (Xu et al., 2020), the current", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "state-of-the-art method. We follow the same three stage process used by both DARTS and RS-WS", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "for search and evaluation. Table 1 displays results on both datasets and demonstrates that GAEA’s", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "parameterization and optimization scheme improves upon PC-DARTS. In fact, GAEA PC-DARTS", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "outperforms all search methods except ProxylessNAS, which uses 1.5 times as many parameters on a", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 275, + 507, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 507, + 289 + ], + "score": 1.0, + "content": "different search space. Thus we improve the state-of-the-art on the DARTS search space. To meet a", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "higher bar for reproducibility on CIFAR-10, in Appendix C we report “broad reproducibility” (Li &", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "Talwalkar, 2019) by repeating our pipeline with new seeds. While GAEA PC-DARTS consistently", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "finds good networks when selecting the best of four independent trials, multiple trials are required", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 503, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 503, + 333 + ], + "score": 1.0, + "content": "due to sensitivity to initialization, as is true for many approaches (Liu et al., 2019; Xu et al., 2020).", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 676, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 78, + 505, + 168 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 78, + 505, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 168 + ], + "score": 0.95, + "type": "image", + "image_path": "256fe57190a403501ca5f7fd2ddac27d71516438ba9a18b29b2b46a4467b7320.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 78, + 505, + 108.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 108.0, + 505, + 138.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 138.0, + 505, + 168.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 175, + 506, + 242 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "Figure 2: NAS-Bench-1Shot1: Online comparison of PC-DARTS and GAEA PC-DARTS in terms", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "of the test regret at each epoch of shared-weights training, i.e. the difference between the ground truth", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "test error of the proposed architecture and that of the best architecture in the search space. The dark", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 209, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 399, + 221 + ], + "score": 1.0, + "content": "lines indicate the mean of four random trials and the light colored bands", + "type": "text" + }, + { + "bbox": [ + 400, + 210, + 410, + 219 + ], + "score": 0.72, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 209, + 506, + 221 + ], + "score": 1.0, + "content": "one standard deviation.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "The dashed line is the final regret of the best weight-sharing method according to Zela et al. (2020b);", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 231, + 502, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 502, + 243 + ], + "score": 1.0, + "content": "note that in our reproduction PC-DARTS performed better than their evaluation on spaces 1 and 3.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "outperforms all search methods except ProxylessNAS, which uses 1.5 times as many parameters on a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 275, + 507, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 507, + 289 + ], + "score": 1.0, + "content": "different search space. Thus we improve the state-of-the-art on the DARTS search space. To meet a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "higher bar for reproducibility on CIFAR-10, in Appendix C we report “broad reproducibility” (Li &", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "Talwalkar, 2019) by repeating our pipeline with new seeds. While GAEA PC-DARTS consistently", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "finds good networks when selecting the best of four independent trials, multiple trials are required", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 320, + 503, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 503, + 333 + ], + "score": 1.0, + "content": "due to sensitivity to initialization, as is true for many approaches (Liu et al., 2019; Xu et al., 2020).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 506, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 212, + 350 + ], + "score": 1.0, + "content": "On ImageNet, we follow", + "type": "text" + }, + { + "bbox": [ + 212, + 337, + 226, + 347 + ], + "score": 0.3, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 336, + 415, + 350 + ], + "score": 1.0, + "content": "et al. (2020) by using subsamples containing", + "type": "text" + }, + { + "bbox": [ + 415, + 336, + 435, + 348 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 336, + 454, + 350 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 336, + 477, + 348 + ], + "score": 0.88, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 347, + 507, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 507, + 361 + ], + "score": 1.0, + "content": "training images from ILSVRC-2012 (Russakovsky et al., 2015) as training and validation sets,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "score": 1.0, + "content": "respectively. We fix architecture parameters for the first 35 epochs, then run GAEA PC-DARTS with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 336, + 381 + ], + "score": 1.0, + "content": "step-size 0.1. All other hyperparameters match those of", + "type": "text" + }, + { + "bbox": [ + 336, + 370, + 351, + 380 + ], + "score": 0.4, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 370, + 506, + 381 + ], + "score": 1.0, + "content": "et al. (2020). Table 1 shows the final", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "performance of both the architecture found by GAEA PC-DARTS on CIFAR-10 and the one found", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 390, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 470, + 405 + ], + "score": 1.0, + "content": "directly on ImageNet when trained from scratch for 250 epochs using the same settings as", + "type": "text" + }, + { + "bbox": [ + 470, + 392, + 484, + 402 + ], + "score": 0.31, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 390, + 507, + 405 + ], + "score": 1.0, + "content": "et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 327, + 415 + ], + "score": 1.0, + "content": "(2020). GAEA PC-DARTS achieves a top-1 test error of", + "type": "text" + }, + { + "bbox": [ + 328, + 402, + 354, + 413 + ], + "score": 0.89, + "content": "2 4 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 402, + 506, + 415 + ], + "score": 1.0, + "content": ", which is state-of-the-art performance", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "in the mobile setting when excluding additional training modifications, e.g. those in the caption.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "Additionally, the architecture found by GAEA PC-DARTS for CIFAR-10 and transferred achieves", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 164, + 447 + ], + "score": 1.0, + "content": "a test error of", + "type": "text" + }, + { + "bbox": [ + 165, + 435, + 192, + 446 + ], + "score": 0.87, + "content": "2 4 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 435, + 272, + 447 + ], + "score": 1.0, + "content": ", comparable to the", + "type": "text" + }, + { + "bbox": [ + 272, + 435, + 300, + 446 + ], + "score": 0.88, + "content": "2 4 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "error of the one found by PC-DARTS directly on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 446, + 503, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 503, + 459 + ], + "score": 1.0, + "content": "ImageNet. Top architectures found by GAEA PC-DARTS are depicted in Figure 3 in the appendix.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 271, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 273, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 273, + 479 + ], + "score": 1.0, + "content": "4.3 GAEA ON NAS-BENCH-1SHOT1", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "NAS-Bench-1Shot1 (Zela et al., 2020a) is a subset of NAS-Bench-101 (Ying et al., 2019) that allows", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "benchmarking weight-sharing methods on three search spaces over CIFAR-10 that differ in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "number of nodes considered and the number of input edges per node. Of the weight-sharing methods", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "benchmarked by Zela et al. (2020a), we found that PC-DARTS achieves the best performance on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "2 of 3 search spaces, so we again evaluate GAEA PC-DARTS here. Figure 2 shows that GAEA", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "PC-DARTS consistently finds better architectures on average than PC-DARTS and thus exceeds the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "performance of the best method from Zela et al. (2020a) on 2 of 3 search spaces. We hypothesize", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "that the benefits of GAEA are limited here due to the near-saturation of NAS methods. 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Dong", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "& Yang (2020) report results for several algorithms in the transfer NAS setting, where search is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "conducted on CIFAR-10 and the resulting networks are trained on a possibly different target dataset.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Table 2 reports a subset of these results alongside evaluations of our implementation of several", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "existing and GAEA-modified NAS methods in both the transfer and direct setting. 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The dark", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 209, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 399, + 221 + ], + "score": 1.0, + "content": "lines indicate the mean of four random trials and the light colored bands", + "type": "text" + }, + { + "bbox": [ + 400, + 210, + 410, + 219 + ], + "score": 0.72, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 209, + 506, + 221 + ], + "score": 1.0, + "content": "one standard deviation.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "The dashed line is the final regret of the best weight-sharing method according to Zela et al. (2020b);", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 231, + 502, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 502, + 243 + ], + "score": 1.0, + "content": "note that in our reproduction PC-DARTS performed better than their evaluation on spaces 1 and 3.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 331 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 105, + 264, + 507, + 333 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 506, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 212, + 350 + ], + "score": 1.0, + "content": "On ImageNet, we follow", + "type": "text" + }, + { + "bbox": [ + 212, + 337, + 226, + 347 + ], + "score": 0.3, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 336, + 415, + 350 + ], + "score": 1.0, + "content": "et al. (2020) by using subsamples containing", + "type": "text" + }, + { + "bbox": [ + 415, + 336, + 435, + 348 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 336, + 454, + 350 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 336, + 477, + 348 + ], + "score": 0.88, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 347, + 507, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 507, + 361 + ], + "score": 1.0, + "content": "training images from ILSVRC-2012 (Russakovsky et al., 2015) as training and validation sets,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "score": 1.0, + "content": "respectively. We fix architecture parameters for the first 35 epochs, then run GAEA PC-DARTS with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 336, + 381 + ], + "score": 1.0, + "content": "step-size 0.1. All other hyperparameters match those of", + "type": "text" + }, + { + "bbox": [ + 336, + 370, + 351, + 380 + ], + "score": 0.4, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 370, + 506, + 381 + ], + "score": 1.0, + "content": "et al. (2020). Table 1 shows the final", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "performance of both the architecture found by GAEA PC-DARTS on CIFAR-10 and the one found", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 390, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 470, + 405 + ], + "score": 1.0, + "content": "directly on ImageNet when trained from scratch for 250 epochs using the same settings as", + "type": "text" + }, + { + "bbox": [ + 470, + 392, + 484, + 402 + ], + "score": 0.31, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 390, + 507, + 405 + ], + "score": 1.0, + "content": "et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 327, + 415 + ], + "score": 1.0, + "content": "(2020). GAEA PC-DARTS achieves a top-1 test error of", + "type": "text" + }, + { + "bbox": [ + 328, + 402, + 354, + 413 + ], + "score": 0.89, + "content": "2 4 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 402, + 506, + 415 + ], + "score": 1.0, + "content": ", which is state-of-the-art performance", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "in the mobile setting when excluding additional training modifications, e.g. those in the caption.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "Additionally, the architecture found by GAEA PC-DARTS for CIFAR-10 and transferred achieves", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 164, + 447 + ], + "score": 1.0, + "content": "a test error of", + "type": "text" + }, + { + "bbox": [ + 165, + 435, + 192, + 446 + ], + "score": 0.87, + "content": "2 4 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 435, + 272, + 447 + ], + "score": 1.0, + "content": ", comparable to the", + "type": "text" + }, + { + "bbox": [ + 272, + 435, + 300, + 446 + ], + "score": 0.88, + "content": "2 4 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "error of the one found by PC-DARTS directly on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 446, + 503, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 503, + 459 + ], + "score": 1.0, + "content": "ImageNet. Top architectures found by GAEA PC-DARTS are depicted in Figure 3 in the appendix.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 336, + 507, + 459 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 466, + 271, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 273, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 273, + 479 + ], + "score": 1.0, + "content": "4.3 GAEA ON NAS-BENCH-1SHOT1", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "NAS-Bench-1Shot1 (Zela et al., 2020a) is a subset of NAS-Bench-101 (Ying et al., 2019) that allows", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "benchmarking weight-sharing methods on three search spaces over CIFAR-10 that differ in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "number of nodes considered and the number of input edges per node. Of the weight-sharing methods", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "benchmarked by Zela et al. (2020a), we found that PC-DARTS achieves the best performance on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "2 of 3 search spaces, so we again evaluate GAEA PC-DARTS here. Figure 2 shows that GAEA", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "PC-DARTS consistently finds better architectures on average than PC-DARTS and thus exceeds the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "performance of the best method from Zela et al. (2020a) on 2 of 3 search spaces. We hypothesize", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "that the benefits of GAEA are limited here due to the near-saturation of NAS methods. In particular,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 231, + 585 + ], + "score": 1.0, + "content": "existing methods obtain within", + "type": "text" + }, + { + "bbox": [ + 232, + 572, + 246, + 583 + ], + "score": 0.91, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "test error of the top network in each space, while the latters’ test", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 484, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 324, + 597 + ], + "score": 1.0, + "content": "errors when evaluated with different initializations are", + "type": "text" + }, + { + "bbox": [ + 325, + 583, + 352, + 594 + ], + "score": 0.83, + "content": "0 . 3 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 582, + 355, + 597 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 356, + 583, + 383, + 594 + ], + "score": 0.83, + "content": "0 . 2 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 582, + 401, + 597 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 401, + 583, + 428, + 594 + ], + "score": 0.89, + "content": "0 . 1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 582, + 484, + 597 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 484, + 506, + 597 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 604, + 254, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 255, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 255, + 617 + ], + "score": 1.0, + "content": "4.4 GAEA ON NAS-BENCH-201", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "NAS-Bench-201 has one search space on three datasets—CIFAR-10, CIFAR-100, and ImageNet-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 380, + 645 + ], + "score": 1.0, + "content": "16-120—that includes 4-node architectures with an operation from", + "type": "text" + }, + { + "bbox": [ + 380, + 633, + 428, + 645 + ], + "score": 0.63, + "content": "O = \\{ \\mathrm { n o n e }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 632, + 506, + 645 + ], + "score": 1.0, + "content": ", skip connect, 1x1", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 226, + 658 + ], + "score": 1.0, + "content": "convolution, 3x3 convolution,", + "type": "text" + }, + { + "bbox": [ + 227, + 644, + 284, + 656 + ], + "score": 0.76, + "content": "3 \\mathrm { x } 3 \\mathrm { a v g } \\mathrm { p o o l } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 642, + 506, + 658 + ], + "score": 1.0, + "content": "on each edge, yielding 15625 possible networks. Dong", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "& Yang (2020) report results for several algorithms in the transfer NAS setting, where search is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "conducted on CIFAR-10 and the resulting networks are trained on a possibly different target dataset.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Table 2 reports a subset of these results alongside evaluations of our implementation of several", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "existing and GAEA-modified NAS methods in both the transfer and direct setting. Both the results", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "from Dong & Yang (2020) and our reproductions show that GDAS is the best previous weight-sharing", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "method; we evaluate GAEA GDAS and find that it achieves better results on CIFAR-100 and similar", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 239, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 239, + 732 + ], + "score": 1.0, + "content": "results on the other two datasets.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 621, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 149, + 505, + 354 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 507, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 507, + 93 + ], + "score": 1.0, + "content": "Table 2: NAS-Bench-201: Results are separated into traditional hyperparameter optimization al-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "score": 1.0, + "content": "gorithms with search run on CIFAR-10 (top block), weight-sharing methods with search run on", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 100, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 104, + 100, + 506, + 117 + ], + "score": 1.0, + "content": "CIFAR-10 (middle block), and weight-sharing methods run directly on the dataset used for training", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "(bottom block). The use of transfer NAS follows the evaluations conducted by Dong & Yang (2020);", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "score": 1.0, + "content": "unless otherwise stated all non-GAEA results are from their paper. The best results in the transfer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 294, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 294, + 147 + ], + "score": 1.0, + "content": "and direct settings on each dataset are bolded.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 149, + 505, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 149, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 354 + ], + "score": 0.986, + "html": "
Search*(seconds)CIFAR-10 (test)CIFAR-100 (test)ImageNet-16-120 (test)
RegularREAN/A93.92 ± 0.3071.84 ± 0.9945.54 ± 1.03
HO,RSN/A93.70±0.3671.04 ± 1.0844.57 ±1.25
search onREINFORCEN/A93.85 ± 0.3771.71 ± 1.0945.25 ± 1.18
CIFAR-10BOHBN/A93.61 ±0.5270.85 ±1.2844.42 ± 1.49
RSPS758787.66 ± 1.6958.33 ± 4.3431.14 ± 3.88
DARTS (bilevel)3578154.30 ±0.0015.61 ±0.0016.32 ± 0.00
WeightSETN3413987.64 ± 0.0059.05 ± 0.2432.52 ±0.21
sharing,GDAS3160993.61 ±0.0970.70 ± 0.3041.71 ± 0.98
search onDARTS‡ (bilevel)10683†54.30 ±0.0015.32 ± 0.0016.38 ± 0.00
CIFAR-10GAEA DARTS (bilevel)7930+91.63 ± 2.5768.39 ± 4.4741.59 ± 4.20
DARTS* (ERM)18112t84.39 ±3.8254.81 ± 7.0831.82 ± 4.78
GAEA DARTS (ERM)9061†94.10±0.2972.60± 0.8945.81 ± 0.51
WeightGDAS27923†93.52 ± 0.1567.52 ± 0.1540.91 ± 0.12
GAEA GDAS16754t93.55 ± 0.1370.47 ± 0.4740.91 ±0.12
sharing,DARTS‡ (bilevel)10683†54.30 ±0.0015.32 ±0.0028.96 ±10.22
directGAEA DARTS (bilevel)7930t91.63 ± 2.5771.87 ± 0.5745.69 ± 0.56
searchDARTS* (ERM)18112†84.39 ±3.8251.26 ±6.1431.35 ± 7.46
GAEA DARTS (ERM)9061†94.10±0.2973.43 ± 0.1346.36 ±0.00
ResNetN/A93.9770.8643.63
OptimalN/A94.3773.5147.31
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We evaluate GAEA DARTS with both single-level (ERM) and bilevel optimization; recall", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "that in the latter case we optimize architecture parameters w.r.t. the validation loss and the shared", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "weights w.r.t. the training loss, whereas in ERM there is no data split. GAEA DARTS (ERM)", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "achieves state-of-the-art performance on all three datasets in both the transfer and direct setting,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "exceeding the test accuracy of both weight-sharing and traditional hyperparameter tuning by a wide", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "score": 1.0, + "content": "margin. GAEA DARTS (bilevel) performs worse but still exceeds all other methods on CIFAR-100", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "and ImageNet-16-120 in the direct search setting. The result thus also confirms the relevance of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "studying the single-level case to understand NAS; notably, the DARTS (ERM) baseline also improves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 525, + 303, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 303, + 537 + ], + "score": 1.0, + "content": "substantially upon the DARTS (bilevel) baseline.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 567, + 195, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 197, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 197, + 583 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "score": 1.0, + "content": "In this paper we take an optimization-based view of NAS, arguing that the design of good NAS", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "score": 1.0, + "content": "algorithms is largely a matter of successfully optimizing and regularizing the supernet. In support of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "this, we develop GAEA, a simple modification of gradient-based NAS that attains state-of-the-art", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "performance on several computer vision benchmarks while enjoying favorable speed and sparsity", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "properties. We believe that obtaining high-performance NAS algorithms for a wide variety of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "applications will continue to require a similar co-design of search space parameterizations and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "optimization methods, and that our geometry-aware framework can help accelerate this process. 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Search*(seconds)CIFAR-10 (test)CIFAR-100 (test)ImageNet-16-120 (test)
RegularREAN/A93.92 ± 0.3071.84 ± 0.9945.54 ± 1.03
HO,RSN/A93.70±0.3671.04 ± 1.0844.57 ±1.25
search onREINFORCEN/A93.85 ± 0.3771.71 ± 1.0945.25 ± 1.18
CIFAR-10BOHBN/A93.61 ±0.5270.85 ±1.2844.42 ± 1.49
RSPS758787.66 ± 1.6958.33 ± 4.3431.14 ± 3.88
DARTS (bilevel)3578154.30 ±0.0015.61 ±0.0016.32 ± 0.00
WeightSETN3413987.64 ± 0.0059.05 ± 0.2432.52 ±0.21
sharing,GDAS3160993.61 ±0.0970.70 ± 0.3041.71 ± 0.98
search onDARTS‡ (bilevel)10683†54.30 ±0.0015.32 ± 0.0016.38 ± 0.00
CIFAR-10GAEA DARTS (bilevel)7930+91.63 ± 2.5768.39 ± 4.4741.59 ± 4.20
DARTS* (ERM)18112t84.39 ±3.8254.81 ± 7.0831.82 ± 4.78
GAEA DARTS (ERM)9061†94.10±0.2972.60± 0.8945.81 ± 0.51
WeightGDAS27923†93.52 ± 0.1567.52 ± 0.1540.91 ± 0.12
GAEA GDAS16754t93.55 ± 0.1370.47 ± 0.4740.91 ±0.12
sharing,DARTS‡ (bilevel)10683†54.30 ±0.0015.32 ±0.0028.96 ±10.22
directGAEA DARTS (bilevel)7930t91.63 ± 2.5771.87 ± 0.5745.69 ± 0.56
searchDARTS* (ERM)18112†84.39 ±3.8251.26 ±6.1431.35 ± 7.46
GAEA DARTS (ERM)9061†94.10±0.2973.43 ± 0.1346.36 ±0.00
ResNetN/A93.9770.8643.63
OptimalN/A94.3773.5147.31
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GAEA DARTS (bilevel) performs worse but still exceeds all other methods on CIFAR-100", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "and ImageNet-16-120 in the direct search setting. The result thus also confirms the relevance of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "studying the single-level case to understand NAS; notably, the DARTS (ERM) baseline also improves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 525, + 303, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 303, + 537 + ], + "score": 1.0, + "content": "substantially upon the DARTS (bilevel) baseline.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 414, + 506, + 537 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 567, + 195, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 566, + 197, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 197, + 583 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "score": 1.0, + "content": "In this paper we take an optimization-based view of NAS, arguing that the design of good NAS", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "score": 1.0, + "content": "algorithms is largely a matter of successfully optimizing and regularizing the supernet. 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This work was", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "supported in part by DARPA under cooperative agreements FA875017C0141 and HR0011202000,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "NSF grants CCF-1535967, CCF-1910321, IIS-1618714, IIS-1705121, IIS-1838017, IIS-1901403, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 150 + ], + "score": 1.0, + "content": "IIS-2046613, a Microsoft Research Faculty Fellowship, a Bloomberg Data Science research grant, an", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "Amazon Research Award, an AWS Machine Learning Research Award, a Facebook Faculty Research", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Award, funding from Booz Allen Hamilton Inc., a Block Center Grant, a Carnegie Bosch Institute", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "Research Award, and a Two Sigma Fellowship Award. Any opinions, findings and conclusions, or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "recommendations expressed in this material are those of the authors and do not necessarily reflect the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 319, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 319, + 205 + ], + "score": 1.0, + "content": "views of DARPA, NSF, or any other funding agency.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 220, + 175, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 220, + 176, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 176, + 234 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 239, + 504, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "Youhei Akimoto, Shinichi Shirakawa, Nozomu Noshinari, Kento Uchida, Shota Saito, and Kouhei", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 116, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 116, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "Nishida. 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For each", + "type": "text" + }, + { + "bbox": [ + 180, + 259, + 207, + 270 + ], + "score": 0.9, + "content": "i \\in [ b ]", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 258, + 333, + 272 + ], + "score": 1.0, + "content": "there exists a stochastic oracle", + "type": "text" + }, + { + "bbox": [ + 334, + 259, + 345, + 270 + ], + "score": 0.89, + "content": "G _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 258, + 402, + 272 + ], + "score": 1.0, + "content": "that for input", + "type": "text" + }, + { + "bbox": [ + 403, + 259, + 430, + 270 + ], + "score": 0.89, + "content": "\\mathbf { x } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "outputs a random", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 142, + 270, + 171, + 282 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 171, + 270, + 205, + 281 + ], + "score": 0.92, + "content": "G _ { i } ( \\mathbf { x } , \\boldsymbol { \\xi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 270, + 225, + 282 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 225, + 270, + 314, + 282 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\xi } G _ { i } ( \\mathbf { x } , \\xi ) \\in \\partial _ { i } f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 270, + 346, + 282 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 346, + 270, + 375, + 281 + ], + "score": 0.92, + "content": "\\partial _ { i } f ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "is the subdifferential set of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 280, + 388, + 296 + ], + "spans": [ + { + "bbox": [ + 142, + 280, + 186, + 296 + ], + "score": 1.0, + "content": "restriction", + "type": "text" + }, + { + "bbox": [ + 186, + 282, + 223, + 294 + ], + "score": 0.89, + "content": "f ( \\cdot , \\mathbf { x } _ { - i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 280, + 235, + 296 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 235, + 283, + 245, + 293 + ], + "score": 0.82, + "content": "\\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 280, + 293, + 296 + ], + "score": 1.0, + "content": ". 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Positivity and homogeneity are trivial. 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For some fixed constants", + "type": "text" + }, + { + "bbox": [ + 252, + 208, + 295, + 219 + ], + "score": 0.91, + "content": "\\gamma _ { i } , L _ { i } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 206, + 331, + 221 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 331, + 207, + 358, + 220 + ], + "score": 0.92, + "content": "i \\in [ b ]", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 206, + 452, + 221 + ], + "score": 1.0, + "content": "we have the following:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 100, + 506, + 221 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 228, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 130, + 228, + 504, + 241 + ], + "spans": [ + { + "bbox": [ + 130, + 228, + 142, + 241 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 230, + 191, + 240 + ], + "score": 0.91, + "content": "f : \\mathcal { X } \\mapsto \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 228, + 384, + 241 + ], + "score": 1.0, + "content": "is everywhere-subdifferentiable with minimum", + "type": "text" + }, + { + "bbox": [ + 384, + 230, + 428, + 240 + ], + "score": 0.92, + "content": "f ^ { * } > - \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 228, + 474, + 241 + ], + "score": 1.0, + "content": "and for all", + "type": "text" + }, + { + "bbox": [ + 474, + 229, + 504, + 240 + ], + "score": 0.85, + "content": "\\mathbf { x } \\in \\mathcal { X }", + "type": "inline_equation" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 240, + 387, + 253 + ], + "spans": [ + { + "bbox": [ + 142, + 240, + 181, + 253 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 181, + 240, + 208, + 252 + ], + "score": 0.92, + "content": "i \\in [ b ]", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 240, + 267, + 253 + ], + "score": 1.0, + "content": "the restriction", + "type": "text" + }, + { + "bbox": [ + 267, + 240, + 305, + 252 + ], + "score": 0.92, + "content": "f ( \\cdot , \\mathbf { x } _ { - i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 240, + 315, + 253 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 315, + 241, + 325, + 252 + ], + "score": 0.82, + "content": "\\gamma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 240, + 372, + 253 + ], + "score": 1.0, + "content": "-RWC w.r.t.", + "type": "text" + }, + { + "bbox": [ + 372, + 241, + 382, + 251 + ], + "score": 0.87, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 240, + 387, + 253 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 128, + 258, + 180, + 272 + ], + "score": 1.0, + "content": "2. 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Moreover,", + "type": "text" + }, + { + "bbox": [ + 293, + 281, + 383, + 295 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\xi } \\| G _ { i } ( \\mathbf { x } , \\xi ) \\| _ { i , * } ^ { 2 } \\le L _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 280, + 388, + 296 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + } + ], + "index": 12, + "bbox_fs": [ + 128, + 228, + 505, + 296 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 284, + 320 + ], + "lines": [ + { + "bbox": [ + 102, + 299, + 281, + 324 + ], + "spans": [ + { + "bbox": [ + 102, + 299, + 134, + 324 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 135, + 308, + 200, + 320 + ], + "score": 0.85, + "content": "\\gamma = \\operatorname* { m a x } _ { i \\in [ b ] } \\gamma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 299, + 218, + 324 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 304, + 281, + 319 + ], + "score": 0.93, + "content": "\\begin{array} { r } { L ^ { 2 } = \\sum _ { i = 1 } ^ { b } L _ { i } ^ { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 102, + 299, + 281, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 231, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 232, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 147, + 335 + ], + "score": 1.0, + "content": "Claim 1.", + "type": "text" + }, + { + "bbox": [ + 147, + 322, + 166, + 334 + ], + "score": 0.89, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 320, + 219, + 335 + ], + "score": 1.0, + "content": "is a norm on", + "type": "text" + }, + { + "bbox": [ + 219, + 322, + 228, + 332 + ], + "score": 0.26, + "content": "\\mathbb { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 320, + 232, + 335 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 320, + 232, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 504, + 369 + ], + "lines": [ + { + "bbox": [ + 104, + 344, + 504, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 344, + 465, + 361 + ], + "score": 1.0, + "content": "Proof. 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Let", + "type": "text" + }, + { + "bbox": [ + 177, + 83, + 184, + 92 + ], + "score": 0.58, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 81, + 329, + 95 + ], + "score": 1.0, + "content": "be the output of Algorithm 2 after", + "type": "text" + }, + { + "bbox": [ + 330, + 83, + 339, + 92 + ], + "score": 0.71, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "iterations with non-increasing step-size", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 384, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 146, + 108 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 93, + 178, + 106 + ], + "score": 0.93, + "content": "\\{ \\eta _ { t } \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 93, + 261, + 108 + ], + "score": 1.0, + "content": ". Then under Setting", + "type": "text" + }, + { + "bbox": [ + 262, + 94, + 267, + 104 + ], + "score": 0.47, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 93, + 302, + 108 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 302, + 94, + 329, + 105 + ], + "score": 0.68, + "content": "\\hat { \\gamma } > \\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 93, + 384, + 108 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 110, + 452, + 145 + ], + "lines": [ + { + "bbox": [ + 158, + 110, + 452, + 145 + ], + "spans": [ + { + "bbox": [ + 158, + 110, + 452, + 145 + ], + "score": 0.92, + "content": "\\mathbb { E } \\Delta _ { \\frac { 1 } { \\hat { \\mathbf { \\xi } } } } ( \\hat { \\mathbf { x } } ) \\leq \\frac { \\hat { \\gamma } b } { \\hat { \\gamma } - \\gamma } \\frac { \\operatorname* { m i n } _ { \\mathbf { u } \\in \\mathcal { X } } f ( \\mathbf { u } ) + \\hat { \\gamma } D _ { \\phi } ( \\mathbf { u } | | \\mathbf { x } ^ { ( 1 ) } ) - f ^ { * } + \\frac { \\hat { \\gamma } L ^ { 2 } } { 2 b } \\sum _ { t = 1 } ^ { T } \\eta _ { t } ^ { 2 } } { \\sum _ { t = 1 } ^ { T } \\eta _ { t } }", + "type": "interline_equation", + "image_path": "3e2c408691e5a4606956e526aece1a38a1f2222cf17c374d9831ef484fbb4a08.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 158, + 110, + 452, + 121.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 158, + 121.66666666666667, + 452, + 133.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 158, + 133.33333333333334, + 452, + 145.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 148, + 388, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 388, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 226, + 162 + ], + "score": 1.0, + "content": "where the expectation is w.r.t.", + "type": "text" + }, + { + "bbox": [ + 227, + 149, + 236, + 160 + ], + "score": 0.88, + "content": "\\xi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 147, + 388, + 162 + ], + "score": 1.0, + "content": "and the randomness of the algorithm.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 100, + 164, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 100, + 164, + 214, + 199 + ], + "score": 1.0, + "content": "Proof. Define transforms be a stochastic oracle tha", + "type": "text" + }, + { + "bbox": [ + 214, + 174, + 262, + 186 + ], + "score": 0.92, + "content": "\\mathbf { U } _ { i } , i \\in \\ [ b ]", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 173, + 332, + 186 + ], + "score": 0.91, + "content": "\\mathbf { U } _ { i } ^ { T } \\mathbf { x } = \\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 172, + 470, + 186 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { x } = \\sum _ { i = 1 } ^ { b } \\mathbf { U } _ { i } \\mathbf { x } _ { i } \\ \\forall \\textbf { x } \\in \\mathcal { X } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 164, + 495, + 199 + ], + "score": 1.0, + "content": ". Let impli", + "type": "text" + }, + { + "bbox": [ + 495, + 174, + 504, + 185 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 262, + 186, + 441, + 198 + ], + "spans": [ + { + "bbox": [ + 262, + 186, + 295, + 196 + ], + "score": 0.89, + "content": "\\textbf { x } \\in { \\mathcal { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 186, + 441, + 198 + ], + "score": 0.91, + "content": "G ( { \\bf x } , i , \\xi ) = b { \\bf U } _ { i } G _ { i } ( { \\bf x } , \\xi )", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 191, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 197, + 398, + 211 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { E } _ { i , \\xi } G ( \\mathbf { x } , i , \\xi ) = \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } b \\mathbf { U } _ { i } \\mathbb { E } _ { \\xi } G _ { i } ( \\mathbf { x } , \\xi ) \\in \\sum _ { i = 1 } ^ { b } \\mathbf { U } _ { i } \\partial _ { i } f ( \\mathbf { x } ) = \\partial f ( \\mathbf { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 191, + 420, + 216 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 420, + 198, + 504, + 211 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { i , \\xi } \\Vert G ( \\mathbf { x } , i , \\xi ) \\Vert _ { * } ^ { 2 } =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 204, + 344, + 230 + ], + "spans": [ + { + "bbox": [ + 107, + 212, + 322, + 226 + ], + "score": 0.76, + "content": "\\begin{array} { r } { \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } b ^ { 2 } \\mathbb { E } _ { \\xi } \\| \\mathbf { U } _ { i } G _ { i } ( \\mathbf { x } , \\boldsymbol { \\xi } ) \\| _ { i , * } ^ { 2 } \\leq b \\sum _ { i = 1 } ^ { b } L _ { i } ^ { 2 } = b L ^ { 2 } . \\mathrm { \\bf { T } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 204, + 344, + 230 + ], + "score": 1.0, + "content": "hen", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 231, + 448, + 320 + ], + "lines": [ + { + "bbox": [ + 160, + 231, + 448, + 320 + ], + "spans": [ + { + "bbox": [ + 160, + 231, + 448, + 320 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbf { x } ^ { ( t + 1 ) } = \\mathbf { U } _ { i } \\left[ \\underset { \\mathbf { u } \\in \\mathcal { X } _ { i } } { \\arg \\operatorname* { m i n } } \\eta _ { t } \\langle g , u \\rangle + D _ { \\phi _ { i } } ( \\mathbf { u } | | \\mathbf { x } _ { i } ^ { ( t ) } ) \\right] } \\\\ & { \\quad \\quad \\quad = \\mathbf { U } _ { i } \\mathbf { U } _ { i } ^ { T } \\left[ \\underset { \\mathbf { u } \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\eta _ { t } \\langle \\mathbf { U } _ { i } G _ { i } ( \\mathbf { x } ^ { ( t ) } , \\xi _ { t } ) , \\mathbf { u } \\rangle + \\underset { i = 1 } { \\overset { b } { \\sum } } D _ { \\phi _ { i } } ( \\mathbf { u } _ { i } | | \\mathbf { x } _ { i } ^ { ( t ) } ) \\right] } \\\\ & { \\quad \\quad \\quad = \\underset { \\mathbf { u } \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\frac { \\eta _ { t } } { b } \\langle G ( \\mathbf { x } , i , \\xi _ { t } ) , \\mathbf { u } \\rangle + D _ { \\phi } ( \\mathbf { u } | | \\mathbf { x } ^ { ( t ) } ) } \\end{array}", + "type": "interline_equation", + "image_path": "070aa552d5e2831a317e3305ef3b56dfc0943f49d02863fcb016a81d0d2c9e57.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 160, + 231, + 448, + 260.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 160, + 260.6666666666667, + 448, + 290.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 160, + 290.33333333333337, + 448, + 320.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 462, + 338 + ], + "score": 1.0, + "content": "Thus Algorithm 2 is equivalent to Zhang & He (2018, Algorithm 1) with stochastic oracle", + "type": "text" + }, + { + "bbox": [ + 463, + 325, + 502, + 337 + ], + "score": 0.92, + "content": "G ( \\mathbf { x } , i , \\xi )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 324, + 506, + 338 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 334, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 183, + 351 + ], + "score": 1.0, + "content": "step-size sequence", + "type": "text" + }, + { + "bbox": [ + 183, + 335, + 225, + 348 + ], + "score": 0.93, + "content": "\\{ \\eta _ { t } / b \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 334, + 345, + 351 + ], + "score": 1.0, + "content": ", and no regularizer. 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The result then follows from Theorem 3.1 of the same.", + "type": "text" + }, + { + "bbox": [ + 495, + 358, + 505, + 368 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 506, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 226, + 389 + ], + "score": 1.0, + "content": "Corollary 1. Under Setting", + "type": "text" + }, + { + "bbox": [ + 227, + 377, + 233, + 386 + ], + "score": 0.52, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 374, + 481, + 389 + ], + "score": 1.0, + "content": "let ˆx be the output of Algorithm 2 with constant step-size", + "type": "text" + }, + { + "bbox": [ + 481, + 377, + 505, + 388 + ], + "score": 0.84, + "content": "\\eta _ { t } ~ =", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 111, + 386, + 358, + 407 + ], + "spans": [ + { + "bbox": [ + 111, + 386, + 358, + 407 + ], + "score": 1.0, + "content": "q 2b(f(1)−f∗)γL2T ∀ t ∈ [T ], where f (1) = f (x(1)). Then we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 412, + 376, + 439 + ], + "lines": [ + { + "bbox": [ + 234, + 412, + 376, + 439 + ], + "spans": [ + { + "bbox": [ + 234, + 412, + 376, + 439 + ], + "score": 0.93, + "content": "\\mathbb { E } \\Delta _ { \\frac { 1 } { 2 \\gamma } } \\big ( \\hat { \\mathbf { x } } \\big ) \\leq 2 L \\sqrt { \\frac { 2 b \\gamma \\big ( f ^ { ( 1 ) } - f ^ { * } \\big ) } { T } }", + "type": "interline_equation", + "image_path": "76c89a24f42e69932dc7143e58b2ac512fa359d39b4fef87e712ac1268659bfd.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 234, + 412, + 376, + 439 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 226, + 457 + ], + "score": 1.0, + "content": "where the expectation is w.r.t.", + "type": "text" + }, + { + "bbox": [ + 226, + 445, + 235, + 456 + ], + "score": 0.86, + "content": "\\xi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "and the randomness of the algorithm. Equivalently, we can reach a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 452, + 403, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 178, + 474 + ], + "score": 1.0, + "content": "point ˆx satisfying", + "type": "text" + }, + { + "bbox": [ + 179, + 455, + 308, + 474 + ], + "score": 0.72, + "content": "\\begin{array} { r } { \\mathbb { E } \\Delta _ { \\frac { 1 } { 2 \\gamma } } ( \\hat { \\mathbf { x } } ) \\leq \\varepsilon i n \\frac { 8 \\gamma b L ^ { 2 } ( f ^ { ( 1 ) } - f ^ { * } ) } { \\varepsilon ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 452, + 403, + 474 + ], + "score": 1.0, + "content": "stochastic oracle calls.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 177, + 95 + ], + "score": 1.0, + "content": "Theorem 2. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 83, + 184, + 92 + ], + "score": 0.58, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 81, + 329, + 95 + ], + "score": 1.0, + "content": "be the output of Algorithm 2 after", + "type": "text" + }, + { + "bbox": [ + 330, + 83, + 339, + 92 + ], + "score": 0.71, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "iterations with non-increasing step-size", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 384, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 146, + 108 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 93, + 178, + 106 + ], + "score": 0.93, + "content": "\\{ \\eta _ { t } \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 93, + 261, + 108 + ], + "score": 1.0, + "content": ". Then under Setting", + "type": "text" + }, + { + "bbox": [ + 262, + 94, + 267, + 104 + ], + "score": 0.47, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 93, + 302, + 108 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 302, + 94, + 329, + 105 + ], + "score": 0.68, + "content": "\\hat { \\gamma } > \\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 93, + 384, + 108 + ], + "score": 1.0, + "content": "we have that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 81, + 505, + 108 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 110, + 452, + 145 + ], + "lines": [ + { + "bbox": [ + 158, + 110, + 452, + 145 + ], + "spans": [ + { + "bbox": [ + 158, + 110, + 452, + 145 + ], + "score": 0.92, + "content": "\\mathbb { E } \\Delta _ { \\frac { 1 } { \\hat { \\mathbf { \\xi } } } } ( \\hat { \\mathbf { x } } ) \\leq \\frac { \\hat { \\gamma } b } { \\hat { \\gamma } - \\gamma } \\frac { \\operatorname* { m i n } _ { \\mathbf { u } \\in \\mathcal { X } } f ( \\mathbf { u } ) + \\hat { \\gamma } D _ { \\phi } ( \\mathbf { u } | | \\mathbf { x } ^ { ( 1 ) } ) - f ^ { * } + \\frac { \\hat { \\gamma } L ^ { 2 } } { 2 b } \\sum _ { t = 1 } ^ { T } \\eta _ { t } ^ { 2 } } { \\sum _ { t = 1 } ^ { T } \\eta _ { t } }", + "type": "interline_equation", + "image_path": "3e2c408691e5a4606956e526aece1a38a1f2222cf17c374d9831ef484fbb4a08.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 158, + 110, + 452, + 121.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 158, + 121.66666666666667, + 452, + 133.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 158, + 133.33333333333334, + 452, + 145.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 148, + 388, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 388, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 226, + 162 + ], + "score": 1.0, + "content": "where the expectation is w.r.t.", + "type": "text" + }, + { + "bbox": [ + 227, + 149, + 236, + 160 + ], + "score": 0.88, + "content": "\\xi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 147, + 388, + 162 + ], + "score": 1.0, + "content": "and the randomness of the algorithm.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 147, + 388, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 100, + 164, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 100, + 164, + 214, + 199 + ], + "score": 1.0, + "content": "Proof. Define transforms be a stochastic oracle tha", + "type": "text" + }, + { + "bbox": [ + 214, + 174, + 262, + 186 + ], + "score": 0.92, + "content": "\\mathbf { U } _ { i } , i \\in \\ [ b ]", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 173, + 332, + 186 + ], + "score": 0.91, + "content": "\\mathbf { U } _ { i } ^ { T } \\mathbf { x } = \\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 172, + 470, + 186 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { x } = \\sum _ { i = 1 } ^ { b } \\mathbf { U } _ { i } \\mathbf { x } _ { i } \\ \\forall \\textbf { x } \\in \\mathcal { X } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 164, + 495, + 199 + ], + "score": 1.0, + "content": ". Let impli", + "type": "text" + }, + { + "bbox": [ + 495, + 174, + 504, + 185 + ], + "score": 0.78, + "content": "G", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 262, + 186, + 441, + 198 + ], + "spans": [ + { + "bbox": [ + 262, + 186, + 295, + 196 + ], + "score": 0.89, + "content": "\\textbf { x } \\in { \\mathcal { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 186, + 441, + 198 + ], + "score": 0.91, + "content": "G ( { \\bf x } , i , \\xi ) = b { \\bf U } _ { i } G _ { i } ( { \\bf x } , \\xi )", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 191, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 197, + 398, + 211 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbb { E } _ { i , \\xi } G ( \\mathbf { x } , i , \\xi ) = \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } b \\mathbf { U } _ { i } \\mathbb { E } _ { \\xi } G _ { i } ( \\mathbf { x } , \\xi ) \\in \\sum _ { i = 1 } ^ { b } \\mathbf { U } _ { i } \\partial _ { i } f ( \\mathbf { x } ) = \\partial f ( \\mathbf { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 191, + 420, + 216 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 420, + 198, + 504, + 211 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { i , \\xi } \\Vert G ( \\mathbf { x } , i , \\xi ) \\Vert _ { * } ^ { 2 } =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 204, + 344, + 230 + ], + "spans": [ + { + "bbox": [ + 107, + 212, + 322, + 226 + ], + "score": 0.76, + "content": "\\begin{array} { r } { \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } b ^ { 2 } \\mathbb { E } _ { \\xi } \\| \\mathbf { U } _ { i } G _ { i } ( \\mathbf { x } , \\boldsymbol { \\xi } ) \\| _ { i , * } ^ { 2 } \\leq b \\sum _ { i = 1 } ^ { b } L _ { i } ^ { 2 } = b L ^ { 2 } . \\mathrm { \\bf { T } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 204, 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) , \\mathbf { u } \\rangle + \\underset { i = 1 } { \\overset { b } { \\sum } } D _ { \\phi _ { i } } ( \\mathbf { u } _ { i } | | \\mathbf { x } _ { i } ^ { ( t ) } ) \\right] } \\\\ & { \\quad \\quad \\quad = \\underset { \\mathbf { u } \\in \\mathcal { X } } { \\arg \\operatorname* { m i n } } \\frac { \\eta _ { t } } { b } \\langle G ( \\mathbf { x } , i , \\xi _ { t } ) , \\mathbf { u } \\rangle + D _ { \\phi } ( \\mathbf { u } | | \\mathbf { x } ^ { ( t ) } ) } \\end{array}", + "type": "interline_equation", + "image_path": "070aa552d5e2831a317e3305ef3b56dfc0943f49d02863fcb016a81d0d2c9e57.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 160, + 231, + 448, + 260.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 160, + 260.6666666666667, + 448, + 290.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 160, + 290.33333333333337, + 448, + 320.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 462, + 338 + ], + "score": 1.0, + "content": "Thus Algorithm 2 is equivalent to Zhang & He (2018, Algorithm 1) with stochastic oracle", + "type": "text" + }, + { + "bbox": [ + 463, + 325, + 502, + 337 + ], + "score": 0.92, + "content": "G ( \\mathbf { x } , i , \\xi )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 324, + 506, + 338 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 334, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 183, + 351 + ], + "score": 1.0, + "content": "step-size sequence", + "type": "text" + }, + { + "bbox": [ + 183, + 335, + 225, + 348 + ], + "score": 0.93, + "content": "\\{ \\eta _ { t } / b \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 334, + 345, + 351 + ], + "score": 1.0, + 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Note that", + "type": "text" + }, + { + "bbox": [ + 345, + 336, + 353, + 347 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 334, + 460, + 351 + ], + "score": 1.0, + "content": "is 1-strongly-convex w.r.t.", + "type": "text" + }, + { + "bbox": [ + 460, + 337, + 478, + 348 + ], + "score": 0.9, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 334, + 496, + 351 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 497, + 337, + 504, + 347 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 115, + 359 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 115, + 348, + 123, + 358 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 347, + 172, + 359 + ], + "score": 1.0, + "content": "-RWC w.r.t.", + "type": "text" + }, + { + "bbox": [ + 173, + 348, + 180, + 359 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 347, + 505, + 359 + ], + "score": 1.0, + "content": ", so in light of Claims 1 and 2 our setup satisfies Assumption 3.1 of Zhang & He", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 359, + 370 + ], + "score": 1.0, + "content": "(2018). The result then follows from Theorem 3.1 of the same.", + "type": "text" + }, + { + "bbox": [ + 495, + 358, + 505, + 368 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 324, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 506, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 226, + 389 + ], + "score": 1.0, + "content": "Corollary 1. Under Setting", + "type": "text" + }, + { + "bbox": [ + 227, + 377, + 233, + 386 + ], + "score": 0.52, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 374, + 481, + 389 + ], + "score": 1.0, + "content": "let ˆx be the output of Algorithm 2 with constant step-size", + "type": "text" + }, + { + "bbox": [ + 481, + 377, + 505, + 388 + ], + "score": 0.84, + "content": "\\eta _ { t } ~ =", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 111, + 386, + 358, + 407 + ], + "spans": [ + { + "bbox": [ + 111, + 386, + 358, + 407 + ], + "score": 1.0, + "content": "q 2b(f(1)−f∗)γL2T ∀ t ∈ [T ], where f (1) = f (x(1)). Then we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 374, + 505, + 407 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 412, + 376, + 439 + ], + "lines": [ + { + "bbox": [ + 234, + 412, + 376, + 439 + ], + "spans": [ + { + "bbox": [ + 234, + 412, + 376, + 439 + ], + "score": 0.93, + "content": "\\mathbb { E } \\Delta _ { \\frac { 1 } { 2 \\gamma } } \\big ( \\hat { \\mathbf { x } } \\big ) \\leq 2 L \\sqrt { \\frac { 2 b \\gamma \\big ( f ^ { ( 1 ) } - f ^ { * } \\big ) } { T } }", + "type": "interline_equation", + "image_path": "76c89a24f42e69932dc7143e58b2ac512fa359d39b4fef87e712ac1268659bfd.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 234, + 412, + 376, + 439 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 226, + 457 + ], + "score": 1.0, + "content": "where the expectation is w.r.t.", + "type": "text" + }, + { + "bbox": [ + 226, + 445, + 235, + 456 + ], + "score": 0.86, + "content": "\\xi _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "and the randomness of the algorithm. Equivalently, we can reach a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 452, + 403, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 178, + 474 + ], + "score": 1.0, + "content": "point ˆx satisfying", + "type": "text" + }, + { + "bbox": [ + 179, + 455, + 308, + 474 + ], + "score": 0.72, + "content": "\\begin{array} { r } { \\mathbb { E } \\Delta _ { \\frac { 1 } { 2 \\gamma } } ( \\hat { \\mathbf { x } } ) \\leq \\varepsilon i n \\frac { 8 \\gamma b L ^ { 2 } ( f ^ { ( 1 ) } - f ^ { * } ) } { \\varepsilon ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 452, + 403, + 474 + ], + "score": 1.0, + "content": "stochastic oracle calls.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 443, + 506, + 474 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 357, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 357, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 357, + 95 + ], + "score": 1.0, + "content": "B.3 A SINGLE-LEVEL ANALYSIS OF ENAS AND DARTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 506, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "In this section we apply our analysis to understanding two existing NAS algorithms, ENAS (Pham", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "et al., 2018) and DARTS (Liu et al., 2019). For simplicity, we assume objectives induced by architec-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 250, + 138 + ], + "score": 1.0, + "content": "tures in the relaxed search space are", + "type": "text" + }, + { + "bbox": [ + 250, + 127, + 257, + 137 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 123, + 507, + 138 + ], + "score": 1.0, + "content": "-smooth, which excludes components such as ReLU. However,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "score": 1.0, + "content": "such cases can be smoothed via Gaussian convolution, i.e. adding noise to every gradient; thus given", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 145, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 507, + 160 + ], + "score": 1.0, + "content": "the noisiness of SGD training we believe the following analysis is still informative (Kleinberg et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 136, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 136, + 171 + ], + "score": 1.0, + "content": "2018).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 175, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 223, + 187 + ], + "score": 1.0, + "content": "ENAS continuously relaxes", + "type": "text" + }, + { + "bbox": [ + 223, + 175, + 232, + 185 + ], + "score": 0.74, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 175, + 437, + 187 + ], + "score": 1.0, + "content": "via a neural controller that samples architectures", + "type": "text" + }, + { + "bbox": [ + 438, + 175, + 466, + 185 + ], + "score": 0.89, + "content": "a \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 175, + 483, + 187 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 483, + 175, + 505, + 186 + ], + "score": 0.84, + "content": "\\Theta =", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 507, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 136, + 198 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { \\mathcal { O } ( h ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 184, + 167, + 202 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 167, + 188, + 174, + 198 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 184, + 507, + 202 + ], + "score": 1.0, + "content": "is the number of hidden units. The controller is trained with Monte Carlo gradients.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "On the other hand, first-order DARTS uses a mixture relaxation similar to the one in Section A but", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 207, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 207, + 393, + 222 + ], + "score": 1.0, + "content": "using a softmax instead of constraining parameters to the simplex. Thus", + "type": "text" + }, + { + "bbox": [ + 393, + 209, + 451, + 219 + ], + "score": 0.92, + "content": "\\Theta = \\mathbb { R } ^ { | E | \\times | O | }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 207, + 466, + 222 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 467, + 210, + 476, + 219 + ], + "score": 0.83, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 207, + 507, + 222 + ], + "score": 1.0, + "content": "the set", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 198, + 235 + ], + "score": 1.0, + "content": "of learnable edges and", + "type": "text" + }, + { + "bbox": [ + 198, + 221, + 207, + 231 + ], + "score": 0.76, + "content": "O", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 219, + 506, + 235 + ], + "score": 1.0, + "content": "the set of possible operations. 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Since ENAS uses a hidden state size of", + "type": "text" + }, + { + "bbox": [ + 452, + 336, + 487, + 347 + ], + "score": 0.89, + "content": "h = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 331, + 509, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 227, + 360 + ], + "score": 1.0, + "content": "the DARTS search space has", + "type": "text" + }, + { + "bbox": [ + 228, + 348, + 267, + 359 + ], + "score": 0.89, + "content": "| E | = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 347, + 312, + 360 + ], + "score": 1.0, + "content": "edges and", + "type": "text" + }, + { + "bbox": [ + 312, + 347, + 346, + 359 + ], + "score": 0.93, + "content": "| O | = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "operations, this also points to DARTS", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 257, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 257, + 371 + ], + "score": 1.0, + "content": "needing fewer iterations to converge.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 357, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 357, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 357, + 95 + ], + "score": 1.0, + "content": "B.3 A SINGLE-LEVEL ANALYSIS OF ENAS AND DARTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 357, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 506, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "In this section we apply our analysis to understanding two existing NAS algorithms, ENAS (Pham", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "et al., 2018) and DARTS (Liu et al., 2019). For simplicity, we assume objectives induced by architec-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 250, + 138 + ], + "score": 1.0, + "content": "tures in the relaxed search space are", + "type": "text" + }, + { + "bbox": [ + 250, + 127, + 257, + 137 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 123, + 507, + 138 + ], + "score": 1.0, + "content": "-smooth, which excludes components such as ReLU. However,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 506, + 149 + ], + "score": 1.0, + "content": "such cases can be smoothed via Gaussian convolution, i.e. adding noise to every gradient; thus given", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 145, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 507, + 160 + ], + "score": 1.0, + "content": "the noisiness of SGD training we believe the following analysis is still informative (Kleinberg et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 136, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 136, + 171 + ], + "score": 1.0, + "content": "2018).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 103, + 507, + 171 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 106, + 175, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 223, + 187 + ], + "score": 1.0, + "content": "ENAS continuously relaxes", + "type": "text" + }, + { + "bbox": [ + 223, + 175, + 232, + 185 + ], + "score": 0.74, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 175, + 437, + 187 + ], + "score": 1.0, + "content": "via a neural controller that samples architectures", + "type": "text" + }, + { + "bbox": [ + 438, + 175, + 466, + 185 + ], + "score": 0.89, + "content": "a \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 175, + 483, + 187 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 483, + 175, + 505, + 186 + ], + "score": 0.84, + "content": "\\Theta =", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 507, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 136, + 198 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { \\mathcal { O } ( h ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 184, + 167, + 202 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 167, + 188, + 174, + 198 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 184, + 507, + 202 + ], + "score": 1.0, + "content": "is the number of hidden units. The controller is trained with Monte Carlo gradients.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "On the other hand, first-order DARTS uses a mixture relaxation similar to the one in Section A but", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 207, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 207, + 393, + 222 + ], + "score": 1.0, + "content": "using a softmax instead of constraining parameters to the simplex. Thus", + "type": "text" + }, + { + "bbox": [ + 393, + 209, + 451, + 219 + ], + "score": 0.92, + "content": "\\Theta = \\mathbb { R } ^ { | E | \\times | O | }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 207, + 466, + 222 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 467, + 210, + 476, + 219 + ], + "score": 0.83, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 207, + 507, + 222 + ], + "score": 1.0, + "content": "the set", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 198, + 235 + ], + "score": 1.0, + "content": "of learnable edges and", + "type": "text" + }, + { + "bbox": [ + 198, + 221, + 207, + 231 + ], + "score": 0.76, + "content": "O", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 219, + 506, + 235 + ], + "score": 1.0, + "content": "the set of possible operations. If we assume that both algorithms use SGD", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "for the architecture parameters then to compare them we are interested in their respective values of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 241, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 120, + 254 + ], + "score": 0.87, + "content": "G _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 241, + 225, + 257 + ], + "score": 1.0, + "content": ", which we will refer to as", + "type": "text" + }, + { + "bbox": [ + 226, + 243, + 253, + 254 + ], + "score": 0.88, + "content": "G _ { \\mathrm { E N A S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 241, + 271, + 257 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 243, + 302, + 254 + ], + "score": 0.79, + "content": "G _ { \\mathrm { D A R T S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 241, + 505, + 257 + ], + "score": 1.0, + "content": ". Before proceeding, we note again that our theory", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "holds only for the single-level objective and when using SGD as the architecture optimizer, whereas", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 473, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 473, + 278 + ], + "score": 1.0, + "content": "both algorithms specify the bilevel objective and Adam (Kingma & Ba, 2015), respectively.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 175, + 507, + 278 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "At a very high level, the Monte Carlo gradients used by ENAS are known to be high-variance, so", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 107, + 293, + 134, + 304 + ], + "score": 0.68, + "content": "G _ { \\mathrm { E N A S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 292, + 236, + 306 + ], + "score": 1.0, + "content": "may be much larger than", + "type": "text" + }, + { + "bbox": [ + 236, + 293, + 267, + 304 + ], + "score": 0.78, + "content": "G _ { \\mathrm { D A R T S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 292, + 506, + 306 + ], + "score": 1.0, + "content": ", yielding faster convergence for DARTS, which is reflected", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "in practice (Liu et al., 2019). We can also do a simple low-level analysis under the assumption that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 340, + 327 + ], + "score": 1.0, + "content": "all architecture gradients are bounded entry-wise, i.e. in", + "type": "text" + }, + { + "bbox": [ + 340, + 315, + 354, + 325 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "-norm, by some constant; then since", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 154, + 338 + ], + "score": 1.0, + "content": "the squared", + "type": "text" + }, + { + "bbox": [ + 155, + 325, + 164, + 336 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 324, + 430, + 338 + ], + "score": 1.0, + "content": "-norm is bounded by the product of the dimension and the squared", + "type": "text" + }, + { + "bbox": [ + 430, + 325, + 444, + 336 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "-norm we have", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 331, + 509, + 354 + ], + "spans": [ + { + "bbox": [ + 107, + 336, + 173, + 348 + ], + "score": 0.92, + "content": "G _ { \\mathrm { E N A S } } ^ { 2 } = \\mathcal { O } ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 331, + 200, + 354 + ], + "score": 1.0, + "content": "while", + "type": "text" + }, + { + "bbox": [ + 200, + 336, + 287, + 348 + ], + "score": 0.91, + "content": "G _ { \\mathrm { D A R T S } } ^ { 2 } = \\mathcal { O } ( | \\bar { E } | | { \\cal O } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 331, + 451, + 354 + ], + "score": 1.0, + "content": ". Since ENAS uses a hidden state size of", + "type": "text" + }, + { + "bbox": [ + 452, + 336, + 487, + 347 + ], + "score": 0.89, + "content": "h = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 331, + 509, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 227, + 360 + ], + "score": 1.0, + "content": "the DARTS search space has", + "type": "text" + }, + { + "bbox": [ + 228, + 348, + 267, + 359 + ], + "score": 0.89, + "content": "| E | = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 347, + 312, + 360 + ], + "score": 1.0, + "content": "edges and", + "type": "text" + }, + { + "bbox": [ + 312, + 347, + 346, + 359 + ], + "score": 0.93, + "content": "| O | = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "operations, this also points to DARTS", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 257, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 257, + 371 + ], + "score": 1.0, + "content": "needing fewer iterations to converge.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 281, + 509, + 371 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 257, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "score": 1.0, + "content": "C EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "We provide additional detail on the experimental setup and hyperparameter settings used for each", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "benchmark studied in Section 4. We also provide a more detailed discussion of how XNAS differs", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 464, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 464, + 140 + ], + "score": 1.0, + "content": "from GAEA, along with empirical results for XNAS on the NAS-Bench-201 benchmark.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 152, + 236, + 164 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 237, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 237, + 165 + ], + "score": 1.0, + "content": "C.1 DARTS SEARCH SPACE", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 173, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "We consider the same search space as DARTS (Liu et al., 2019), which has become one of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "score": 1.0, + "content": "standard search spaces for CNN cell search (Xie et al., 2019; Nayman et al., 2019; Chen et al., 2019;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "Noy et al., 2019; Liang et al., 2019). Following the evaluation procedure used in Liu et al. (2019) and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 205, + 424, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 424, + 219 + ], + "score": 1.0, + "content": "Xu et al. (2020) , our evaluation of GAEA PC-DARTS consists of three stages:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "• Stage 1: In the search phase, we run GAEA PC-DARTS with 5 random seeds to reduce variance", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 237, + 357, + 249 + ], + "spans": [ + { + "bbox": [ + 115, + 237, + 357, + 249 + ], + "score": 1.0, + "content": "from different initialization of the shared-weights network.3", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "• Stage 2: We evaluate the best architecture identified by each search run by training from scratch.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "• Stage 3: We perform a more thorough evaluation of the best architecture from stage 2 by training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 275, + 299, + 286 + ], + "spans": [ + { + "bbox": [ + 115, + 275, + 299, + 286 + ], + "score": 1.0, + "content": "with ten different random seed initializations.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 306 + ], + "score": 1.0, + "content": "For completeness, we describe the convolutional neural network search space considered. A cell", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "consists of 2 input nodes and 4 intermediate nodes for a total of 6 nodes. The nodes are ordered", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 317, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 457, + 328 + ], + "score": 1.0, + "content": "and subsequent nodes can receive the output of prior nodes as input so for a given node", + "type": "text" + }, + { + "bbox": [ + 457, + 317, + 464, + 326 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 317, + 505, + 328 + ], + "score": 1.0, + "content": ", there are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 107, + 328, + 131, + 338 + ], + "score": 0.87, + "content": "k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 328, + 245, + 340 + ], + "score": 1.0, + "content": "possible input edges to node", + "type": "text" + }, + { + "bbox": [ + 245, + 328, + 252, + 338 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 328, + 374, + 340 + ], + "score": 1.0, + "content": ". Therefore, there are a total of", + "type": "text" + }, + { + "bbox": [ + 374, + 328, + 455, + 338 + ], + "score": 0.91, + "content": "2 + 3 + 4 + 5 = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "edges in the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 206, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 206, + 351 + ], + "score": 1.0, + "content": "weight-sharing network.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "An architecture is defined by selecting 2 input edges per intermediate node and also selecting a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 367, + 504, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 351, + 379 + ], + "score": 1.0, + "content": "single operation per edge from the following 8 operations: (1)", + "type": "text" + }, + { + "bbox": [ + 352, + 367, + 375, + 377 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 367, + 480, + 379 + ], + "score": 1.0, + "content": "separable convolution, (2)", + "type": "text" + }, + { + "bbox": [ + 480, + 367, + 504, + 377 + ], + "score": 0.87, + "content": "5 \\times 5", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 212, + 390 + ], + "score": 1.0, + "content": "separable convolution, (3)", + "type": "text" + }, + { + "bbox": [ + 212, + 378, + 236, + 388 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 377, + 331, + 390 + ], + "score": 1.0, + "content": "dilated convolution, (4)", + "type": "text" + }, + { + "bbox": [ + 332, + 378, + 355, + 388 + ], + "score": 0.9, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "dilated convolution, (5) max pooling,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "(6) average pooling, (7) identity (8) zero. We use the same search space to design a “normal” cell and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 504, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 504, + 410 + ], + "score": 1.0, + "content": "a “reduction” cell; the normal cells have stride 1 operations that do not change the dimension of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "input, while the reduction cells have stride 2 operations that half the length and width dimensions of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "the input. In the experiments, for both cell types, , after which the output of all intermediate nodes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 294, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 294, + 443 + ], + "score": 1.0, + "content": "are concatenated to form the output of the cell.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 455, + 290, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 291, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 291, + 468 + ], + "score": 1.0, + "content": "C.1.1 STAGE 1: ARCHITECTURE SEARCH", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "For stage 1, as is done by DARTS and PC-DARTS, we use GAEA PC-DARTS to update architecture", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "parameters for a smaller shared-weights network in the search phase with 8 layers and 16 initial", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "channels. 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However, since PC-DARTS", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 389, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 389, + 732 + ], + "score": 1.0, + "content": "is significantly faster than DARTS, the cost of an additional seed is negligible.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 257, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "score": 1.0, + "content": "C EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "We provide additional detail on the experimental setup and hyperparameter settings used for each", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "benchmark studied in Section 4. We also provide a more detailed discussion of how XNAS differs", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 464, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 464, + 140 + ], + "score": 1.0, + "content": "from GAEA, along with empirical results for XNAS on the NAS-Bench-201 benchmark.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 105, + 505, + 140 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 152, + 236, + 164 + ], + "lines": [ + { + "bbox": [ + 106, + 151, + 237, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 237, + 165 + ], + "score": 1.0, + "content": "C.1 DARTS SEARCH SPACE", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 173, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "We consider the same search space as DARTS (Liu et al., 2019), which has become one of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 196 + ], + "score": 1.0, + "content": "standard search spaces for CNN cell search (Xie et al., 2019; Nayman et al., 2019; Chen et al., 2019;", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "Noy et al., 2019; Liang et al., 2019). Following the evaluation procedure used in Liu et al. (2019) and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 205, + 424, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 424, + 219 + ], + "score": 1.0, + "content": "Xu et al. 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A cell", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "consists of 2 input nodes and 4 intermediate nodes for a total of 6 nodes. The nodes are ordered", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 317, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 457, + 328 + ], + "score": 1.0, + "content": "and subsequent nodes can receive the output of prior nodes as input so for a given node", + "type": "text" + }, + { + "bbox": [ + 457, + 317, + 464, + 326 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 317, + 505, + 328 + ], + "score": 1.0, + "content": ", there are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 107, + 328, + 131, + 338 + ], + "score": 0.87, + "content": "k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 328, + 245, + 340 + ], + "score": 1.0, + "content": "possible input edges to node", + "type": "text" + }, + { + "bbox": [ + 245, + 328, + 252, + 338 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 328, + 374, + 340 + ], + "score": 1.0, + "content": ". Therefore, there are a total of", + "type": "text" + }, + { + "bbox": [ + 374, + 328, + 455, + 338 + ], + "score": 0.91, + "content": "2 + 3 + 4 + 5 = 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "edges in the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 206, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 206, + 351 + ], + "score": 1.0, + "content": "weight-sharing network.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 295, + 505, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 355, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "score": 1.0, + "content": "An architecture is defined by selecting 2 input edges per intermediate node and also selecting a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 367, + 504, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 351, + 379 + ], + "score": 1.0, + "content": "single operation per edge from the following 8 operations: (1)", + "type": "text" + }, + { + "bbox": [ + 352, + 367, + 375, + 377 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 367, + 480, + 379 + ], + "score": 1.0, + "content": "separable convolution, (2)", + "type": "text" + }, + { + "bbox": [ + 480, + 367, + 504, + 377 + ], + "score": 0.87, + "content": "5 \\times 5", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 212, + 390 + ], + "score": 1.0, + "content": "separable convolution, (3)", + "type": "text" + }, + { + "bbox": [ + 212, + 378, + 236, + 388 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 377, + 331, + 390 + ], + "score": 1.0, + "content": "dilated convolution, (4)", + "type": "text" + }, + { + "bbox": [ + 332, + 378, + 355, + 388 + ], + "score": 0.9, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "dilated convolution, (5) max pooling,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "(6) average pooling, (7) identity (8) zero. 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Set 1 (Reported)Set2Set3
2.50 ± 0.072.50±0.092.60±0.09
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PC-DARTS for 2 additional sets of random seeds from stage 1 search. The performance", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "of GAEA PC-DARTS for one set is similar to that reported in Table 1, while the other is on par", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "with the performance reported for PC-DARTS in Xu et al. (2020). We do observe non-negligible", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "variance in the performance of the architecture found by different random seed initializations of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "shared-weights network, necessitating running multiple searches before selecting an architecture.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "We also found that it was possible to identify and eliminate poor performing architectures in just 20", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "epochs of training during stage 2 intermediate evaluation, thereby reducing the total training cost by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 421, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 126, + 531 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 127, + 518, + 146, + 529 + ], + "score": 0.87, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 518, + 421, + 531 + ], + "score": 1.0, + "content": "(we only trained 3 out of 10 architectures for the entire 600 epochs).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 430, + 506, + 531 + ] + }, + { + "type": "table", + "bbox": [ + 205, + 551, + 406, + 575 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 267, + 540, + 344, + 550 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 266, + 538, + 345, + 552 + ], + "spans": [ + { + "bbox": [ + 266, + 538, + 345, + 552 + ], + "score": 1.0, + "content": "Stage 3 Evaluation", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "table_body", + "bbox": [ + 205, + 551, + 406, + 575 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 205, + 551, + 406, + 575 + ], + "spans": [ + { + "bbox": [ + 205, + 551, + 406, + 575 + ], + "score": 0.951, + "html": "
Set 1 (Reported)Set2Set3
2.50 ± 0.072.50±0.092.60±0.09
", + "type": "table", + "image_path": "793f07b1b6970a1cc0cd4a37b8b093cbbbe52f440c45bf771c47941791617714.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 205, + 551, + 406, + 575 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 154, + 583, + 456, + 595 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 153, + 582, + 457, + 596 + ], + "spans": [ + { + "bbox": [ + 153, + 582, + 457, + 596 + ], + "score": 1.0, + "content": "Table 3: GAEA PC-DARTS Stage 3 Evaluation for 3 sets of random seeds.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 606, + 504, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "We depict the top architectures found by GAEA PC-DARTS for CIFAR-10 and ImageNet in Figure 3", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 618, + 259, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 259, + 630 + ], + "score": 1.0, + "content": "and detailed results in Tables 4 and 5.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 606, + 505, + 630 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 110, + 80, + 501, + 217 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 110, + 80, + 501, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 80, + 501, + 217 + ], + "spans": [ + { + "bbox": [ + 110, + 80, + 501, + 217 + ], + "score": 0.984, + "html": "
ArchitectureTest Error AverageParams (M)Search Cost (GPU Days)Comparable Search SpaceSearch Method
Shake-Shake(DeVries&Taylor,2017)Best N/A2.5626.2--manual
PyramidNet (Yamada et al.,2018)2.31N/A261-manual
NASNet-A* (Zoph et al.,2018)N/A2.653.32000NRL
AmoebaNet-B*(Real et al.,2019)N/A2.55± 0.052.83150Nevolution
ProxylessNAS* (Cai et al.,2019)2.08N/A5.74Ngradient
ENAS (Pham et al.,2018)2.89N/A4.60.5YRL
Random search Wst (Li& Talwalkar,2019)2.712.85 ±0.083.80.7Yrandom
ASNG-NAS (Akimoto et al.,2019)N/A2.83 ± 0.143.90.1Ygradient
SNAS (Xie et al.,2019)N/A2.85 ±0.022.81.5Ygradient
DARTS (lst-order)†(Liu et al.,2019)N/A3.00 ±0.143.30.4Ygradient
DARTS (2nd-order)†(Liu et al.,2019)N/A2.76±0.093.31Ygradient
PDARTS# (Chen et al., 2019)2.50N/A3.40.3Ygradient
PC-DARTS† (Xu et al.,2020)N/A2.57 ±0.073.60.1Ygradient
GAEA PC-DARTS† (Ours)2.392.50±0.063.70.1Ygradient
", + "type": "table", + "image_path": "ce6d48674ef443b3731e351bae3f145e958b8559063ac5fc11eb15040c8f7056.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 80, + 501, + 125.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 125.66666666666666, + 501, + 171.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 171.33333333333331, + 501, + 216.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 114, + 218, + 501, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 219, + 324, + 227 + ], + "spans": [ + { + "bbox": [ + 122, + 219, + 324, + 227 + ], + "score": 1.0, + "content": "We show results for networks with a comparable number of parameters.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 117, + 224, + 477, + 246 + ], + "spans": [ + { + "bbox": [ + 117, + 224, + 477, + 246 + ], + "score": 1.0, + "content": "For fair comparison to other work, we show the search cost for training the shared-weights network with a single initialization. ‡ Search space and backbone architecture (PyramidNet) differ from the DARTS setting.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 123, + 242, + 501, + 252 + ], + "spans": [ + { + "bbox": [ + 123, + 242, + 501, + 252 + ], + "score": 1.0, + "content": "PDARTS results not reported for multiple seeds. Additionally, PDARTS uses deeper weight-sharing networks during search, on which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 119, + 249, + 501, + 261 + ], + "spans": [ + { + "bbox": [ + 119, + 249, + 501, + 261 + ], + "score": 1.0, + "content": "PC-DARTS has also been shown to improve performance (Xu et al., 2020), so we GAEA PC-DARTS to further improve if modified", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 119, + 257, + 150, + 269 + ], + "spans": [ + { + "bbox": [ + 119, + 257, + 150, + 269 + ], + "score": 1.0, + "content": "similarly.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 110, + 270, + 501, + 347 + ], + "lines": [ + { + "bbox": [ + 109, + 270, + 501, + 283 + ], + "spans": [ + { + "bbox": [ + 109, + 270, + 501, + 283 + ], + "score": 1.0, + "content": "Table 4: DARTS (CIFAR-10): Comparison with manually designed networks and those found by", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 110, + 281, + 501, + 294 + ], + "spans": [ + { + "bbox": [ + 110, + 281, + 501, + 294 + ], + "score": 1.0, + "content": "SOTA NAS methods, mainly on the DARTS search space (Liu et al., 2019). Results grouped by the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 110, + 291, + 502, + 305 + ], + "spans": [ + { + "bbox": [ + 110, + 291, + 502, + 305 + ], + "score": 1.0, + "content": "type of search method: manually designed, full-evaluation NAS, and weight-sharing NAS. All test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 108, + 302, + 501, + 316 + ], + "spans": [ + { + "bbox": [ + 108, + 302, + 501, + 316 + ], + "score": 1.0, + "content": "errors are for models trained with auxiliary towers and cutout (parameter counts exclude auxiliary", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 109, + 313, + 502, + 327 + ], + "spans": [ + { + "bbox": [ + 109, + 313, + 502, + 327 + ], + "score": 1.0, + "content": "weights). Test errors we report are averaged over 10 seeds. “-” indicates that the field does not", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 108, + 324, + 502, + 338 + ], + "spans": [ + { + "bbox": [ + 108, + 324, + 502, + 338 + ], + "score": 1.0, + "content": "apply while “N/A” indicates unknown. Note that search cost is hardware-dependent; our results", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 110, + 336, + 208, + 347 + ], + "spans": [ + { + "bbox": [ + 110, + 336, + 208, + 347 + ], + "score": 1.0, + "content": "used Tesla V100 GPUs.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "table", + "bbox": [ + 124, + 418, + 486, + 539 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 124, + 418, + 486, + 539 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 124, + 418, + 486, + 539 + ], + "spans": [ + { + "bbox": [ + 124, + 418, + 486, + 539 + ], + "score": 0.985, + "html": "
Test ErrorParams (M)Search Cost (GPU Days)Search Method
ArchitectureSourcetop-1top-5
MobileNet(Howard et al.,2017)29.410.54.2-manual
ShuffleNet V2 2x(Ma et al.,2018)25.1N/A~5-manual
NASNet-A*(Zoph et al.,2018)26.08.45.31800RL
AmoebaNet-C*(Real et al.,2019)24.37.66.43150evolution
DARTS(Liu et al.,2019)26.78.74.74.0gradient
SNAS(Xie et al.,2019)27.39.24.31.5gradient
ProxylessNAS‡(Cai et al.,2019)24.97.57.18.3gradient
PDARTS#(Chen et al., 2019)24.47.44.90.3gradient
PC-DARTS (CIFAR-10)+(Xu et al.,2020)25.17.85.30.1gradient
PC-DARTS (ImageNet)t(Xu et al.,2020)24.27.35.33.8gradient
GAEA PC-DARTS (CIFAR-10)†Ours24.37.35.30.1gradient
GAEA PC-DARTS (ImageNet)†Ours24.07.35.63.8gradient
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ArchitectureTest Error AverageParams (M)Search Cost (GPU Days)Comparable Search SpaceSearch Method
Shake-Shake(DeVries&Taylor,2017)Best N/A2.5626.2--manual
PyramidNet (Yamada et al.,2018)2.31N/A261-manual
NASNet-A* (Zoph et al.,2018)N/A2.653.32000NRL
AmoebaNet-B*(Real et al.,2019)N/A2.55± 0.052.83150Nevolution
ProxylessNAS* (Cai et al.,2019)2.08N/A5.74Ngradient
ENAS (Pham et al.,2018)2.89N/A4.60.5YRL
Random search Wst (Li& Talwalkar,2019)2.712.85 ±0.083.80.7Yrandom
ASNG-NAS (Akimoto et al.,2019)N/A2.83 ± 0.143.90.1Ygradient
SNAS (Xie et al.,2019)N/A2.85 ±0.022.81.5Ygradient
DARTS (lst-order)†(Liu et al.,2019)N/A3.00 ±0.143.30.4Ygradient
DARTS (2nd-order)†(Liu et al.,2019)N/A2.76±0.093.31Ygradient
PDARTS# (Chen et al., 2019)2.50N/A3.40.3Ygradient
PC-DARTS† (Xu et al.,2020)N/A2.57 ±0.073.60.1Ygradient
GAEA PC-DARTS† (Ours)2.392.50±0.063.70.1Ygradient
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Test ErrorParams (M)Search Cost (GPU Days)Search Method
ArchitectureSourcetop-1top-5
MobileNet(Howard et al.,2017)29.410.54.2-manual
ShuffleNet V2 2x(Ma et al.,2018)25.1N/A~5-manual
NASNet-A*(Zoph et al.,2018)26.08.45.31800RL
AmoebaNet-C*(Real et al.,2019)24.37.66.43150evolution
DARTS(Liu et al.,2019)26.78.74.74.0gradient
SNAS(Xie et al.,2019)27.39.24.31.5gradient
ProxylessNAS‡(Cai et al.,2019)24.97.57.18.3gradient
PDARTS#(Chen et al., 2019)24.47.44.90.3gradient
PC-DARTS (CIFAR-10)+(Xu et al.,2020)25.17.85.30.1gradient
PC-DARTS (ImageNet)t(Xu et al.,2020)24.27.35.33.8gradient
GAEA PC-DARTS (CIFAR-10)†Ours24.37.35.30.1gradient
GAEA PC-DARTS (ImageNet)†Ours24.07.35.63.8gradient
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(2020a), PC-DARTS had the most robust performance", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 562, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 506, + 573 + ], + "score": 1.0, + "content": "across the three search spaces and converged to the best architecture in search spaces 1 and 3. GDAS,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 572, + 507, + 586 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 507, + 586 + ], + "score": 1.0, + "content": "a probabilistic gradient NAS method, achieved the best performance on search space 2. Hence,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "we focused on applying a geometry-aware approach to PC-DARTS. We implemented GAEA PC-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 592, + 507, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 507, + 608 + ], + "score": 1.0, + "content": "DARTS within the repository provided by the authors of Zela et al. (2020a) available at https:", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "//github.com/automl/nasbench-1shot1. We used the same hyperparameter settings for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "training the weight-sharing network as that used by Zela et al. (2020a) for PC-DARTS. Similar to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "the previous benchmark, we initialize architecture parameters to allocate equal weight to all options.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "For the architecture updates, the only hyperparameter for GAEA PC-DARTS is the learning rate for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 649, + 286, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 286, + 661 + ], + "score": 1.0, + "content": "exponentiated gradient, which we set to 0.1.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 549, + 507, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "As mentioned in Section 4, the search spaces considered in this benchmark differ in that operations", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "are applied after aggregating all edge inputs to a node instead of per edge input as in the DARTS", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "and NAS-Bench-201 search spaces. This structure inherently limits the size of the weight-sharing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 413, + 712 + ], + "score": 1.0, + "content": "network to scale with the number of nodes instead of the number of edges", + "type": "text" + }, + { + "bbox": [ + 413, + 698, + 466, + 711 + ], + "score": 0.86, + "content": "( \\mathcal { O } ( | \\mathrm { n o d e s } | ^ { \\bar { 2 } } ) ;", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 697, + 505, + 712 + ], + "score": 1.0, + "content": ", thereby", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "limiting the degree of overparameterization. Understanding the impact of overparameterization on", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 721, + 426, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 426, + 733 + ], + "score": 1.0, + "content": "the performance of weight-sharing NAS methods is a direction for future study.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 665, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 208, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 210, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 210, + 96 + ], + "score": 1.0, + "content": "C.3 NAS-BENCH-201", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "score": 1.0, + "content": "The NAS-Bench-201 benchmark (Dong & Yang, 2020) evaluates a single search space across", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "3 datasets: CIFAR-10, CIFAR-100, and a miniature version of ImageNet (ImageNet-16-120).", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 369, + 137 + ], + "score": 1.0, + "content": "ImageNet-16-120 is a downsampled version of ImageNet with", + "type": "text" + }, + { + "bbox": [ + 369, + 125, + 404, + 136 + ], + "score": 0.9, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 125, + 505, + 137 + ], + "score": 1.0, + "content": "images and 120 classes", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 134, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 158, + 150 + ], + "score": 1.0, + "content": "for a total of", + "type": "text" + }, + { + "bbox": [ + 159, + 136, + 187, + 146 + ], + "score": 0.38, + "content": "1 5 1 . 7 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 134, + 506, + 150 + ], + "score": 1.0, + "content": "training images, 3k validation images, and 3k test images. The authors of Dong", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "& Yang (2020) evaluated the architecture search performance of multiple weight-sharing methods and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "traditional hyperparameter optimization methods on all three datasets. According to the results from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 183 + ], + "score": 1.0, + "content": "Dong & Yang (2020), GDAS outperformed other weight-sharing methods by a large margin. Hence,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "we first evaluated the performance of GAEA GDAS on each of the three datasets. Our implementation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 203 + ], + "score": 1.0, + "content": "of GAEA GDAS uses an architecture learning rate of 0.1, which matches the learning rate used for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "score": 1.0, + "content": "GAEA approaches in the previous two benchmarks. Additionally, we run GAEA GDAS for 150", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "epochs instead of 250 epochs used for GDAS in the original benchmarked results; this is why the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "search cost is lower for GAEA GDAS. All other hyperparameter settings are the same. Our results", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 235, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 506, + 247 + ], + "score": 1.0, + "content": "for GAEA GDAS is comparable to the reported results for GDAS on CIFAR-10 and CIFAR-100 but", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "slightly lower on ImageNet-16-120. Compared to our reproduced results for GDAS, GAEA GDAS", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 256, + 458, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 458, + 269 + ], + "score": 1.0, + "content": "outperforms GDAS on CIFAR-100 and matches it on CIFAR-10 and ImageNet-16-120.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "Next, to see if we can use GAEA to further improve the performance of weight-sharing methods, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "evaluated GAEA DARTS (first order) applied to both the single-level (ERM) and bi-level optimization", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "problems. Again, we used a learning rate of 0.1 and trained GAEA DARTS for 25 epochs on each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "dataset. The one additional modification we made was to exclude the zero operation, which limits", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "GAEA DARTS to a subset of the search space. To isolate the impact of this modification, we also", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "evaluated first-order DARTS with this modification. Similar to (Dong & Yang, 2020), we observe", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "DARTS with this modification to also converge to architectures with nearly all skip connections,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 348, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 364 + ], + "score": 1.0, + "content": "resulting in similar performance as that reported in Dong & Yang (2020). We present the learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "curves of the oracle architecture recommended by DARTS and GAEA DARTS (when excluding zero", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "operation) over the training horizon for 4 different runs in Figure 4. For GAEA GDAS and GAEA", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 383, + 438, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 438, + 396 + ], + "score": 1.0, + "content": "DARTS, we train the weight-sharing network with the following hyperparameters:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 108, + 405, + 294, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 145, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 145, + 416 + ], + "score": 1.0, + "content": "train:", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 130, + 415, + 235, + 426 + ], + "spans": [ + { + "bbox": [ + 130, + 415, + 235, + 426 + ], + "score": 1.0, + "content": "scheduler: cosine", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 130, + 425, + 246, + 437 + ], + "spans": [ + { + "bbox": [ + 130, + 425, + 246, + 437 + ], + "score": 1.0, + "content": "lr_anneal_cycles: 1", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 130, + 437, + 294, + 448 + ], + "spans": [ + { + "bbox": [ + 130, + 437, + 294, + 448 + ], + "score": 1.0, + "content": "smooth_cross_entropy: false", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 130, + 448, + 217, + 459 + ], + "spans": [ + { + "bbox": [ + 130, + 448, + 217, + 459 + ], + "score": 1.0, + "content": "batch_size: 64", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 130, + 459, + 252, + 470 + ], + "spans": [ + { + "bbox": [ + 130, + 459, + 252, + 470 + ], + "score": 1.0, + "content": "learning_rate: 0.025", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 129, + 470, + 276, + 480 + ], + "spans": [ + { + "bbox": [ + 129, + 470, + 276, + 480 + ], + "score": 1.0, + "content": "learning_rate_min: 0.001", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 129, + 481, + 211, + 491 + ], + "spans": [ + { + "bbox": [ + 129, + 481, + 211, + 491 + ], + "score": 1.0, + "content": "momentum: 0.9", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 130, + 491, + 252, + 503 + ], + "spans": [ + { + "bbox": [ + 130, + 491, + 252, + 503 + ], + "score": 1.0, + "content": "weight_decay: 0.0003", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 130, + 503, + 235, + 514 + ], + "spans": [ + { + "bbox": [ + 130, + 503, + 235, + 514 + ], + "score": 1.0, + "content": "init_channels: 16", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 130, + 514, + 187, + 524 + ], + "spans": [ + { + "bbox": [ + 130, + 514, + 187, + 524 + ], + "score": 1.0, + "content": "layers: 5", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 130, + 524, + 240, + 536 + ], + "spans": [ + { + "bbox": [ + 130, + 524, + 240, + 536 + ], + "score": 1.0, + "content": "autoaugment: false", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 131, + 536, + 211, + 546 + ], + "spans": [ + { + "bbox": [ + 131, + 536, + 211, + 546 + ], + "score": 1.0, + "content": "cutout: false", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 131, + 547, + 228, + 557 + ], + "spans": [ + { + "bbox": [ + 131, + 547, + 228, + 557 + ], + "score": 1.0, + "content": "auxiliary: false", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 130, + 557, + 258, + 569 + ], + "spans": [ + { + "bbox": [ + 130, + 557, + 258, + 569 + ], + "score": 1.0, + "content": "auxiliary_weight: 0.4", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 130, + 568, + 235, + 580 + ], + "spans": [ + { + "bbox": [ + 130, + 568, + 235, + 580 + ], + "score": 1.0, + "content": "drop_path_prob: 0", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 130, + 580, + 205, + 591 + ], + "spans": [ + { + "bbox": [ + 130, + 580, + 205, + 591 + ], + "score": 1.0, + "content": "grad_clip: 5", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "Surprisingly, we observe single-level optimization to yield better performance than solving the bi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "level problem with GAEA DARTS on this search space. In fact, the performance of GAEA DARTS", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "(ERM) not only exceeds that of GDAS, but also outperforms traditional hyperparameter optimization", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "approaches on all three datasets, nearly reaching the optimal accuracy on all three datasets. In", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "contrast, GAEA DARTS (bi-level) outperforms GDAS on CIFAR-100 and ImageNet-16-120 but", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "underperforms slightly on CIFAR-10. The single-level results on this benchmark provides concrete", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "support for our convergence analysis, which only applies to the ERM problem. As noted in Section 4,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "the search space considered in this benchmark differs from the prior two in that there is no subsequent", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "edge pruning. Additionally, the search space is fairly small with only 3 nodes for which architecture", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "decisions must be made. The success of GAEA DARTS (ERM) on this benchmark indicate the need", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "for a better understanding of when single-level optimization should be used in favor of the default", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 721, + 417, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 417, + 733 + ], + "score": 1.0, + "content": "bi-level optimization problem and how the search space impacts the decision.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49.5 + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 208, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 210, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 210, + 96 + ], + "score": 1.0, + "content": "C.3 NAS-BENCH-201", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "score": 1.0, + "content": "The NAS-Bench-201 benchmark (Dong & Yang, 2020) evaluates a single search space across", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "3 datasets: CIFAR-10, CIFAR-100, and a miniature version of ImageNet (ImageNet-16-120).", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 369, + 137 + ], + "score": 1.0, + "content": "ImageNet-16-120 is a downsampled version of ImageNet with", + "type": "text" + }, + { + "bbox": [ + 369, + 125, + 404, + 136 + ], + "score": 0.9, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 125, + 505, + 137 + ], + "score": 1.0, + "content": "images and 120 classes", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 134, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 158, + 150 + ], + "score": 1.0, + "content": "for a total of", + "type": "text" + }, + { + "bbox": [ + 159, + 136, + 187, + 146 + ], + "score": 0.38, + "content": "1 5 1 . 7 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 134, + 506, + 150 + ], + "score": 1.0, + "content": "training images, 3k validation images, and 3k test images. The authors of Dong", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "& Yang (2020) evaluated the architecture search performance of multiple weight-sharing methods and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "traditional hyperparameter optimization methods on all three datasets. According to the results from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 183 + ], + "score": 1.0, + "content": "Dong & Yang (2020), GDAS outperformed other weight-sharing methods by a large margin. Hence,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "we first evaluated the performance of GAEA GDAS on each of the three datasets. Our implementation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 203 + ], + "score": 1.0, + "content": "of GAEA GDAS uses an architecture learning rate of 0.1, which matches the learning rate used for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "score": 1.0, + "content": "GAEA approaches in the previous two benchmarks. Additionally, we run GAEA GDAS for 150", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "epochs instead of 250 epochs used for GDAS in the original benchmarked results; this is why the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "search cost is lower for GAEA GDAS. All other hyperparameter settings are the same. Our results", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 235, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 506, + 247 + ], + "score": 1.0, + "content": "for GAEA GDAS is comparable to the reported results for GDAS on CIFAR-10 and CIFAR-100 but", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "slightly lower on ImageNet-16-120. Compared to our reproduced results for GDAS, GAEA GDAS", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 256, + 458, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 458, + 269 + ], + "score": 1.0, + "content": "outperforms GDAS on CIFAR-100 and matches it on CIFAR-10 and ImageNet-16-120.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 102, + 506, + 269 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "Next, to see if we can use GAEA to further improve the performance of weight-sharing methods, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "evaluated GAEA DARTS (first order) applied to both the single-level (ERM) and bi-level optimization", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "problems. Again, we used a learning rate of 0.1 and trained GAEA DARTS for 25 epochs on each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "dataset. The one additional modification we made was to exclude the zero operation, which limits", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "GAEA DARTS to a subset of the search space. To isolate the impact of this modification, we also", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "evaluated first-order DARTS with this modification. Similar to (Dong & Yang, 2020), we observe", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "DARTS with this modification to also converge to architectures with nearly all skip connections,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 348, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 364 + ], + "score": 1.0, + "content": "resulting in similar performance as that reported in Dong & Yang (2020). We present the learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "curves of the oracle architecture recommended by DARTS and GAEA DARTS (when excluding zero", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 384 + ], + "score": 1.0, + "content": "operation) over the training horizon for 4 different runs in Figure 4. 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In fact, the performance of GAEA DARTS", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "(ERM) not only exceeds that of GDAS, but also outperforms traditional hyperparameter optimization", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "approaches on all three datasets, nearly reaching the optimal accuracy on all three datasets. In", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "contrast, GAEA DARTS (bi-level) outperforms GDAS on CIFAR-100 and ImageNet-16-120 but", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "underperforms slightly on CIFAR-10. The single-level results on this benchmark provides concrete", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "support for our convergence analysis, which only applies to the ERM problem. As noted in Section 4,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "the search space considered in this benchmark differs from the prior two in that there is no subsequent", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "edge pruning. Additionally, the search space is fairly small with only 3 nodes for which architecture", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "decisions must be made. 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for iteration t=1,...,TdoInput: nitialization(w(1),(1)）∈Rd×,strongly-convex DGF𝜙:→R,numberofiterations T≥1,step-size n>0
sample bt~Unif{w,0} if block bt = w then//randomly select update block
w(t+1) ←w(t)-nwf(w(t),0(t))// SGD update to shared weights
θ(t+1)←0(t)// no update to architecture params
else w(t+1）←w(t)
// no update to shared weights
θ(t+1)← argminuεe nθf(w(t),θ(t))·u + D(ul|θ(t))// update architecture params
Output:(w(r),θ(r)) for r~ Unif{1,...,T}
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Search Method (source)CIFAR-10 ErrorSearch Cost (GPU Days)ImageNet Error top-1Search Cost (GPU Days)method
BestAveragetop-5
NASNet-A* (Zoph et al.,2018)-2.65200026.08.41800RL
AmoebaNet-B* (Real et al.,2019)12.55± 0.05315024.37.63150evolution
ProxylessNAS*(Cai et al.,2019)2.08424.97.58.3gradient (WS)
ENAS (Pham et al., 2018)2.890.5--1RL (WS)
RS-Wst (Li& Talwalkar,2019)2.712.85±0.080.7-1-random (WS)
ASNG (Akimoto et al.,2019)12.83 ± 0.140.1--gradient (WS)
SNAS (Xie et al., 2019)=2.85±0.021.527.39.21.5gradient (WS)
DARTS (1st)† (Liu et al.,2019)=3.00 ±0.140.4---gradient (WS)
DARTS (2nd)† (Liu et al., 2019)2.76 ±0.09126.78.74.0gradient (WS)
PDARTS (Chen et al.,2019)2.50=0.324.47.40.3gradient (WS)
PC-DARTSt (Xu et al.,2020)-2.57 ±0.070.124.27.33.8gradient (WS)
GAEA PC-DARTS† (ours)2.392.50 ±0.060.124.07.33.8gradient (WS)
PC-DARTSt (Xu et al.,2020)(search on CIFAR-1O,train on ImageNet)25.17.80.1gradient (WS)
GAEA PC-DARTS† (ours)(search on CIFAR-10, train on ImageNet)24.37.30.1gradient (WS)
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Search*(seconds)CIFAR-10 (test)CIFAR-100 (test)ImageNet-16-120 (test)
RegularREAN/A93.92 ± 0.3071.84 ± 0.9945.54 ± 1.03
HO,RSN/A93.70±0.3671.04 ± 1.0844.57 ±1.25
search onREINFORCEN/A93.85 ± 0.3771.71 ± 1.0945.25 ± 1.18
CIFAR-10BOHBN/A93.61 ±0.5270.85 ±1.2844.42 ± 1.49
RSPS758787.66 ± 1.6958.33 ± 4.3431.14 ± 3.88
DARTS (bilevel)3578154.30 ±0.0015.61 ±0.0016.32 ± 0.00
WeightSETN3413987.64 ± 0.0059.05 ± 0.2432.52 ±0.21
sharing,GDAS3160993.61 ±0.0970.70 ± 0.3041.71 ± 0.98
search onDARTS‡ (bilevel)10683†54.30 ±0.0015.32 ± 0.0016.38 ± 0.00
CIFAR-10GAEA DARTS (bilevel)7930+91.63 ± 2.5768.39 ± 4.4741.59 ± 4.20
DARTS* (ERM)18112t84.39 ±3.8254.81 ± 7.0831.82 ± 4.78
GAEA DARTS (ERM)9061†94.10±0.2972.60± 0.8945.81 ± 0.51
WeightGDAS27923†93.52 ± 0.1567.52 ± 0.1540.91 ± 0.12
GAEA GDAS16754t93.55 ± 0.1370.47 ± 0.4740.91 ±0.12
sharing,DARTS‡ (bilevel)10683†54.30 ±0.0015.32 ±0.0028.96 ±10.22
directGAEA DARTS (bilevel)7930t91.63 ± 2.5771.87 ± 0.5745.69 ± 0.56
searchDARTS* (ERM)18112†84.39 ±3.8251.26 ±6.1431.35 ± 7.46
GAEA DARTS (ERM)9061†94.10±0.2973.43 ± 0.1346.36 ±0.00
ResNetN/A93.9770.8643.63
OptimalN/A94.3773.5147.31
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Set 1 (Reported)Set2Set3
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Test ErrorParams (M)Search Cost (GPU Days)Search Method
ArchitectureSourcetop-1top-5
MobileNet(Howard et al.,2017)29.410.54.2-manual
ShuffleNet V2 2x(Ma et al.,2018)25.1N/A~5-manual
NASNet-A*(Zoph et al.,2018)26.08.45.31800RL
AmoebaNet-C*(Real et al.,2019)24.37.66.43150evolution
DARTS(Liu et al.,2019)26.78.74.74.0gradient
SNAS(Xie et al.,2019)27.39.24.31.5gradient
ProxylessNAS‡(Cai et al.,2019)24.97.57.18.3gradient
PDARTS#(Chen et al., 2019)24.47.44.90.3gradient
PC-DARTS (CIFAR-10)+(Xu et al.,2020)25.17.85.30.1gradient
PC-DARTS (ImageNet)t(Xu et al.,2020)24.27.35.33.8gradient
GAEA PC-DARTS (CIFAR-10)†Ours24.37.35.30.1gradient
GAEA PC-DARTS (ImageNet)†Ours24.07.35.63.8gradient
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ArchitectureTest Error AverageParams (M)Search Cost (GPU Days)Comparable Search SpaceSearch Method
Shake-Shake(DeVries&Taylor,2017)Best N/A2.5626.2--manual
PyramidNet (Yamada et al.,2018)2.31N/A261-manual
NASNet-A* (Zoph et al.,2018)N/A2.653.32000NRL
AmoebaNet-B*(Real et al.,2019)N/A2.55± 0.052.83150Nevolution
ProxylessNAS* (Cai et al.,2019)2.08N/A5.74Ngradient
ENAS (Pham et al.,2018)2.89N/A4.60.5YRL
Random search Wst (Li& Talwalkar,2019)2.712.85 ±0.083.80.7Yrandom
ASNG-NAS (Akimoto et al.,2019)N/A2.83 ± 0.143.90.1Ygradient
SNAS (Xie et al.,2019)N/A2.85 ±0.022.81.5Ygradient
DARTS (lst-order)†(Liu et al.,2019)N/A3.00 ±0.143.30.4Ygradient
DARTS (2nd-order)†(Liu et al.,2019)N/A2.76±0.093.31Ygradient
PDARTS# (Chen et al., 2019)2.50N/A3.40.3Ygradient
PC-DARTS† (Xu et al.,2020)N/A2.57 ±0.073.60.1Ygradient
GAEA PC-DARTS† (Ours)2.392.50±0.063.70.1Ygradient
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jsnoek@google.com + +Jeremiah Zhe Liu Google Research & Harvard University jereliu@google.com + +Balaji Lakshminarayanan Google Research balajiln@google.com + +Andrew M. Dai Google Research adai@google.com + +Dustin Tran +Google Research +trandustin@google.com + +# ABSTRACT + +Recent approaches to efficiently ensemble neural networks have shown that strong robustness and uncertainty performance can be achieved with a negligible gain in parameters over the original network. However, these methods still require multiple forward passes for prediction, leading to a significant computational cost. In this work, we show a surprising result: the benefits of using multiple predictions can be achieved ‘for free’ under a single model’s forward pass. In particular, we show that, using a multi-input multi-output (MIMO) configuration, one can utilize a single model’s capacity to train multiple subnetworks that independently learn the task at hand. By ensembling the predictions made by the subnetworks, we improve model robustness without increasing compute. We observe a significant improvement in negative log-likelihood, accuracy, and calibration error on CIFAR10, CIFAR100, ImageNet, and their out-of-distribution variants compared to previous methods. + +# 1 INTRODUCTION + +Uncertainty estimation and out-of-distribution robustness are critical problems in machine learning. In medical applications, a confident misprediction may be a misdiagnosis that is not referred to a physician as during decision-making with a “human-in-the-loop.” This can have disastrous consequences, and the problem is particularly challenging as patient data deviates significantly from the training set such as in demographics, disease types, epidemics, and hospital locations (Dusenberry et al., 2020b; Filos et al., 2019). + +Using a distribution over neural networks is a popular solution stemming from classic Bayesian and ensemble learning literature (Hansen & Salamon, 1990; Neal, 1996), and recent advances such as BatchEnsemble and extensions thereof achieve strong uncertainty and robustness performance (Wen et al., 2020; Dusenberry et al., 2020a; Wenzel et al., 2020). These methods demonstrate that significant gains can be had with negligible additional parameters compared to the original model. However, these methods still require multiple (typically, 4-10) forward passes for prediction, leading to a significant runtime cost. In this work, we show a surprising result: the benefits of using multiple predictions can be achieved “for free” under a single model’s forward pass. + +The insight we build on comes from sparsity. Neural networks are heavily overparameterized models. The lottery ticket hypothesis (Frankle & Carbin, 2018) and other works on model pruning (Molchanov et al., 2016; Zhu & Gupta, 2017) show that one can prune away $70 \%$ of the connections in a neural network without adversely affecting performance. The remaining sparse subnetwork, called the winning ticket, retains its predictive accuracy. This suggests that a neural network has sufficient capacity to fit 3-4 independent subnetworks simultaneously. We show that, using a multi-input multioutput (MIMO) configuration, we can concurrently train multiple independent subnetworks within one network. These subnetworks co-habit the network without explicit separation. The advantage of doing this is that at test time, we can evaluate all of the subnetworks at the same time, leveraging the benefits of ensembles in a single forward pass. + +![](images/366458d83f05825bbd16f241a272a1ee0ccb4fcd319ea87221c83bf3a6f42580.jpg) +Figure 1: In the multi-input multi-output (MIMO) configuration, the network takes $M = 3$ inputs and gives $M$ outputs. The hidden layers remain unchanged. The black connections are shared by all subnetworks, while the colored connections are for individual subnetworks. (a) During training, the inputs are independently sampled from the training set and the outputs are trained to classify their corresponding inputs. (b) During testing, the same input is repeated $M$ times and the outputs are averaged in an ensemble to obtain the final prediction. + +Our proposed MIMO configuration only requires two changes to a neural network architecture. First, replace the input layer: instead of taking a single datapoint as input, take $M$ datapoints as inputs, where $M$ is the desired number of ensemble members. Second, replace the output layer: instead of a single head, use $M$ heads that make $M$ predictions based on the last hidden layer. During training, the inputs are sampled independently from the training set and each of the $M$ heads is trained to predict its matching input (Figure 1a). Since, the features derived from the other inputs are not useful for predicting the matching input, the heads learn to ignore the other inputs and make their predictions independently. At test time, the same input is repeated $M$ times. That is, the heads make $M$ independent predictions on the same input, forming an ensemble for a single robust prediction that can be computed in a single forward pass (Figure 1b). + +The core component of an ensemble’s robustness such as in Deep Ensembles is the diversity of its ensemble members (Fort et al., 2019). While it is possible that a single network makes a confident misprediction, it is less likely that multiple independently trained networks make the same mistake. Our model operates on the same principle. By realizing multiple independent winning lottery tickets, we are reducing the impact of one of them making a confident misprediction. For this method to be effective, it is essential that the subnetworks make independent predictions. We empirically show that the subnetworks use disjoint parts of the network and that the functions they represent have the same diversity as the diversity between independently trained neural networks. + +# Summary of contributions. + +1. We propose a multi-input multi-output (MIMO) configuration to network architectures, enabling multiple independent predictions in a single forward pass “for free.” Ensembling these predictions significantly improves uncertainty estimation and robustness with minor changes to the number of parameters and compute cost. +2. We analyze the diversity of the individual members and show that they are as diverse as independently trained neural networks. +3. We demonstrate that when adjusting for wall-clock time, MIMO networks achieve new state-ofthe-art on CIFAR10, CIFAR100, ImageNet, and their out-of-distribution variants. + +# 2 MULTI-INPUT MULTI-OUTPUT NETWORKS + +The MIMO model is applicable in a supervised classification or regression setting. Denote the set of training examples $\mathbb { X } = \{ ( \pmb { x } ^ { ( n ) } , \pmb { y } ^ { ( n ) } ) \} _ { n = 1 } ^ { N }$ where $\pmb { x } ^ { ( n ) }$ is the $n ^ { t h }$ datapoint with the corresponding label $\pmb { y } ^ { ( n ) }$ and $N$ is the size of the training set. In the usual setting, for an input $_ { \textbf { \em x } }$ , the output of the neural network $\hat { \mathbf { y } }$ is a probability distribution $p _ { \theta } ( \hat { \mathbf { y } } | \boldsymbol { x } )$ ,1 which captures the uncertainty in the predictions of the network. The network parameters $\theta$ are trained using stochastic gradient descent (SGD) to minimize the loss $L ( \theta )$ on the training set, where the loss usually includes the negative log-likelihood and a regularization term $R$ (such as the L2 regularization): $L ( \theta ) =$ $\mathbb { E } ( \mathbf { x } , \mathbf { y } ) \mathrm { { \dot { e } x } } [ - \log { \bar { p _ { \theta } } ( \mathbf { y } | \mathbf { x } ) } ] + R ( \theta )$ . + +![](images/e60e924d6286d74022e9d183724ef389bf138dd336575a224d0b64380843e397.jpg) +Figure 2: Illustration of MIMO applied to a synthetic regression problem. (left) Example of MIMO learning $M = 3$ diverse predictors. As $M$ increases, predicting with MIMO comes with a higher bias but a smaller variance (two middle panels respectively). Despite the slight increase in bias, the decrease in variance translates into an improved generalization performance (right). + +In the MIMO configuration, the network takes $M$ inputs and returns $M$ outputs (Figure 1), where each output is a prediction for the corresponding input. This requires two small changes to the architecture. At the input layer, the $M$ inputs $\mathbf { \bar { \{ x } } _ { 1 } , \dots , \mathbf { x } _ { M } \}$ are concatenated before the first hidden layer is applied and at the output layer, the network gives $M$ predictive distributions $\left\{ p _ { \boldsymbol \theta } ( \mathbf { y } _ { 1 } | \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { M } ) , \ldots , p _ { \boldsymbol \theta } ( \mathbf { y } _ { M } | \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { M } ) \right\}$ correspondingly. Having $M$ input and $M$ outputs require additional model parameters. The additional weights used in the MIMO configuration account for just a $0 . 0 3 \%$ increase in the total number of parameters and $0 . 0 1 \%$ increase in floating-point operations (FLOPs).2 + +The network is trained similarly to a traditional neural network, with a few key modifications to account for the $M$ inputs and $M$ outputs (Figure 1a). During training, the inputs $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { M }$ are sampled independently from the training set. The loss is the sum of the negative log-likelihoods of the predictions and the regularization term: + +$$ +L _ { M } ( \theta ) = \underset { ( \mathbf { x } _ { 1 } , \mathbf { y } _ { 1 } ) \in \mathbb { X } } { \mathbb { E } } \left[ \sum _ { m = 1 } ^ { M } - \log p _ { \theta } ( \pmb { y _ { m } } | \pmb { x } _ { 1 } , \dots , \pmb { x _ { M } } ) \right] + R ( \theta ) , +$$ + +which is optimized using stochastic gradient descent. Note that the sum of the log-likelihoods is equal to the log-likelihood of the joint distribution $\begin{array} { r l } { \sum _ { m = 1 } ^ { M } \log p _ { \boldsymbol \theta } ( \pmb y _ { m } | \pmb x _ { 1 } , \dots , \pmb x _ { M } ) } & { = } \end{array}$ $\log p _ { \boldsymbol { \theta } } ( \pmb { y } _ { 1 } , \dots , \pmb { y } _ { M } | \pmb { x } _ { 1 } , \dots , \pmb { x } _ { M } )$ since the input-output pairs are independent. Hence a second interpretation of MIMO is that it is simply training a traditional neural network over $M$ -tuples of independently sampled datapoints. + +At evaluation time, the network is used to make a prediction on a previously unseen input $\mathbf { x } ^ { \prime }$ . The input $\mathbf { x } ^ { \prime }$ is tiled $M$ times, so $\pmb { x } _ { 1 } = . . . = \pmb { x } _ { M } = \pmb { x } ^ { \prime }$ (Figure 1b). Since all of the inputs are $\mathbf { x } ^ { \prime }$ , each of the outputs independently approximate the predictive distribution $p _ { \boldsymbol \theta } ( \mathbf { y } _ { m } | \mathbf { x } ^ { \prime } , \ldots , \mathbf { x } ^ { \prime } ) \approx p ( \mathbf { y } ^ { \prime } | \mathbf { x } ^ { \prime } )$ (for $m = 1 \ldots M )$ . As an ensembl combined output $\begin{array} { r } { p _ { \boldsymbol { \theta } } ( \mathbf { y } ^ { \prime } | \mathbf { x } ^ { \prime } ) = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p _ { \boldsymbol { \theta } } ( \mathbf { y } _ { m } | \mathbf { x } ^ { \prime } , \dots , \mathbf { x } ^ { \prime } ) } \end{array}$ e predictive performance,. + +Unlike Bayesian methods requiring multiple weight samples, or even parameter-efficient methods like BatchEnsemble, MIMO’s advantage is that all of the ensemble members can be calculated in a single forward pass. As a result, MIMO’s wall-clock time is almost equivalent to a standard neural network. + +![](images/ae5a07d29a346ba35da6faa7d3f8fdfd07778953474a355fcadfbdd21235922b.jpg) +Figure 3: Accuracy landscape and function space landscape comparison of individual subnetworks for MIMO (top row) and the naive multiheaded architecture (bottom row). (left): The test accuracy in the weight space section containing $M = 3$ trained subnetworks and the origin. For the MIMO architecture, the individual subnetworks converge to three distinct low-loss basins, while naive multihead leads to the same mode. (middleleft to right): The blue, red and green panels show the disagreement between the three trained subnetworks for the same section of the weight space. For the MIMO architecture, the subnetworks often disagree, while for the naive multihead architecture they are all essentially equivalent. + +# 2.1 ILLUSTRATION OF MIMO ON A SYNTHETIC REGRESSION EXAMPLE + +Before applying MIMO to large-scale vision models, we first illustrate its behavior on a simple one-dimensional regression problem. We consider the noisy function from Blundell et al. (2015) (see Figure 2, left), with a training and test set of $N = 6 4$ and 3000 observations respectively. We train a multilayer perceptron with two hidden-layers, composed of (32, 128) units and ReLU activations. 3 + +For different ensemble sizes $M \in \{ 1 , \ldots , 5 \}$ , we evaluate the resulting models in terms of expected mean squared error $\mathcal { E } _ { M }$ . If we denote by $\hat { f } _ { M }$ the regressor with $M$ ensemble members learned over $\mathbb { X }$ , we recall that $\mathcal { E } _ { M } = \mathbb { E } ( \pmb { x } ^ { \prime } , \pmb { y } ^ { \prime } ) \in \mathbb { X } _ { \mathrm { t e s t } } [ \mathbb { E } \pmb { \mathbb { X } } [ ( \hat { f } _ { M } ( \pmb { x } ^ { \prime } , \dots , \pmb { x } ^ { \prime } ) - \pmb { y } ^ { \prime } ) ^ { 2 } ] ]$ , where $\mathbb { E } \mathbb { x } [ \cdot ]$ denotes the expectation over training sets of size $N$ . We make two main observations. + +First, in the example of $M = 3$ in Figure 2 (left), we can see that MIMO can learn a diverse set of predictors. Second, the diverse predictors obtained by MIMO translate into improved performance, as seen in Figure 2 (right). Moreover, in the regression setting, we can decompose ${ \mathcal { E } } _ { M }$ into its (squared) bias and variance components (Sec. 2.5 in Hastie et al. (2009)). More formally, we have + +$$ +\mathcal { E } _ { M } = \underbrace { \mathbb { E } } _ { ( \mathbf { x } ^ { \prime } , y ^ { \prime } ) \in \mathrm { X } _ { \mathrm { e x t } } } \left[ ( \widetilde { f } _ { M } ( \mathbf { x } ^ { \prime } , \dots , \mathbf { x } ^ { \prime } ) - y ^ { \prime } ) ^ { 2 } \right] + \underbrace { \mathbb { E } } _ { ( \mathbf { x } ^ { \prime } , y ^ { \prime } ) \in \mathbb { X } _ { \mathrm { e x t } } } \left[ \mathbb { E } \left[ ( \widetilde { f } _ { M } ( \mathbf { x } ^ { \prime } , \dots , \mathbf { x } ^ { \prime } ) - \hat { f } _ { M } ( \mathbf { x } ^ { \prime } , \dots , \mathbf { x } ^ { \prime } ) ) ^ { 2 } \right] \right] +$$ + +where $\bar { f } _ { M } = \mathbb { E } _ { \mathbb { X } } [ \hat { f } _ { M } ]$ . The bias-variance decomposition nicely captures the strength of MIMO. While learning a neural network over $M$ -tuples induces a slight bias compared to a standard model with $M = 1$ , i.e. the individual members perform slightly worse (Figure 2, middle-left), this is compensated by the diverse predictions of the ensemble members that lead to lower variance (Figure 2, middle-right). MIMO yields an improvement when the model has sufficient capacity to fit $M > 1$ diverse, well-performing ensemble members. + +
DdisagreementDKL
Naivemultihead0.0000.000
TreeNet0.0100.010
BatchEnsemble0.0140.020
MIMO0.0320.086
Deep Ensemble0.0320.086
+ +![](images/28e46c252ba3873e62fbc944c1a9ce57e97aeed69f7fcf0b4e2c68c81b914f57.jpg) +Figure 4: Analyzing the subnetworks on the CIFAR10 dataset. (left): Histogram of the conditional variances of the pre-activations w.r.t. each input $M = 2$ , ResNet28-10). (middle-left): Scatter plot of the conditional variances of the pre-activations w.r.t. each input. Almost all the pre-activations only have variance with respect to one of the inputs: the subnetwork they that are part of $M = 3$ , ResNet28-10). (middle-right): Training trajectories of the subnetworks. The subnetworks converge to different local optima $M = 3$ , SmallCNN). (right): Diversity of the members $( \mathcal { D } _ { D } )$ in different efficient ensemble models (ResNet 28-10). + +# 3 UNDERSTANDING THE SUBNETWORKS + +The mapping of each input-output pair in a MIMO configuration is referred to as a subnetwork. In this section, we show that the subnetworks converge to distinct local optima and they functionally behave as independently trained neural networks. + +# 3.1 LOSS-LANDSCAPE ANALYSIS + +Using multiple inputs is the key to training diverse subnetworks. The subnetworks learn independently, since features derived from each input are only useful for the corresponding output. In contrast, in a naive multiheaded architecture, where the input is shared, but the model has separate $M$ outputs, the outputs rely on the same features for prediction, which leads to very low diversity. + +To showcase this, we replicate a study from Fort et al. (2019). We look at the SmallCNN model (3 convolutional layers with 16, 32, 32 filters respectively) trained on CIFAR-10, and linearly interpolate between the three subnetworks in weight space. In the case of MIMO, we interpolate the input and output layers, since the body of the network is shared, and for the naive multiheaded model, we only interpolate the output layers, since the input and the body of the network is shared. Analogously to Deep Ensembles, the subnetworks trained using MIMO converge to disconnected modes in weight space due to differences in initialization, while in the case of the naive multiheaded model, the subnetworks end up in the same mode (Figure 3, left). Figure 3 (right) shows the disagreement i.e. the probability that the subnetworks disagree on the predicted class. MIMO’s disconnected modes yield diverse predictions, while the predictions of the naive multiheaded model are highly correlated. + +# 3.2 FUNCTION SPACE ANALYSIS + +We visualize the training trajectories of the subnetworks in MIMO in function space (similarly to Fort et al. (2019)). As we train the SmallCNN architecture ( $M = 3$ ), we periodically save the predictions on the test set. Once training is finished, we plot the t-SNE projection (Maaten & Hinton, 2008) of the predictions. We observe that the trajectories converge to distinct local optima (Figure 4). + +For a quantitative measurement of diversity in large scale networks, we look at the average pairwise similarity of the subnetwork’s predictions at test time, and compare against other efficient ensemble methods. The average pairwise similarity is + +$$ +\mathcal { D } _ { D } = \mathbb { E } \left[ D \left( p _ { \theta } ( \mathbf { y } _ { 1 } | \mathbf { x } , \mathbf { x } \ldots \mathbf { x } ) , p _ { \theta } ( \mathbf { y } _ { 2 } | \mathbf { x } , \mathbf { x } \ldots \ldots \mathbf { x } ) \right) \right] , +$$ + +where $D$ is a distance metric between predictive distributions and $( \mathbf { x } , \mathbf { y } ) \in \mathbb { X }$ . We consider two distance metrics. Disagreement: whether the predicted classes agree, $D _ { \mathrm { d i s a g r e e m e n t } } ( P _ { 1 } , P _ { 2 } ) \ =$ $\begin{array} { r c l } { \operatorname { I } ( \arg \operatorname* { m a x } _ { \hat { y } } P _ { 1 } ( \hat { y } ) } & { = } & { \arg \operatorname* { m a x } _ { \hat { y } } P _ { 2 } ( \hat { y } ) ) } \end{array}$ and Kullback–Leibler divergence: $\begin{array} { r l } { D _ { \mathrm { K L } } ( P _ { 1 } , P _ { 2 } ) } & { { } = } \end{array}$ $\mathbb { E } _ { P _ { 1 } } \left[ \log P _ { 1 } ( y ) - \log P _ { 2 } ( y ) \right]$ . When the ensemble members give the same prediction at all test points, both their disagreement and KL divergence are 0. + +The first efficient ensemble approach we compare against is the aforementioned naive multiheaded model, where the input and the body of the neural network is shared by the ensemble members, but each member has its own output layer. Next, TreeNet (Lee et al., 2015), where the input and the first two residual groups are shared, but the final residual group and the output layer are trained separately for each member. Finally, BatchEnsemble (Wen et al., 2020), where the members share network parameters up to a rank-1 perturbation, which changes information flow through the full network. + +![](images/4b208d0efc4979d83270a16d261e7758938409355e703f1ad219a174f6c536d8.jpg) +Figure 5: The performance of the subnetworks and the ensemble of the subnetworks as the number of subnetworks $( M )$ varies. $M = 1$ is equivalent to a standard neural network (ResNet-28-10). + +We find that the naive multiheaded model fails to induce diversity: the predictions of its subnetworks are nearly identical on all test points as shown in Figure 4 (right). TreeNet and BatchEnsemble have more diversity, although there is still significant correlation in the predictions. MIMO has better diversity than prior efficient ensemble approaches and it matches the diversity of independently trained neural networks. + +These results allow us to pinpoint the source of robustness: The robustness of MIMO comes from ensembling the diverse predictions made by the subnetworks, thus MIMO faithfully replicates the behaviour of a Deep Ensemble within one network. + +# 3.3 SEPARATION OF THE SUBNETWORKS + +To show that the subnetworks utilize separate parts of the network, we look at the activations and measure how they react to changes in each of the $M$ inputs. Namely, we calculate the conditional variance of each pre-activation in the network with respect to each individual input. For input $\mathbf { x } _ { 1 }$ : + +$$ +\mathrm { 7 a r } ( a _ { i } | \mathbf { x } _ { 2 } ) = \underset { \mathbf { x } _ { 2 } } { \mathbb { E } } [ \mathrm { V a r } ( a _ { i } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } ) ] \quad ( M = 2 ) , \ \mathrm { a n d } \ \mathrm { V a r } ( a _ { i } | \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 } ) = \underset { \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 } } { \mathbb { E } } [ \mathrm { V a r } ( a _ { i } ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 } ) ] \quad ( M = 3 ) +$$ + +where $a _ { i }$ is the value of $i$ -th pre-activation in the function of the $M$ inputs. For reference, there are 8190 pre-activations in a ResNet28-10. We can estimate the conditional variance by fixing $\mathbf { x } _ { 1 }$ and calculating the variance of $a _ { i }$ w.r.t. $\mathbf { x } _ { 2 } \in \mathbb { X }$ (when $M = 2$ , $\mathbf { x } _ { 2 } , \mathbf { x } _ { 3 } \in \mathbb { X }$ when $M = 3$ ) and finally averaging over the possible values $\mathbf { x } _ { 1 } \in \mathbb { X }$ . The conditional variance is analogously defined for $\mathbf { x } _ { 2 } , \ldots , \mathbf { x } _ { M }$ . If the conditional variance of an activation is non-zero w.r.t. an input, that means that the activation changes as the input changes and therefore we consider it part of the subnetwork corresponding to the input. If the subnetworks are independent within the network, we expect that the conditional variance of each activation is non-zero w.r.t. one of the inputs, the subnetwork to which it belongs, and close-to zero w.r.t. all the other inputs. + +When we plot the conditional variances, this is exactly what we see. In Figure 4 (left), we see the histogram of the pre-activations in the network. Each point has two corresponding values: the conditional variance w.r.t. the two inputs. As we can see, all activations have non-zero conditional variance w.r.t. one of the inputs and close-to zero w.r.t. the other. Figure 4 (middle-left) shows the scatterplot of the activations for $M = 3$ . We see that, similarly to $M = 2$ , almost all activations have non-zero conditional variance w.r.t. one of the inputs and close-to zero conditional variance w.r.t. the others. Since almost all activations are part of exactly one of the subnetworks, which we can identify by calculating the conditional variances, we conclude that the subnetworks separate within the network. This implies an extension to Frankle & Carbin (2018): the subnetworks realize separate winning lottery tickets within a single network instance. + +# 3.4 THE OPTIMAL NUMBER OF SUBNETWORKS + +A natural question that arises is the ideal number of subnetworks $M$ to fit in a network. Too few subnetworks do not fully leverage the benefits of ensembling, while having too many quickly reaches the network capacity, hurting their individual performances. Ideally, we are looking to fit as many subnetworks as possible without significantly impacting their individual performances. + +Figure 5 shows the performance of both the individual subnetworks and the ensemble as $M$ varies. $M = 1$ is equivalent to a traditionally trained network i.e. the performance of the subnetwork matches the performance of the ensemble since there is only one subnetwork. As $M$ grows, we can see that the performance of the subnetworks slowly declines as they utilize more and more of the network capacity. The performance of the ensemble, however, peaks between $M = 2$ and $M = 4$ , where the benefits of ensembling outweigh the slight decrease in performance of the individual subnetworks. Interestingly, the accuracy peaks earlier than the log-likelihood, which suggests that ensembling is more beneficial for the latter. + +![](images/ffa34da70b55fac06c97da7044f3b96b71a936e0c540a4edbcc3bb7ef0a7bacf.jpg) +Figure 6: (a) Performance of MIMO $M = 2$ ) as a function of $\rho$ on ImageNet. At $\rho = 0$ , the subnetworks are independent and they are limited by the network capacity. With $\rho > 0$ , the subnetworks are able to share features and better utilize the network capacity. Wide ResNet has $2 \times$ more filters. (b) Repeating examples in the same batch improves convergence and yields a slight boost in performance. + +In Appendix C, we further illustrate how MIMO exploits the capacity of the network. In particular, we study the performance of MIMO when the regularization increases (both in terms of L1 and L2 regularization), i.e. when the capacity of the network is increasingly constrained. In agreement with our hypothesis that MIMO better utilizes capacity, we observe that its performance degrades more quickly as the regularization intensifies. Moreover, the larger $M$ , the stronger the effect. Interestingly, for the L1 regularization, we can relate the performance of MIMO with the sparsity of the network, strengthening the connection to Frankle & Carbin (2018). + +# 3.5 INPUT AND BATCH REPETITION + +MIMO works well by simply adding the multi-input and multi-output configuration to an existing baseline, and varying only one additional hyperparameter (Section 3.4’s number of subnetworks $M$ ). We found two additional hyperparameters can further improve performance, and they can be important when the network capacity is limited. + +Input repetition Selecting independent examples for the multi-input configuration during training forces the subnetworks not to share any features. This is beneficial when the network has sufficient capacity, but when the network has limited excess capacity, we found that relaxing independence is beneficial. For example, ResNet50 on ImageNet (He et al., 2016) does not have sufficient capacity to support two independent subnetworks $M = 2$ ) in MIMO configuration. + +Our proposed solution is to relax independence between the inputs. Instead of independently sampling $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ from the training set during training, they share the same value with probability $\rho$ . That is, $\mathbf { x } _ { 1 }$ is sampled from the training set and $\mathbf { x } _ { 2 }$ is set to be equal to $\mathbf { x } _ { 1 }$ with probability $\rho$ or sampled from the training set with probability $1 - \rho$ . Note that this does not affect the marginal distributions of $\mathbf { x } _ { 1 }$ and $\mathbf { x } _ { 2 }$ , it merely introduces a correlation in their joint distribution. + +Figure 6a shows the performance as $\rho$ varies. At $\rho = 0$ , the subnetworks are independent but their performance is limited by the network capacity. As $\rho$ grows, the subnetworks share increasingly more features, which improves their performance. However, as $\rho$ approaches 1, the subnetworks become highly correlated and the benefit of ensembling is lost. Unlike ResNet50, Wide ResNet50 has more capacity and benefits less from input repetition (roughly $78 \text{‰}$ vs $7 4 \mathrm { - } 7 7 \%$ ). + +Batch repetition For stochastic models, most notably MC dropout and variational Bayesian neural nets, drawing multiple approximate posterior samples for each example during training can improve performance as it reduces gradient noise w.r.t. the network’s model uncertainty, e.g., Dusenberry et al. (2020a). We achieve a similar effect by repeating examples in the minibatch: this forms a new minibatch size of, e.g., $5 1 2 \cdot 5$ (batch size and number of batch repetitions respectively). Like the choice of batch size which determines the number of unique examples in the SGD step (Shallue et al., 2018), varying the number of repetitions has an implicit regularization effect. Figure 6b shows performance over the number of batch repetitions, where each batch repetition setting indicates a box plot over a sweep of 12 settings of batch size, learning rate, and ensemble size. Higher repetitions typically yield a slight boost. + +# 4 BENCHMARKS + +We described and analyzed MIMO. In this section, we compare MIMO on benchmarks building on Uncertainty Baselines.4 This framework allows us to benchmark the performance and to compare against high-quality, well-optimized implementations of baseline methods (see framework for further baselines than ones highlighted here). We looked at three model/dataset combinations: ResNet28- 10/CIFAR10, ResNet28-10/CIFAR100, and ResNet50/ImageNet. MIMO’s code is open-sourced. 5 + +Baselines Our baselines include the reference implementations of a deterministic deep neural network (trained with SGD), MC-Dropout, BatchEnsemble, and ensemble models, as well as two related models, Naive multihead and TreeNet. Thin networks use half the number of convolutional filters while wide models use double. See Appendix B for the details on the hyperparameters. + +Metrics To measure robustness, we look at accuracy, negative log-likelihood (NLL), and expected calibration error (ECE) on the IID test set as well as a corrupted test set where the test images are perturbed (e.g. added blur, compression artifacts, frost effects) (Hendrycks & Dietterich, 2019). Appendix D includes ImageNet results for 5 additional out-of-distribution datasets. To measure computational cost, we look at how long it takes to evaluate the model on a TPUv2 core, measured in ms per example. + +Table 1: ResNet28-10/CIFAR10: The best single forward pass results are highlighted in bold + +
NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic960.1590.02376.11.0500.1530.6321
Dropout95.90.1600.02468.81.2700.1660.6561
Naive mutlihead (M= 3)95.90.1610.02276.60.9690.1440.6361
MIMO (M= 3)(This work)96.40.1230.01076.60.9270.1120.6391
TreeNet (M= 3)95.90.1580.01875.50.9690.1370.9611.5
BatchEnsemble (M=4)96.20.1430.02177.51.0200.1292.5524
Thin Ensemble (M=4)96.30.1150.00877.20.8400.0890.8234
Ensemble (M=4)96.60.1140.01077.90.8100.0872.5364
+ +
NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic79.80.8750.08651.42.7000.2390.6321
Monte Carlo Dropout79.60.8300.05042.62.9000.2020.6561
Naive mutlihead (M= 3)79.50.8340.04852.12.3390.1560.6361
MIMO (M= 3)(This work)82.00.6900.02253.72.2840.1290.6391
TreeNet (M= 3)80.80.7770.04753.52.2950.1760.9611.5
BatchEnsemble (M= 4)81.50.7400.05654.12.4900.1912.5524
Thin Ensemble (M= 4)81.50.6940.01753.72.1900.1110.8234
Ensemble (M=4)82.70.6660.02154.12.2700.1382.5364
+ +Table 2: ResNet28-10/CIFAR100: The best single forward pass results are highlighted in bold. + +
NameAccuracy (↑)NLL (↓)ECE(↓)cAcc (↑)cNLL (↓)cECE(↓)Prediction time (↓)#Forward passes (↓)
Deterministic76.1000.9430.03940.5003.2000.1050.6401
Naive mutlihead (M = 3)76.6110.9290.04340.6163.2500.1220.6381
MIMO (M=2)(p=0.6)(This work)77.5000.8870.03743.3003.0300.1060.6351
TreeNet(M=2)78.1390.8520.01742.4203.0520.0730.8481.5
BatchEnsemble (M =4)76.7000.9440.04941.8003.1800.1102.5924
Ensemble (M=4)77.5000.8770.03142.1002.9900.0512.6244
WideDeterministic77.8850.9380.07245.0003.1000.1501.6741
Wide MIMO (M=2)(p= 0.6)(This work)79.3000.8430.06145.7913.0480.1471.7061
+ +Table 3: ResNet50/ImageNet: The best single forward pass results are highlighted in bold. + +The metrics show that MIMO significantly outperforms other single forward pass methods on all three benchmarks. It approaches the robustness of a Deep Ensemble, without increasing the computational costs. + +# 5 RELATED WORK + +Multi-headed networks have been previously studied by Lee et al. (2015); Osband et al. (2016); Tran et al. (2020). In this approach to efficient ensembles, the input and part of the network are shared by the members while the final few layers and the outputs are separate. The advantage of the approach is that the computational cost is reduced compared to typical ensembles, since many layers are shared, but the ensemble diversity (and resulting performance) is quite lacking. MIMO’s multi-input configuration makes a significant impact as each ensemble member may take different paths throughout the full network. Further, MIMO has lower computational cost than multi-headed approaches, since all of the layers except the first and last are shared. + +Related efficient ensemble approaches include BatchEnsemble, Rank-1 BNNs, and hyper batch ensembles (Wen et al., 2020; Dusenberry et al., 2020a; Wenzel et al., 2020). In these methods, most of the model parameters are shared among the members, which reduces the memory requirement of the model, but the evaluation cost still requires multiple forward passes. Interestingly, like MIMO, these methods also apply a multi-input multi-output configuration, treating an ensemble of networks as a single bigger network; however, MIMO still outperforms BatchEnsemble. We believe important insights such as Section 3.5’s input independence may also improve these methods. + +Finally, there are simple heuristics which also retain efficient compute such as data augmentation, temperature scaling, label smoothing, contrastive training. These methods are orthogonal to MIMO and they can provide a performance boost, without increasing the computational cost. + +# 6 CONCLUSIONS + +We propose MIMO, a novel approach for training multiple independent subnetworks within a network. We show that the subnetworks separate within the model and that they behave as independently trained neural networks. The key benefits of MIMO are its simplicity, since it does not require significant modifications to the network architecture and it has few hyperparameters, and also its computational efficiency, since it can be evaluated in a single forward pass. Our empirical results confirm that MIMO improves performance and robustness with minor changes to the number of parameters and the compute cost. + +# ACKNOWLEDGEMENTS + +Marton Havasi is funded by EPSRC. We thank Ellen Jiang for helping with writing. + +# REFERENCES + +Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks. In Proceedings of the 32nd International Conference on International Conference on Machine Learning-Volume 37, pp. 1613–1622, 2015. + +Michael W Dusenberry, Ghassen Jerfel, Yeming Wen, Yi-an Ma, Jasper Snoek, Katherine Heller, Balaji Lakshminarayanan, and Dustin Tran. Efficient and scalable Bayesian neural nets with rank-1 factors. In ICML, 2020a. + +Michael W Dusenberry, Dustin Tran, Edward Choi, Jonas Kemp, Jeremy Nixon, Ghassen Jerfel, Katherine Heller, and Andrew M Dai. Analyzing the role of model uncertainty for electronic health records. In Proceedings of the ACM Conference on Health, Inference, and Learning, pp. 204–213, 2020b. + +Angelos Filos, Sebastian Farquhar, Aidan N Gomez, Tim GJ Rudner, Zachary Kenton, Lewis Smith, Milad Alizadeh, Arnoud de Kroon, and Yarin Gal. A systematic comparison of Bayesian deep learning robustness in diabetic retinopathy tasks. arXiv preprint arXiv:1912.10481, 2019. + +Stanislav Fort, Huiyi Hu, and Balaji Lakshminarayanan. Deep ensembles: A loss landscape perspective. arXiv preprint arXiv:1912.02757, 2019. + +Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. arXiv preprint arXiv:1803.03635, 2018. + +Lars Kai Hansen and Peter Salamon. Neural network ensembles. IEEE transactions on pattern analysis and machine intelligence, 12(10):993–1001, 1990. + +Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The elements of statistical learning: data mining, inference, and prediction. Springer Science & Business Media, 2009. + +Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. + +Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In ICLR, 2019. + +Stefan Lee, Senthil Purushwalkam, Michael Cogswell, David Crandall, and Dhruv Batra. Why M heads are better than one: Training a diverse ensemble of deep networks. arXiv preprint arXiv:1511.06314, 2015. + +Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-SNE. Journal of machine learning research, 9(Nov):2579–2605, 2008. + +Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz. Pruning convolutional neural networks for resource efficient inference. arXiv preprint arXiv:1611.06440, 2016. + +Radford M. Neal. Bayesian Learning for Neural Networks. Springer-Verlag New York, Inc., 1996. + +Ian Osband, Charles Blundell, Alexander Pritzel, and Benjamin Van Roy. Deep exploration via bootstrapped DQN. In NeurIPS, 2016. + +Christopher J Shallue, Jaehoon Lee, Joseph Antognini, Jascha Sohl-Dickstein, Roy Frostig, and George E Dahl. Measuring the effects of data parallelism on neural network training. arXiv preprint arXiv:1811.03600, 2018. + +Linh Tran, Bastiaan S Veeling, Kevin Roth, Jakub Swiatkowski, Joshua V Dillon, Jasper Snoek, Stephan Mandt, Tim Salimans, Sebastian Nowozin, and Rodolphe Jenatton. Hydra: Preserving ensemble diversity for model distillation. arXiv preprint arXiv:2001.04694, 2020. + +Yeming Wen, Dustin Tran, and Jimmy Ba. BatchEnsemble: an alternative approach to efficient ensemble and lifelong learning. In International Conference on Learning Representations, 2020. + +Florian Wenzel, Jasper Snoek, Dustin Tran, and Rodolphe Jenatton. Hyperparameter ensembles for robustness and uncertainty quantification. In Neural Information Processing Systems, 2020. + +Michael Zhu and Suyog Gupta. To prune, or not to prune: exploring the efficacy of pruning for model compression. arXiv preprint arXiv:1710.01878, 2017. + +A PSEUDOCODE + +# $\overline { { \mathbf { A l g o r i t h m 1 T r a i n ( \mathbb { X } ) } } }$ + +
1:for t=1...Niter do
2:(x1:M,Y1:M) ~ U(X)
3:Pe(yi|x1:M)...Pe(ym|x1:m) ← MIMO(x1:M)
4:
5:0←0-∈VLM(0)
6: end for
+ +.  is the learning rate. + +# Algorithm 2 Evaluate(x0) + +# B HYPERPARAMETERS + +For the ResNet28-10/CIFAR models, we use a batch-size of 512, a decaying learning rate of 0.1 (decay rate 0.1) and L2 regularization 2e-4. The Deterministic, Dropout and Ensemble models are trained for 200 epochs while BatchEnsemble, Naive multihead and TreeNet are trained for 250 epochs. + +For the ResNet50/ImageNet models, we use a batch-size of 4096 and a decaying learning rate of 0.1 (decay rate 0.1) and L2 regularization 1e-4. The Deterministic, Dropout and Ensemble models are trained for 90 epochs, the BatchEnsemble model is trained for 135 epochs and Naive multihead and TreeNet are trained for 150 epochs. + +Regarding model specific hyperparameters, Dropout uses a $10 \%$ dropout rate and a single forward pass at evaluation time. Both Ensemble and BatchEnsemble models use $M = 4$ members, since this provides most of the benefits of ensembling without significantly increasing the computational costs. The TreeNet architecture + +MIMO For MIMO, we use the hyperparameters of the baseline implementations wherever possible. For the ResNet28-10/CIFAR models, we use a batch-size of 512 with decaying learning rate of 0.1 (decay rate 0.1), L2 regularization 3e-4, 250 training epochs, and a batch repetition of 4. For the ResNet50/ImageNet models, we use a batch-size of 4096 with decaying learning rate of 0.1 (decay rate 0.1), L2 regularization 1e-4, 150 training epochs, and batch repetition of 2. This makes the training cost of MIMO comparable to that of BatchEnsemble and Ensemble models. For the ResNet28-10/CIFAR experiments, we use $M = 3$ subnetworks because it performs well in both accuracy and log-likelihood. For ResNet50/ImageNet the model has lower capacity so we used $M = 2$ with $\rho = 0 . 6$ . + +# C MIMO BETTER EXPLOITS THE NETWORK CAPACITY: PERFORMANCE VERSUS REGULARIZATION STRENGTH + +In this section, we further illustrate how MIMO better exploits the capacity of the network, through the lens of its sensitivity to regularization. + +Our experimental protocol is guided by the following rationale + +• Regularization controls the capacity of the network: the higher the regularization, the more constrained the capacity. +MIMO makes better use of the capacity of the network: the more ensemble members, the more capacity is exploited. +• As a result, MIMO should be more sensitive to the constraining of the capacity of the network. And the more ensemble members (i.e., the larger $M$ ), the stronger the effect should be. + +We consider the ResNet28-10/CIFAR10 and ResNet28-10/CIFAR100 settings used in the main paper where we additionally vary the L1 (respectively L2) regularization while keeping the other L2 (respectively L1) term equal to zero. We display in Figures 7-8 the accuracy and log-likelihood over those regularization paths (averaged over three repetitions of the experiments). + +![](images/1f53db65d310420f227d0b4bea343170a48e5bd096f71a180d96b8dd18946127.jpg) +Figure 7: Accuracy and log-likelihood versus varying L1 regularization for $\mathrm { R e s N e t } 2 8 – 1 0$ on CIFAR10 (top row) and CIFAR100 (bottom row). Since MIMO better exploits the capacity of the network, its performance is more sensitive to the constraining of the capacity as the regularization increases. The larger the ensemble size, the stronger the effect. + +As previously hypothesised, we observe that MIMO is indeed more sensitive to the constraining of the capacity of the network as the regularization increases. Moreover, the larger the ensemble size, the stronger the effect. In the case of the L1 regularization, we can show how the accuracy and log-likelihood evolve with respect to the sparsity of the network6. We report those results in Figure 9 where we can observe the same phenomenon. + +# D ADDITIONAL IMAGENET OOD RESULTS + +In the following table, we evaluate trained ResNet-50 models on 7 datasets. ImageNet, ImageNet-C, ImageNet-A, and ImageNetV2 each display three metrics: negative log-likelihood, accuracy, and expected calibration error respectively. ImageNet-C further includes mCE (mean corruption error) in parentheses. ImageNet-Vid-Robust, YTTB-Robust, and ObjectNet use their own pre-defined stability metrics. + +These experiments were expensive to run, so we were only able to obtain them for a smaller set of methods. We find these results are consistent with the main text’s benchmarks, showing MIMO consistently outperforms methods not only on corrupted images, but also across distribution shifts. + +
NameImageNetImageNet-CImageNet-A
Deterministic0.939/76.2%/0.0323.21/40.5%/0.103 (75.4%)8.09/ 0.7% /0.425
MIMO (M= 2,ρ = 0.6) Wide MIMO (M=2,ρ = 0.6)0.887 / 77.5% /0.037 0.843 /79.3% /0.0613.03 /43.3% / 0.106 (71.7%) 3.1/45.0% /0.150 (69.6%)7.76 /1.4% /0.432 7.52 /3.3% /0.46
NameImageNetV2YTTB-Robust ObjectNet
Deterministic1.58 / 64.4% / 0.074ImageNet-Vid-Robust 29.9%21.7%
1.51/ 65.7% / 0.08425.9%
MIMO (M=2,𝜌= 0.6)31.8%22.2% 28.1%
Wide MIMO (M= 2,p= 0.6)1.49 / 67.9% / 0.10935.3%22.9% 29.5%
+ +Table 4: ResNet50/ImageNet & ImageNet OOD: The best single forward pass results are highlighted in bold. + +![](images/b6e50068fe1b93227b39763564b73716e6c68696c27fab24cd629b763c14da7e.jpg) +Figure 8: Accuracy and log-likelihood versus varying L2 regularization for $\mathrm { R e s N e t } 2 8 – 1 0$ on CIFAR10 (top row) and CIFAR100 (bottom row). Since MIMO better exploits the capacity of the network, its performance is more sensitive to the constraining of the capacity as the regularization increases. The larger the ensemble size, the stronger the effect. + +![](images/02f085b3dad9a2331db91e625cfda9b2552d84239b8e9f1437efb2da74a4c0b5.jpg) +Figure 9: Accuracy and log-likelihood versus varying sparsity of a $\mathrm { R e s N e t } 2 8 – 1 0$ on CIFAR10 (top row) and CIFAR100 (bottom row). Since MIMO better exploits the capacity of the network, its performance is more sensitive to the sparsification of the network (as induced by an increasing L1 regularization). The larger the ensemble size, the stronger the effect. \ No newline at end of file diff --git a/parse/train/OGg9XnKxFAH/OGg9XnKxFAH_content_list.json b/parse/train/OGg9XnKxFAH/OGg9XnKxFAH_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..43700d7500aa568c0157348a7d161aade2cc48e7 --- /dev/null +++ b/parse/train/OGg9XnKxFAH/OGg9XnKxFAH_content_list.json @@ -0,0 +1,1557 @@ +[ + { + "type": "text", + "text": "TRAINING INDEPENDENT SUBNETWORKS FOR ROBUST PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 101, + 825, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Rodolphe Jenatton Google Research rjenatton@google.com ", + "bbox": [ + 395, + 170, + 594, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Marton Havasi∗ Department of Engineering University of Cambridge mh740@cam.ac.uk ", + "bbox": [ + 183, + 170, + 364, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Stanislav Fort Stanford University sfort1@stanford.edu ", + "bbox": [ + 625, + 170, + 813, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jasper Snoek Google Research jsnoek@google.com ", + "bbox": [ + 408, + 247, + 580, + 289 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jeremiah Zhe Liu Google Research & Harvard University jereliu@google.com ", + "bbox": [ + 183, + 247, + 362, + 304 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Balaji Lakshminarayanan Google Research balajiln@google.com ", + "bbox": [ + 624, + 247, + 813, + 290 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Andrew M. Dai Google Research adai@google.com ", + "bbox": [ + 183, + 324, + 334, + 366 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dustin Tran \nGoogle Research \ntrandustin@google.com ", + "bbox": [ + 477, + 324, + 686, + 367 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 404, + 544, + 417 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent approaches to efficiently ensemble neural networks have shown that strong robustness and uncertainty performance can be achieved with a negligible gain in parameters over the original network. However, these methods still require multiple forward passes for prediction, leading to a significant computational cost. In this work, we show a surprising result: the benefits of using multiple predictions can be achieved ‘for free’ under a single model’s forward pass. In particular, we show that, using a multi-input multi-output (MIMO) configuration, one can utilize a single model’s capacity to train multiple subnetworks that independently learn the task at hand. By ensembling the predictions made by the subnetworks, we improve model robustness without increasing compute. We observe a significant improvement in negative log-likelihood, accuracy, and calibration error on CIFAR10, CIFAR100, ImageNet, and their out-of-distribution variants compared to previous methods. ", + "bbox": [ + 233, + 435, + 766, + 602 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 623, + 336, + 640 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Uncertainty estimation and out-of-distribution robustness are critical problems in machine learning. In medical applications, a confident misprediction may be a misdiagnosis that is not referred to a physician as during decision-making with a “human-in-the-loop.” This can have disastrous consequences, and the problem is particularly challenging as patient data deviates significantly from the training set such as in demographics, disease types, epidemics, and hospital locations (Dusenberry et al., 2020b; Filos et al., 2019). ", + "bbox": [ + 174, + 647, + 825, + 731 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Using a distribution over neural networks is a popular solution stemming from classic Bayesian and ensemble learning literature (Hansen & Salamon, 1990; Neal, 1996), and recent advances such as BatchEnsemble and extensions thereof achieve strong uncertainty and robustness performance (Wen et al., 2020; Dusenberry et al., 2020a; Wenzel et al., 2020). These methods demonstrate that significant gains can be had with negligible additional parameters compared to the original model. However, these methods still require multiple (typically, 4-10) forward passes for prediction, leading to a significant runtime cost. In this work, we show a surprising result: the benefits of using multiple predictions can be achieved “for free” under a single model’s forward pass. ", + "bbox": [ + 174, + 738, + 825, + 849 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The insight we build on comes from sparsity. Neural networks are heavily overparameterized models. The lottery ticket hypothesis (Frankle & Carbin, 2018) and other works on model pruning (Molchanov et al., 2016; Zhu & Gupta, 2017) show that one can prune away $70 \\%$ of the connections in a neural network without adversely affecting performance. The remaining sparse subnetwork, called the winning ticket, retains its predictive accuracy. This suggests that a neural network has sufficient capacity to fit 3-4 independent subnetworks simultaneously. We show that, using a multi-input multioutput (MIMO) configuration, we can concurrently train multiple independent subnetworks within one network. These subnetworks co-habit the network without explicit separation. The advantage of doing this is that at test time, we can evaluate all of the subnetworks at the same time, leveraging the benefits of ensembles in a single forward pass. ", + "bbox": [ + 176, + 857, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/366458d83f05825bbd16f241a272a1ee0ccb4fcd319ea87221c83bf3a6f42580.jpg", + "image_caption": [ + "Figure 1: In the multi-input multi-output (MIMO) configuration, the network takes $M = 3$ inputs and gives $M$ outputs. The hidden layers remain unchanged. The black connections are shared by all subnetworks, while the colored connections are for individual subnetworks. (a) During training, the inputs are independently sampled from the training set and the outputs are trained to classify their corresponding inputs. (b) During testing, the same input is repeated $M$ times and the outputs are averaged in an ensemble to obtain the final prediction. " + ], + "image_footnote": [], + "bbox": [ + 184, + 102, + 812, + 233 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 320, + 825, + 417 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our proposed MIMO configuration only requires two changes to a neural network architecture. First, replace the input layer: instead of taking a single datapoint as input, take $M$ datapoints as inputs, where $M$ is the desired number of ensemble members. Second, replace the output layer: instead of a single head, use $M$ heads that make $M$ predictions based on the last hidden layer. During training, the inputs are sampled independently from the training set and each of the $M$ heads is trained to predict its matching input (Figure 1a). Since, the features derived from the other inputs are not useful for predicting the matching input, the heads learn to ignore the other inputs and make their predictions independently. At test time, the same input is repeated $M$ times. That is, the heads make $M$ independent predictions on the same input, forming an ensemble for a single robust prediction that can be computed in a single forward pass (Figure 1b). ", + "bbox": [ + 173, + 424, + 825, + 564 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The core component of an ensemble’s robustness such as in Deep Ensembles is the diversity of its ensemble members (Fort et al., 2019). While it is possible that a single network makes a confident misprediction, it is less likely that multiple independently trained networks make the same mistake. Our model operates on the same principle. By realizing multiple independent winning lottery tickets, we are reducing the impact of one of them making a confident misprediction. For this method to be effective, it is essential that the subnetworks make independent predictions. We empirically show that the subnetworks use disjoint parts of the network and that the functions they represent have the same diversity as the diversity between independently trained neural networks. ", + "bbox": [ + 174, + 570, + 825, + 683 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Summary of contributions. ", + "text_level": 1, + "bbox": [ + 174, + 689, + 362, + 704 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We propose a multi-input multi-output (MIMO) configuration to network architectures, enabling multiple independent predictions in a single forward pass “for free.” Ensembling these predictions significantly improves uncertainty estimation and robustness with minor changes to the number of parameters and compute cost. \n2. We analyze the diversity of the individual members and show that they are as diverse as independently trained neural networks. \n3. We demonstrate that when adjusting for wall-clock time, MIMO networks achieve new state-ofthe-art on CIFAR10, CIFAR100, ImageNet, and their out-of-distribution variants. ", + "bbox": [ + 183, + 712, + 826, + 840 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 MULTI-INPUT MULTI-OUTPUT NETWORKS ", + "text_level": 1, + "bbox": [ + 173, + 854, + 553, + 869 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The MIMO model is applicable in a supervised classification or regression setting. Denote the set of training examples $\\mathbb { X } = \\{ ( \\pmb { x } ^ { ( n ) } , \\pmb { y } ^ { ( n ) } ) \\} _ { n = 1 } ^ { N }$ where $\\pmb { x } ^ { ( n ) }$ is the $n ^ { t h }$ datapoint with the corresponding label $\\pmb { y } ^ { ( n ) }$ and $N$ is the size of the training set. In the usual setting, for an input $_ { \\textbf { \\em x } }$ , the output of the neural network $\\hat { \\mathbf { y } }$ is a probability distribution $p _ { \\theta } ( \\hat { \\mathbf { y } } | \\boldsymbol { x } )$ ,1 which captures the uncertainty in the predictions of the network. The network parameters $\\theta$ are trained using stochastic gradient descent (SGD) to minimize the loss $L ( \\theta )$ on the training set, where the loss usually includes the negative log-likelihood and a regularization term $R$ (such as the L2 regularization): $L ( \\theta ) =$ $\\mathbb { E } ( \\mathbf { x } , \\mathbf { y } ) \\mathrm { { \\dot { e } x } } [ - \\log { \\bar { p _ { \\theta } } ( \\mathbf { y } | \\mathbf { x } ) } ] + R ( \\theta )$ . ", + "bbox": [ + 174, + 878, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/e60e924d6286d74022e9d183724ef389bf138dd336575a224d0b64380843e397.jpg", + "image_caption": [ + "Figure 2: Illustration of MIMO applied to a synthetic regression problem. (left) Example of MIMO learning $M = 3$ diverse predictors. As $M$ increases, predicting with MIMO comes with a higher bias but a smaller variance (two middle panels respectively). Despite the slight increase in bias, the decrease in variance translates into an improved generalization performance (right). " + ], + "image_footnote": [], + "bbox": [ + 179, + 102, + 818, + 200 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 318, + 825, + 388 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the MIMO configuration, the network takes $M$ inputs and returns $M$ outputs (Figure 1), where each output is a prediction for the corresponding input. This requires two small changes to the architecture. At the input layer, the $M$ inputs $\\mathbf { \\bar { \\{ x } } _ { 1 } , \\dots , \\mathbf { x } _ { M } \\}$ are concatenated before the first hidden layer is applied and at the output layer, the network gives $M$ predictive distributions $\\left\\{ p _ { \\boldsymbol \\theta } ( \\mathbf { y } _ { 1 } | \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { M } ) , \\ldots , p _ { \\boldsymbol \\theta } ( \\mathbf { y } _ { M } | \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { M } ) \\right\\}$ correspondingly. Having $M$ input and $M$ outputs require additional model parameters. The additional weights used in the MIMO configuration account for just a $0 . 0 3 \\%$ increase in the total number of parameters and $0 . 0 1 \\%$ increase in floating-point operations (FLOPs).2 ", + "bbox": [ + 173, + 393, + 825, + 507 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The network is trained similarly to a traditional neural network, with a few key modifications to account for the $M$ inputs and $M$ outputs (Figure 1a). During training, the inputs $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { M }$ are sampled independently from the training set. The loss is the sum of the negative log-likelihoods of the predictions and the regularization term: ", + "bbox": [ + 174, + 512, + 825, + 569 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6dc9ee4c5684657a26706bb528fee89a0f7b50fd1fcbf334be9dfe9533948fc4.jpg", + "text": "$$\nL _ { M } ( \\theta ) = \\underset { ( \\mathbf { x } _ { 1 } , \\mathbf { y } _ { 1 } ) \\in \\mathbb { X } } { \\mathbb { E } } \\left[ \\sum _ { m = 1 } ^ { M } - \\log p _ { \\theta } ( \\pmb { y _ { m } } | \\pmb { x } _ { 1 } , \\dots , \\pmb { x _ { M } } ) \\right] + R ( \\theta ) ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 612, + 714, + 667 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "which is optimized using stochastic gradient descent. Note that the sum of the log-likelihoods is equal to the log-likelihood of the joint distribution $\\begin{array} { r l } { \\sum _ { m = 1 } ^ { M } \\log p _ { \\boldsymbol \\theta } ( \\pmb y _ { m } | \\pmb x _ { 1 } , \\dots , \\pmb x _ { M } ) } & { = } \\end{array}$ $\\log p _ { \\boldsymbol { \\theta } } ( \\pmb { y } _ { 1 } , \\dots , \\pmb { y } _ { M } | \\pmb { x } _ { 1 } , \\dots , \\pmb { x } _ { M } )$ since the input-output pairs are independent. Hence a second interpretation of MIMO is that it is simply training a traditional neural network over $M$ -tuples of independently sampled datapoints. ", + "bbox": [ + 173, + 708, + 825, + 782 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "At evaluation time, the network is used to make a prediction on a previously unseen input $\\mathbf { x } ^ { \\prime }$ . The input $\\mathbf { x } ^ { \\prime }$ is tiled $M$ times, so $\\pmb { x } _ { 1 } = . . . = \\pmb { x } _ { M } = \\pmb { x } ^ { \\prime }$ (Figure 1b). Since all of the inputs are $\\mathbf { x } ^ { \\prime }$ , each of the outputs independently approximate the predictive distribution $p _ { \\boldsymbol \\theta } ( \\mathbf { y } _ { m } | \\mathbf { x } ^ { \\prime } , \\ldots , \\mathbf { x } ^ { \\prime } ) \\approx p ( \\mathbf { y } ^ { \\prime } | \\mathbf { x } ^ { \\prime } )$ (for $m = 1 \\ldots M )$ . As an ensembl combined output $\\begin{array} { r } { p _ { \\boldsymbol { \\theta } } ( \\mathbf { y } ^ { \\prime } | \\mathbf { x } ^ { \\prime } ) = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } p _ { \\boldsymbol { \\theta } } ( \\mathbf { y } _ { m } | \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) } \\end{array}$ e predictive performance,. ", + "bbox": [ + 173, + 787, + 825, + 864 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Unlike Bayesian methods requiring multiple weight samples, or even parameter-efficient methods like BatchEnsemble, MIMO’s advantage is that all of the ensemble members can be calculated in a single forward pass. As a result, MIMO’s wall-clock time is almost equivalent to a standard neural network. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/ae5a07d29a346ba35da6faa7d3f8fdfd07778953474a355fcadfbdd21235922b.jpg", + "image_caption": [ + "Figure 3: Accuracy landscape and function space landscape comparison of individual subnetworks for MIMO (top row) and the naive multiheaded architecture (bottom row). (left): The test accuracy in the weight space section containing $M = 3$ trained subnetworks and the origin. For the MIMO architecture, the individual subnetworks converge to three distinct low-loss basins, while naive multihead leads to the same mode. (middleleft to right): The blue, red and green panels show the disagreement between the three trained subnetworks for the same section of the weight space. For the MIMO architecture, the subnetworks often disagree, while for the naive multihead architecture they are all essentially equivalent. " + ], + "image_footnote": [], + "bbox": [ + 178, + 102, + 816, + 347 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.1 ILLUSTRATION OF MIMO ON A SYNTHETIC REGRESSION EXAMPLE ", + "text_level": 1, + "bbox": [ + 176, + 473, + 681, + 487 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Before applying MIMO to large-scale vision models, we first illustrate its behavior on a simple one-dimensional regression problem. We consider the noisy function from Blundell et al. (2015) (see Figure 2, left), with a training and test set of $N = 6 4$ and 3000 observations respectively. We train a multilayer perceptron with two hidden-layers, composed of (32, 128) units and ReLU activations. 3 ", + "bbox": [ + 174, + 494, + 825, + 551 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For different ensemble sizes $M \\in \\{ 1 , \\ldots , 5 \\}$ , we evaluate the resulting models in terms of expected mean squared error $\\mathcal { E } _ { M }$ . If we denote by $\\hat { f } _ { M }$ the regressor with $M$ ensemble members learned over $\\mathbb { X }$ , we recall that $\\mathcal { E } _ { M } = \\mathbb { E } ( \\pmb { x } ^ { \\prime } , \\pmb { y } ^ { \\prime } ) \\in \\mathbb { X } _ { \\mathrm { t e s t } } [ \\mathbb { E } \\pmb { \\mathbb { X } } [ ( \\hat { f } _ { M } ( \\pmb { x } ^ { \\prime } , \\dots , \\pmb { x } ^ { \\prime } ) - \\pmb { y } ^ { \\prime } ) ^ { 2 } ] ]$ , where $\\mathbb { E } \\mathbb { x } [ \\cdot ]$ denotes the expectation over training sets of size $N$ . We make two main observations. ", + "bbox": [ + 174, + 556, + 825, + 618 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "First, in the example of $M = 3$ in Figure 2 (left), we can see that MIMO can learn a diverse set of predictors. Second, the diverse predictors obtained by MIMO translate into improved performance, as seen in Figure 2 (right). Moreover, in the regression setting, we can decompose ${ \\mathcal { E } } _ { M }$ into its (squared) bias and variance components (Sec. 2.5 in Hastie et al. (2009)). More formally, we have ", + "bbox": [ + 173, + 625, + 826, + 681 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a3b0a1fe094346eb892c51e369ea85d25eeca19035fb8d4c22d8ce903db10ce7.jpg", + "text": "$$\n\\mathcal { E } _ { M } = \\underbrace { \\mathbb { E } } _ { ( \\mathbf { x } ^ { \\prime } , y ^ { \\prime } ) \\in \\mathrm { X } _ { \\mathrm { e x t } } } \\left[ ( \\widetilde { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) - y ^ { \\prime } ) ^ { 2 } \\right] + \\underbrace { \\mathbb { E } } _ { ( \\mathbf { x } ^ { \\prime } , y ^ { \\prime } ) \\in \\mathbb { X } _ { \\mathrm { e x t } } } \\left[ \\mathbb { E } \\left[ ( \\widetilde { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) - \\hat { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) ) ^ { 2 } \\right] \\right]\n$$", + "text_format": "latex", + "bbox": [ + 173, + 691, + 816, + 726 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\bar { f } _ { M } = \\mathbb { E } _ { \\mathbb { X } } [ \\hat { f } _ { M } ]$ . The bias-variance decomposition nicely captures the strength of MIMO. While learning a neural network over $M$ -tuples induces a slight bias compared to a standard model with $M = 1$ , i.e. the individual members perform slightly worse (Figure 2, middle-left), this is compensated by the diverse predictions of the ensemble members that lead to lower variance (Figure 2, middle-right). MIMO yields an improvement when the model has sufficient capacity to fit $M > 1$ diverse, well-performing ensemble members. ", + "bbox": [ + 173, + 752, + 826, + 838 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/a64c0f7625d2a786d35c9f22ab77d03c1917327224ba2725ff2b32d3f1daa501.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DdisagreementDKL
Naivemultihead0.0000.000
TreeNet0.0100.010
BatchEnsemble0.0140.020
MIMO0.0320.086
Deep Ensemble0.0320.086
", + "bbox": [ + 651, + 127, + 820, + 180 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/28e46c252ba3873e62fbc944c1a9ce57e97aeed69f7fcf0b4e2c68c81b914f57.jpg", + "image_caption": [ + "Figure 4: Analyzing the subnetworks on the CIFAR10 dataset. (left): Histogram of the conditional variances of the pre-activations w.r.t. each input $M = 2$ , ResNet28-10). (middle-left): Scatter plot of the conditional variances of the pre-activations w.r.t. each input. Almost all the pre-activations only have variance with respect to one of the inputs: the subnetwork they that are part of $M = 3$ , ResNet28-10). (middle-right): Training trajectories of the subnetworks. The subnetworks converge to different local optima $M = 3$ , SmallCNN). (right): Diversity of the members $( \\mathcal { D } _ { D } )$ in different efficient ensemble models (ResNet 28-10). " + ], + "image_footnote": [], + "bbox": [ + 179, + 98, + 640, + 202 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 UNDERSTANDING THE SUBNETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 295, + 526, + 310 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The mapping of each input-output pair in a MIMO configuration is referred to as a subnetwork. In this section, we show that the subnetworks converge to distinct local optima and they functionally behave as independently trained neural networks. ", + "bbox": [ + 174, + 318, + 825, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 LOSS-LANDSCAPE ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 373, + 415, + 387 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Using multiple inputs is the key to training diverse subnetworks. The subnetworks learn independently, since features derived from each input are only useful for the corresponding output. In contrast, in a naive multiheaded architecture, where the input is shared, but the model has separate $M$ outputs, the outputs rely on the same features for prediction, which leads to very low diversity. ", + "bbox": [ + 174, + 395, + 825, + 450 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To showcase this, we replicate a study from Fort et al. (2019). We look at the SmallCNN model (3 convolutional layers with 16, 32, 32 filters respectively) trained on CIFAR-10, and linearly interpolate between the three subnetworks in weight space. In the case of MIMO, we interpolate the input and output layers, since the body of the network is shared, and for the naive multiheaded model, we only interpolate the output layers, since the input and the body of the network is shared. Analogously to Deep Ensembles, the subnetworks trained using MIMO converge to disconnected modes in weight space due to differences in initialization, while in the case of the naive multiheaded model, the subnetworks end up in the same mode (Figure 3, left). Figure 3 (right) shows the disagreement i.e. the probability that the subnetworks disagree on the predicted class. MIMO’s disconnected modes yield diverse predictions, while the predictions of the naive multiheaded model are highly correlated. ", + "bbox": [ + 174, + 457, + 825, + 597 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 FUNCTION SPACE ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 609, + 408, + 623 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We visualize the training trajectories of the subnetworks in MIMO in function space (similarly to Fort et al. (2019)). As we train the SmallCNN architecture ( $M = 3$ ), we periodically save the predictions on the test set. Once training is finished, we plot the t-SNE projection (Maaten & Hinton, 2008) of the predictions. We observe that the trajectories converge to distinct local optima (Figure 4). ", + "bbox": [ + 174, + 630, + 825, + 686 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For a quantitative measurement of diversity in large scale networks, we look at the average pairwise similarity of the subnetwork’s predictions at test time, and compare against other efficient ensemble methods. The average pairwise similarity is ", + "bbox": [ + 173, + 693, + 825, + 736 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ee92d6152d59576cf3ddca017e34afa5e1a0d5423d7b202ec13919f7fd05a8aa.jpg", + "text": "$$\n\\mathcal { D } _ { D } = \\mathbb { E } \\left[ D \\left( p _ { \\theta } ( \\mathbf { y } _ { 1 } | \\mathbf { x } , \\mathbf { x } \\ldots \\mathbf { x } ) , p _ { \\theta } ( \\mathbf { y } _ { 2 } | \\mathbf { x } , \\mathbf { x } \\ldots \\ldots \\mathbf { x } ) \\right) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 741, + 678, + 757 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $D$ is a distance metric between predictive distributions and $( \\mathbf { x } , \\mathbf { y } ) \\in \\mathbb { X }$ . We consider two distance metrics. Disagreement: whether the predicted classes agree, $D _ { \\mathrm { d i s a g r e e m e n t } } ( P _ { 1 } , P _ { 2 } ) \\ =$ $\\begin{array} { r c l } { \\operatorname { I } ( \\arg \\operatorname* { m a x } _ { \\hat { y } } P _ { 1 } ( \\hat { y } ) } & { = } & { \\arg \\operatorname* { m a x } _ { \\hat { y } } P _ { 2 } ( \\hat { y } ) ) } \\end{array}$ and Kullback–Leibler divergence: $\\begin{array} { r l } { D _ { \\mathrm { K L } } ( P _ { 1 } , P _ { 2 } ) } & { { } = } \\end{array}$ $\\mathbb { E } _ { P _ { 1 } } \\left[ \\log P _ { 1 } ( y ) - \\log P _ { 2 } ( y ) \\right]$ . When the ensemble members give the same prediction at all test points, both their disagreement and KL divergence are 0. ", + "bbox": [ + 173, + 762, + 825, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first efficient ensemble approach we compare against is the aforementioned naive multiheaded model, where the input and the body of the neural network is shared by the ensemble members, but each member has its own output layer. Next, TreeNet (Lee et al., 2015), where the input and the first two residual groups are shared, but the final residual group and the output layer are trained separately for each member. Finally, BatchEnsemble (Wen et al., 2020), where the members share network parameters up to a rank-1 perturbation, which changes information flow through the full network. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/4b208d0efc4979d83270a16d261e7758938409355e703f1ad219a174f6c536d8.jpg", + "image_caption": [ + "Figure 5: The performance of the subnetworks and the ensemble of the subnetworks as the number of subnetworks $( M )$ varies. $M = 1$ is equivalent to a standard neural network (ResNet-28-10). " + ], + "image_footnote": [], + "bbox": [ + 179, + 103, + 812, + 229 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We find that the naive multiheaded model fails to induce diversity: the predictions of its subnetworks are nearly identical on all test points as shown in Figure 4 (right). TreeNet and BatchEnsemble have more diversity, although there is still significant correlation in the predictions. MIMO has better diversity than prior efficient ensemble approaches and it matches the diversity of independently trained neural networks. ", + "bbox": [ + 174, + 272, + 825, + 342 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "These results allow us to pinpoint the source of robustness: The robustness of MIMO comes from ensembling the diverse predictions made by the subnetworks, thus MIMO faithfully replicates the behaviour of a Deep Ensemble within one network. ", + "bbox": [ + 173, + 348, + 825, + 390 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 SEPARATION OF THE SUBNETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 404, + 467, + 417 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To show that the subnetworks utilize separate parts of the network, we look at the activations and measure how they react to changes in each of the $M$ inputs. Namely, we calculate the conditional variance of each pre-activation in the network with respect to each individual input. For input $\\mathbf { x } _ { 1 }$ : ", + "bbox": [ + 174, + 425, + 825, + 467 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6f60e436b57962dfafeaa0e3562ba12a7bd59ed6c22bb775eabe5b3286c6566e.jpg", + "text": "$$\n\\mathrm { 7 a r } ( a _ { i } | \\mathbf { x } _ { 2 } ) = \\underset { \\mathbf { x } _ { 2 } } { \\mathbb { E } } [ \\mathrm { V a r } ( a _ { i } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) ] \\quad ( M = 2 ) , \\ \\mathrm { a n d } \\ \\mathrm { V a r } ( a _ { i } | \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } ) = \\underset { \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } } { \\mathbb { E } } [ \\mathrm { V a r } ( a _ { i } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } ) ] \\quad ( M = 3 )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 472, + 826, + 496 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $a _ { i }$ is the value of $i$ -th pre-activation in the function of the $M$ inputs. For reference, there are 8190 pre-activations in a ResNet28-10. We can estimate the conditional variance by fixing $\\mathbf { x } _ { 1 }$ and calculating the variance of $a _ { i }$ w.r.t. $\\mathbf { x } _ { 2 } \\in \\mathbb { X }$ (when $M = 2$ , $\\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } \\in \\mathbb { X }$ when $M = 3$ ) and finally averaging over the possible values $\\mathbf { x } _ { 1 } \\in \\mathbb { X }$ . The conditional variance is analogously defined for $\\mathbf { x } _ { 2 } , \\ldots , \\mathbf { x } _ { M }$ . If the conditional variance of an activation is non-zero w.r.t. an input, that means that the activation changes as the input changes and therefore we consider it part of the subnetwork corresponding to the input. If the subnetworks are independent within the network, we expect that the conditional variance of each activation is non-zero w.r.t. one of the inputs, the subnetwork to which it belongs, and close-to zero w.r.t. all the other inputs. ", + "bbox": [ + 173, + 500, + 825, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "When we plot the conditional variances, this is exactly what we see. In Figure 4 (left), we see the histogram of the pre-activations in the network. Each point has two corresponding values: the conditional variance w.r.t. the two inputs. As we can see, all activations have non-zero conditional variance w.r.t. one of the inputs and close-to zero w.r.t. the other. Figure 4 (middle-left) shows the scatterplot of the activations for $M = 3$ . We see that, similarly to $M = 2$ , almost all activations have non-zero conditional variance w.r.t. one of the inputs and close-to zero conditional variance w.r.t. the others. Since almost all activations are part of exactly one of the subnetworks, which we can identify by calculating the conditional variances, we conclude that the subnetworks separate within the network. This implies an extension to Frankle & Carbin (2018): the subnetworks realize separate winning lottery tickets within a single network instance. ", + "bbox": [ + 173, + 631, + 825, + 771 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.4 THE OPTIMAL NUMBER OF SUBNETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 785, + 509, + 797 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A natural question that arises is the ideal number of subnetworks $M$ to fit in a network. Too few subnetworks do not fully leverage the benefits of ensembling, while having too many quickly reaches the network capacity, hurting their individual performances. Ideally, we are looking to fit as many subnetworks as possible without significantly impacting their individual performances. ", + "bbox": [ + 174, + 805, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 5 shows the performance of both the individual subnetworks and the ensemble as $M$ varies. $M = 1$ is equivalent to a traditionally trained network i.e. the performance of the subnetwork matches the performance of the ensemble since there is only one subnetwork. As $M$ grows, we can see that the performance of the subnetworks slowly declines as they utilize more and more of the network capacity. The performance of the ensemble, however, peaks between $M = 2$ and $M = 4$ , where the benefits of ensembling outweigh the slight decrease in performance of the individual subnetworks. Interestingly, the accuracy peaks earlier than the log-likelihood, which suggests that ensembling is more beneficial for the latter. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/ffa34da70b55fac06c97da7044f3b96b71a936e0c540a4edbcc3bb7ef0a7bacf.jpg", + "image_caption": [ + "Figure 6: (a) Performance of MIMO $M = 2$ ) as a function of $\\rho$ on ImageNet. At $\\rho = 0$ , the subnetworks are independent and they are limited by the network capacity. With $\\rho > 0$ , the subnetworks are able to share features and better utilize the network capacity. Wide ResNet has $2 \\times$ more filters. (b) Repeating examples in the same batch improves convergence and yields a slight boost in performance. " + ], + "image_footnote": [], + "bbox": [ + 178, + 102, + 813, + 244 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 325, + 825, + 381 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Appendix C, we further illustrate how MIMO exploits the capacity of the network. In particular, we study the performance of MIMO when the regularization increases (both in terms of L1 and L2 regularization), i.e. when the capacity of the network is increasingly constrained. In agreement with our hypothesis that MIMO better utilizes capacity, we observe that its performance degrades more quickly as the regularization intensifies. Moreover, the larger $M$ , the stronger the effect. Interestingly, for the L1 regularization, we can relate the performance of MIMO with the sparsity of the network, strengthening the connection to Frankle & Carbin (2018). ", + "bbox": [ + 174, + 387, + 825, + 486 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.5 INPUT AND BATCH REPETITION ", + "text_level": 1, + "bbox": [ + 176, + 498, + 429, + 512 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "MIMO works well by simply adding the multi-input and multi-output configuration to an existing baseline, and varying only one additional hyperparameter (Section 3.4’s number of subnetworks $M$ ). We found two additional hyperparameters can further improve performance, and they can be important when the network capacity is limited. ", + "bbox": [ + 174, + 520, + 825, + 575 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Input repetition Selecting independent examples for the multi-input configuration during training forces the subnetworks not to share any features. This is beneficial when the network has sufficient capacity, but when the network has limited excess capacity, we found that relaxing independence is beneficial. For example, ResNet50 on ImageNet (He et al., 2016) does not have sufficient capacity to support two independent subnetworks $M = 2$ ) in MIMO configuration. ", + "bbox": [ + 174, + 582, + 825, + 652 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our proposed solution is to relax independence between the inputs. Instead of independently sampling $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ from the training set during training, they share the same value with probability $\\rho$ . That is, $\\mathbf { x } _ { 1 }$ is sampled from the training set and $\\mathbf { x } _ { 2 }$ is set to be equal to $\\mathbf { x } _ { 1 }$ with probability $\\rho$ or sampled from the training set with probability $1 - \\rho$ . Note that this does not affect the marginal distributions of $\\mathbf { x } _ { 1 }$ and $\\mathbf { x } _ { 2 }$ , it merely introduces a correlation in their joint distribution. ", + "bbox": [ + 174, + 659, + 825, + 728 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 6a shows the performance as $\\rho$ varies. At $\\rho = 0$ , the subnetworks are independent but their performance is limited by the network capacity. As $\\rho$ grows, the subnetworks share increasingly more features, which improves their performance. However, as $\\rho$ approaches 1, the subnetworks become highly correlated and the benefit of ensembling is lost. Unlike ResNet50, Wide ResNet50 has more capacity and benefits less from input repetition (roughly $78 \\text{‰}$ vs $7 4 \\mathrm { - } 7 7 \\%$ ). ", + "bbox": [ + 174, + 736, + 825, + 805 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Batch repetition For stochastic models, most notably MC dropout and variational Bayesian neural nets, drawing multiple approximate posterior samples for each example during training can improve performance as it reduces gradient noise w.r.t. the network’s model uncertainty, e.g., Dusenberry et al. (2020a). We achieve a similar effect by repeating examples in the minibatch: this forms a new minibatch size of, e.g., $5 1 2 \\cdot 5$ (batch size and number of batch repetitions respectively). Like the choice of batch size which determines the number of unique examples in the SGD step (Shallue et al., 2018), varying the number of repetitions has an implicit regularization effect. Figure 6b shows performance over the number of batch repetitions, where each batch repetition setting indicates a box plot over a sweep of 12 settings of batch size, learning rate, and ensemble size. Higher repetitions typically yield a slight boost. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 BENCHMARKS ", + "text_level": 1, + "bbox": [ + 176, + 147, + 328, + 162 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We described and analyzed MIMO. In this section, we compare MIMO on benchmarks building on Uncertainty Baselines.4 This framework allows us to benchmark the performance and to compare against high-quality, well-optimized implementations of baseline methods (see framework for further baselines than ones highlighted here). We looked at three model/dataset combinations: ResNet28- 10/CIFAR10, ResNet28-10/CIFAR100, and ResNet50/ImageNet. MIMO’s code is open-sourced. 5 ", + "bbox": [ + 174, + 170, + 825, + 251 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Baselines Our baselines include the reference implementations of a deterministic deep neural network (trained with SGD), MC-Dropout, BatchEnsemble, and ensemble models, as well as two related models, Naive multihead and TreeNet. Thin networks use half the number of convolutional filters while wide models use double. See Appendix B for the details on the hyperparameters. ", + "bbox": [ + 174, + 261, + 825, + 318 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Metrics To measure robustness, we look at accuracy, negative log-likelihood (NLL), and expected calibration error (ECE) on the IID test set as well as a corrupted test set where the test images are perturbed (e.g. added blur, compression artifacts, frost effects) (Hendrycks & Dietterich, 2019). Appendix D includes ImageNet results for 5 additional out-of-distribution datasets. To measure computational cost, we look at how long it takes to evaluate the model on a TPUv2 core, measured in ms per example. ", + "bbox": [ + 173, + 324, + 825, + 409 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/bf8b3739919ab54d05a5cef77be93056281ae7ed0028380a2fcdd5171a036291.jpg", + "table_caption": [ + "Table 1: ResNet28-10/CIFAR10: The best single forward pass results are highlighted in bold " + ], + "table_footnote": [], + "table_body": "
NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic960.1590.02376.11.0500.1530.6321
Dropout95.90.1600.02468.81.2700.1660.6561
Naive mutlihead (M= 3)95.90.1610.02276.60.9690.1440.6361
MIMO (M= 3)(This work)96.40.1230.01076.60.9270.1120.6391
TreeNet (M= 3)95.90.1580.01875.50.9690.1370.9611.5
BatchEnsemble (M=4)96.20.1430.02177.51.0200.1292.5524
Thin Ensemble (M=4)96.30.1150.00877.20.8400.0890.8234
Ensemble (M=4)96.60.1140.01077.90.8100.0872.5364
", + "bbox": [ + 176, + 421, + 821, + 525 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/f4618bbae9d4fa7023a1567a2e5b780e5e0ec99a1372e1611b1a2ae1a79b6791.jpg", + "table_caption": [], + "table_footnote": [ + "Table 2: ResNet28-10/CIFAR100: The best single forward pass results are highlighted in bold. " + ], + "table_body": "
NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic79.80.8750.08651.42.7000.2390.6321
Monte Carlo Dropout79.60.8300.05042.62.9000.2020.6561
Naive mutlihead (M= 3)79.50.8340.04852.12.3390.1560.6361
MIMO (M= 3)(This work)82.00.6900.02253.72.2840.1290.6391
TreeNet (M= 3)80.80.7770.04753.52.2950.1760.9611.5
BatchEnsemble (M= 4)81.50.7400.05654.12.4900.1912.5524
Thin Ensemble (M= 4)81.50.6940.01753.72.1900.1110.8234
Ensemble (M=4)82.70.6660.02154.12.2700.1382.5364
", + "bbox": [ + 176, + 566, + 820, + 670 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/47542d8d4eb93cc9556ca9f7c4db896d76a3173bfca5d07d739ac8e34081d10e.jpg", + "table_caption": [], + "table_footnote": [ + "Table 3: ResNet50/ImageNet: The best single forward pass results are highlighted in bold. " + ], + "table_body": "
NameAccuracy (↑)NLL (↓)ECE(↓)cAcc (↑)cNLL (↓)cECE(↓)Prediction time (↓)#Forward passes (↓)
Deterministic76.1000.9430.03940.5003.2000.1050.6401
Naive mutlihead (M = 3)76.6110.9290.04340.6163.2500.1220.6381
MIMO (M=2)(p=0.6)(This work)77.5000.8870.03743.3003.0300.1060.6351
TreeNet(M=2)78.1390.8520.01742.4203.0520.0730.8481.5
BatchEnsemble (M =4)76.7000.9440.04941.8003.1800.1102.5924
Ensemble (M=4)77.5000.8770.03142.1002.9900.0512.6244
WideDeterministic77.8850.9380.07245.0003.1000.1501.6741
Wide MIMO (M=2)(p= 0.6)(This work)79.3000.8430.06145.7913.0480.1471.7061
", + "bbox": [ + 176, + 712, + 821, + 804 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The metrics show that MIMO significantly outperforms other single forward pass methods on all three benchmarks. It approaches the robustness of a Deep Ensemble, without increasing the computational costs. ", + "bbox": [ + 176, + 839, + 825, + 881 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 102, + 341, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Multi-headed networks have been previously studied by Lee et al. (2015); Osband et al. (2016); Tran et al. (2020). In this approach to efficient ensembles, the input and part of the network are shared by the members while the final few layers and the outputs are separate. The advantage of the approach is that the computational cost is reduced compared to typical ensembles, since many layers are shared, but the ensemble diversity (and resulting performance) is quite lacking. MIMO’s multi-input configuration makes a significant impact as each ensemble member may take different paths throughout the full network. Further, MIMO has lower computational cost than multi-headed approaches, since all of the layers except the first and last are shared. ", + "bbox": [ + 174, + 126, + 825, + 238 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Related efficient ensemble approaches include BatchEnsemble, Rank-1 BNNs, and hyper batch ensembles (Wen et al., 2020; Dusenberry et al., 2020a; Wenzel et al., 2020). In these methods, most of the model parameters are shared among the members, which reduces the memory requirement of the model, but the evaluation cost still requires multiple forward passes. Interestingly, like MIMO, these methods also apply a multi-input multi-output configuration, treating an ensemble of networks as a single bigger network; however, MIMO still outperforms BatchEnsemble. We believe important insights such as Section 3.5’s input independence may also improve these methods. ", + "bbox": [ + 174, + 244, + 825, + 343 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, there are simple heuristics which also retain efficient compute such as data augmentation, temperature scaling, label smoothing, contrastive training. These methods are orthogonal to MIMO and they can provide a performance boost, without increasing the computational cost. ", + "bbox": [ + 174, + 349, + 825, + 391 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 405, + 330, + 421 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose MIMO, a novel approach for training multiple independent subnetworks within a network. We show that the subnetworks separate within the model and that they behave as independently trained neural networks. The key benefits of MIMO are its simplicity, since it does not require significant modifications to the network architecture and it has few hyperparameters, and also its computational efficiency, since it can be evaluated in a single forward pass. Our empirical results confirm that MIMO improves performance and robustness with minor changes to the number of parameters and the compute cost. ", + "bbox": [ + 174, + 430, + 825, + 527 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 544, + 367, + 558 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Marton Havasi is funded by EPSRC. We thank Ellen Jiang for helping with writing. ", + "bbox": [ + 174, + 566, + 720, + 582 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 597, + 285, + 611 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks. In Proceedings of the 32nd International Conference on International Conference on Machine Learning-Volume 37, pp. 1613–1622, 2015. 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Why M heads are better than one: Training a diverse ensemble of deep networks. arXiv preprint arXiv:1511.06314, 2015. ", + "bbox": [ + 174, + 267, + 825, + 310 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-SNE. Journal of machine learning research, 9(Nov):2579–2605, 2008. ", + "bbox": [ + 173, + 319, + 821, + 349 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz. Pruning convolutional neural networks for resource efficient inference. arXiv preprint arXiv:1611.06440, 2016. ", + "bbox": [ + 173, + 357, + 820, + 387 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Radford M. Neal. Bayesian Learning for Neural Networks. Springer-Verlag New York, Inc., 1996. 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", + "bbox": [ + 173, + 636, + 823, + 665 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A PSEUDOCODE ", + "bbox": [ + 176, + 102, + 326, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "$\\overline { { \\mathbf { A l g o r i t h m 1 T r a i n ( \\mathbb { X } ) } } }$ ", + "text_level": 1, + "bbox": [ + 174, + 131, + 321, + 145 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/fa036dec41089bfa6a25f40733c9a77878e13169df927d18b45db36286063757.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
1:for t=1...Niter do
2:(x1:M,Y1:M) ~ U(X)
3:Pe(yi|x1:M)...Pe(ym|x1:m) ← MIMO(x1:M)
4:
5:0←0-∈VLM(0)
6: end for
", + "bbox": [ + 179, + 150, + 553, + 236 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": ". \u000f is the learning rate. ", + "bbox": [ + 674, + 209, + 825, + 223 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Algorithm 2 Evaluate(x0) ", + "text_level": 1, + "bbox": [ + 174, + 252, + 348, + 267 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 324, + 380, + 340 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For the ResNet28-10/CIFAR models, we use a batch-size of 512, a decaying learning rate of 0.1 (decay rate 0.1) and L2 regularization 2e-4. The Deterministic, Dropout and Ensemble models are trained for 200 epochs while BatchEnsemble, Naive multihead and TreeNet are trained for 250 epochs. ", + "bbox": [ + 173, + 348, + 825, + 405 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For the ResNet50/ImageNet models, we use a batch-size of 4096 and a decaying learning rate of 0.1 (decay rate 0.1) and L2 regularization 1e-4. The Deterministic, Dropout and Ensemble models are trained for 90 epochs, the BatchEnsemble model is trained for 135 epochs and Naive multihead and TreeNet are trained for 150 epochs. ", + "bbox": [ + 174, + 411, + 825, + 467 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Regarding model specific hyperparameters, Dropout uses a $10 \\%$ dropout rate and a single forward pass at evaluation time. Both Ensemble and BatchEnsemble models use $M = 4$ members, since this provides most of the benefits of ensembling without significantly increasing the computational costs. The TreeNet architecture ", + "bbox": [ + 173, + 474, + 825, + 530 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "MIMO For MIMO, we use the hyperparameters of the baseline implementations wherever possible. For the ResNet28-10/CIFAR models, we use a batch-size of 512 with decaying learning rate of 0.1 (decay rate 0.1), L2 regularization 3e-4, 250 training epochs, and a batch repetition of 4. For the ResNet50/ImageNet models, we use a batch-size of 4096 with decaying learning rate of 0.1 (decay rate 0.1), L2 regularization 1e-4, 150 training epochs, and batch repetition of 2. This makes the training cost of MIMO comparable to that of BatchEnsemble and Ensemble models. For the ResNet28-10/CIFAR experiments, we use $M = 3$ subnetworks because it performs well in both accuracy and log-likelihood. For ResNet50/ImageNet the model has lower capacity so we used $M = 2$ with $\\rho = 0 . 6$ . ", + "bbox": [ + 173, + 536, + 825, + 662 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "C MIMO BETTER EXPLOITS THE NETWORK CAPACITY: PERFORMANCE VERSUS REGULARIZATION STRENGTH ", + "text_level": 1, + "bbox": [ + 176, + 678, + 779, + 710 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In this section, we further illustrate how MIMO better exploits the capacity of the network, through the lens of its sensitivity to regularization. ", + "bbox": [ + 176, + 718, + 823, + 747 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Our experimental protocol is guided by the following rationale ", + "bbox": [ + 174, + 753, + 584, + 768 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "• Regularization controls the capacity of the network: the higher the regularization, the more constrained the capacity. \nMIMO makes better use of the capacity of the network: the more ensemble members, the more capacity is exploited. \n• As a result, MIMO should be more sensitive to the constraining of the capacity of the network. And the more ensemble members (i.e., the larger $M$ ), the stronger the effect should be. ", + "bbox": [ + 215, + 779, + 825, + 885 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We consider the ResNet28-10/CIFAR10 and ResNet28-10/CIFAR100 settings used in the main paper where we additionally vary the L1 (respectively L2) regularization while keeping the other L2 (respectively L1) term equal to zero. We display in Figures 7-8 the accuracy and log-likelihood over those regularization paths (averaged over three repetitions of the experiments). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/1f53db65d310420f227d0b4bea343170a48e5bd096f71a180d96b8dd18946127.jpg", + "image_caption": [ + "Figure 7: Accuracy and log-likelihood versus varying L1 regularization for $\\mathrm { R e s N e t } 2 8 – 1 0$ on CIFAR10 (top row) and CIFAR100 (bottom row). Since MIMO better exploits the capacity of the network, its performance is more sensitive to the constraining of the capacity as the regularization increases. The larger the ensemble size, the stronger the effect. " + ], + "image_footnote": [], + "bbox": [ + 183, + 103, + 812, + 388 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 462, + 820, + 491 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "As previously hypothesised, we observe that MIMO is indeed more sensitive to the constraining of the capacity of the network as the regularization increases. Moreover, the larger the ensemble size, the stronger the effect. In the case of the L1 regularization, we can show how the accuracy and log-likelihood evolve with respect to the sparsity of the network6. We report those results in Figure 9 where we can observe the same phenomenon. ", + "bbox": [ + 173, + 497, + 825, + 568 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "D ADDITIONAL IMAGENET OOD RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 582, + 549, + 598 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In the following table, we evaluate trained ResNet-50 models on 7 datasets. ImageNet, ImageNet-C, ImageNet-A, and ImageNetV2 each display three metrics: negative log-likelihood, accuracy, and expected calibration error respectively. ImageNet-C further includes mCE (mean corruption error) in parentheses. ImageNet-Vid-Robust, YTTB-Robust, and ObjectNet use their own pre-defined stability metrics. ", + "bbox": [ + 174, + 606, + 825, + 676 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "These experiments were expensive to run, so we were only able to obtain them for a smaller set of methods. We find these results are consistent with the main text’s benchmarks, showing MIMO consistently outperforms methods not only on corrupted images, but also across distribution shifts. ", + "bbox": [ + 174, + 683, + 825, + 726 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/d44ddbf5de5f95748b6a7bb2a3ef80bcbe51c492686a85668293f77ccca1dbc0.jpg", + "table_caption": [], + "table_footnote": [ + "Table 4: ResNet50/ImageNet & ImageNet OOD: The best single forward pass results are highlighted in bold. " + ], + "table_body": "
NameImageNetImageNet-CImageNet-A
Deterministic0.939/76.2%/0.0323.21/40.5%/0.103 (75.4%)8.09/ 0.7% /0.425
MIMO (M= 2,ρ = 0.6) Wide MIMO (M=2,ρ = 0.6)0.887 / 77.5% /0.037 0.843 /79.3% /0.0613.03 /43.3% / 0.106 (71.7%) 3.1/45.0% /0.150 (69.6%)7.76 /1.4% /0.432 7.52 /3.3% /0.46
NameImageNetV2YTTB-Robust ObjectNet
Deterministic1.58 / 64.4% / 0.074ImageNet-Vid-Robust 29.9%21.7%
1.51/ 65.7% / 0.08425.9%
MIMO (M=2,𝜌= 0.6)31.8%22.2% 28.1%
Wide MIMO (M= 2,p= 0.6)1.49 / 67.9% / 0.10935.3%22.9% 29.5%
", + "bbox": [ + 176, + 734, + 821, + 838 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/b6e50068fe1b93227b39763564b73716e6c68696c27fab24cd629b763c14da7e.jpg", + "image_caption": [ + "Figure 8: Accuracy and log-likelihood versus varying L2 regularization for $\\mathrm { R e s N e t } 2 8 – 1 0$ on CIFAR10 (top row) and CIFAR100 (bottom row). Since MIMO better exploits the capacity of the network, its performance is more sensitive to the constraining of the capacity as the regularization increases. The larger the ensemble size, the stronger the effect. " + ], + "image_footnote": [], + "bbox": [ + 183, + 118, + 812, + 405 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/02f085b3dad9a2331db91e625cfda9b2552d84239b8e9f1437efb2da74a4c0b5.jpg", + "image_caption": [ + "Figure 9: Accuracy and log-likelihood versus varying sparsity of a $\\mathrm { R e s N e t } 2 8 – 1 0$ on CIFAR10 (top row) and CIFAR100 (bottom row). Since MIMO better exploits the capacity of the network, its performance is more sensitive to the sparsification of the network (as induced by an increasing L1 regularization). The larger the ensemble size, the stronger the effect. 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In particular, we show that,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 142, + 412, + 470, + 423 + ], + "spans": [ + { + "bbox": [ + 142, + 412, + 470, + 423 + ], + "score": 1.0, + "content": "using a multi-input multi-output (MIMO) configuration, one can utilize a single", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 423, + 470, + 434 + ], + "spans": [ + { + "bbox": [ + 142, + 423, + 470, + 434 + ], + "score": 1.0, + "content": "model’s capacity to train multiple subnetworks that independently learn the task at", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 433, + 469, + 445 + ], + "spans": [ + { + "bbox": [ + 141, + 433, + 469, + 445 + ], + "score": 1.0, + "content": "hand. By ensembling the predictions made by the subnetworks, we improve model", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 444, + 469, + 456 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 469, + 456 + ], + "score": 1.0, + "content": "robustness without increasing compute. We observe a significant improvement in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 455, + 471, + 467 + ], + "spans": [ + { + "bbox": [ + 141, + 455, + 471, + 467 + ], + "score": 1.0, + "content": "negative log-likelihood, accuracy, and calibration error on CIFAR10, CIFAR100,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 466, + 461, + 478 + ], + "spans": [ + { + "bbox": [ + 141, + 466, + 461, + 478 + ], + "score": 1.0, + "content": "ImageNet, and their out-of-distribution variants compared to previous methods.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34.5, + "bbox_fs": [ + 141, + 344, + 471, + 478 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 494, + 206, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 208, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 208, + 510 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 507, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 507, + 528 + ], + "score": 1.0, + "content": "Uncertainty estimation and out-of-distribution robustness are critical problems in machine learning.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 525, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 536 + ], + "score": 1.0, + "content": "In medical applications, a confident misprediction may be a misdiagnosis that is not referred to a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 548 + ], + "score": 1.0, + "content": "physician as during decision-making with a “human-in-the-loop.” This can have disastrous conse-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "quences, and the problem is particularly challenging as patient data deviates significantly from the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "training set such as in demographics, disease types, epidemics, and hospital locations (Dusenberry", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 569, + 235, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 235, + 580 + ], + "score": 1.0, + "content": "et al., 2020b; Filos et al., 2019).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 512, + 507, + 580 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "Using a distribution over neural networks is a popular solution stemming from classic Bayesian", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "score": 1.0, + "content": "and ensemble learning literature (Hansen & Salamon, 1990; Neal, 1996), and recent advances such", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "as BatchEnsemble and extensions thereof achieve strong uncertainty and robustness performance", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "(Wen et al., 2020; Dusenberry et al., 2020a; Wenzel et al., 2020). These methods demonstrate that", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "significant gains can be had with negligible additional parameters compared to the original model.", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "However, these methods still require multiple (typically, 4-10) forward passes for prediction, leading", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 652, + 504, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 504, + 663 + ], + "score": 1.0, + "content": "to a significant runtime cost. 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That is, the heads make", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 107, + 425, + 118, + 434 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 424, + 505, + 438 + ], + "score": 1.0, + "content": "independent predictions on the same input, forming an ensemble for a single robust prediction", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 435, + 341, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 341, + 448 + ], + "score": 1.0, + "content": "that can be computed in a single forward pass (Figure 1b).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "The core component of an ensemble’s robustness such as in Deep Ensembles is the diversity of its", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "ensemble members (Fort et al., 2019). While it is possible that a single network makes a confident", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "misprediction, it is less likely that multiple independently trained networks make the same mistake.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "Our model operates on the same principle. By realizing multiple independent winning lottery tickets,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "we are reducing the impact of one of them making a confident misprediction. 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Denote the set of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 102, + 702, + 508, + 726 + ], + "spans": [ + { + "bbox": [ + 102, + 702, + 181, + 726 + ], + "score": 1.0, + "content": "training examples", + "type": "text" + }, + { + "bbox": [ + 182, + 707, + 277, + 720 + ], + "score": 0.92, + "content": "\\mathbb { X } = \\{ ( \\pmb { x } ^ { ( n ) } , \\pmb { y } ^ { ( n ) } ) \\} _ { n = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 702, + 306, + 726 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 307, + 707, + 325, + 718 + ], + "score": 0.9, + "content": "\\pmb { x } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 702, + 351, + 726 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 352, + 708, + 367, + 718 + ], + "score": 0.88, + "content": "n ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 702, + 508, + 726 + ], + "score": 1.0, + "content": "datapoint with the corresponding", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 718, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 129, + 734 + ], + "score": 1.0, + "content": "label", + "type": "text" + }, + { + "bbox": [ + 130, + 719, + 149, + 732 + ], + "score": 0.92, + "content": "\\pmb { y } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 718, + 169, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 721, + 180, + 730 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 718, + 448, + 734 + ], + "score": 1.0, + "content": "is the size of the training set. 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(left): The test accuracy in the weight space", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 177, + 323 + ], + "score": 1.0, + "content": "section containing", + "type": "text" + }, + { + "bbox": [ + 177, + 311, + 207, + 321 + ], + "score": 0.9, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 310, + 506, + 323 + ], + "score": 1.0, + "content": "trained subnetworks and the origin. For the MIMO architecture, the individual", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "subnetworks converge to three distinct low-loss basins, while naive multihead leads to the same mode. (middle-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "left to right): The blue, red and green panels show the disagreement between the three trained subnetworks for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "the same section of the weight space. For the MIMO architecture, the subnetworks often disagree, while for the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 351, + 334, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 334, + 363 + ], + "score": 1.0, + "content": "naive multihead architecture they are all essentially equivalent.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 417, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 418, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 418, + 388 + ], + "score": 1.0, + "content": "2.1 ILLUSTRATION OF MIMO ON A SYNTHETIC REGRESSION EXAMPLE", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "score": 1.0, + "content": "Before applying MIMO to large-scale vision models, we first illustrate its behavior on a simple", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "one-dimensional regression problem. We consider the noisy function from Blundell et al. (2015) (see", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 284, + 426 + ], + "score": 1.0, + "content": "Figure 2, left), with a training and test set of", + "type": "text" + }, + { + "bbox": [ + 284, + 414, + 317, + 424 + ], + "score": 0.9, + "content": "N = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "and 3000 observations respectively. We train a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 425, + 503, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 503, + 437 + ], + "score": 1.0, + "content": "multilayer perceptron with two hidden-layers, composed of (32, 128) units and ReLU activations. 3", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 220, + 454 + ], + "score": 1.0, + "content": "For different ensemble sizes", + "type": "text" + }, + { + "bbox": [ + 221, + 442, + 286, + 454 + ], + "score": 0.91, + "content": "M \\in \\{ 1 , \\ldots , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 440, + 506, + 454 + ], + "score": 1.0, + "content": ", we evaluate the resulting models in terms of expected", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 183, + 467 + ], + "score": 1.0, + "content": "mean squared error", + "type": "text" + }, + { + "bbox": [ + 184, + 455, + 199, + 466 + ], + "score": 0.87, + "content": "\\mathcal { E } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 454, + 266, + 467 + ], + "score": 1.0, + "content": ". If we denote by", + "type": "text" + }, + { + "bbox": [ + 267, + 453, + 281, + 466 + ], + "score": 0.88, + "content": "\\hat { f } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 454, + 353, + 467 + ], + "score": 1.0, + "content": "the regressor with", + "type": "text" + }, + { + "bbox": [ + 354, + 455, + 366, + 465 + ], + "score": 0.68, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 454, + 494, + 467 + ], + "score": 1.0, + "content": "ensemble members learned over", + "type": "text" + }, + { + "bbox": [ + 494, + 455, + 502, + 465 + ], + "score": 0.29, + "content": "\\mathbb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 454, + 506, + 467 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 162, + 482 + ], + "score": 1.0, + "content": "we recall that", + "type": "text" + }, + { + "bbox": [ + 162, + 466, + 357, + 480 + ], + "score": 0.93, + "content": "\\mathcal { E } _ { M } = \\mathbb { E } ( \\pmb { x } ^ { \\prime } , \\pmb { y } ^ { \\prime } ) \\in \\mathbb { X } _ { \\mathrm { t e s t } } [ \\mathbb { E } \\pmb { \\mathbb { X } } [ ( \\hat { f } _ { M } ( \\pmb { x } ^ { \\prime } , \\dots , \\pmb { x } ^ { \\prime } ) - \\pmb { y } ^ { \\prime } ) ^ { 2 } ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 466, + 387, + 482 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 387, + 467, + 409, + 479 + ], + "score": 0.9, + "content": "\\mathbb { E } \\mathbb { x } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "denotes the expectation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 478, + 356, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 206, + 491 + ], + "score": 1.0, + "content": "over training sets of size", + "type": "text" + }, + { + "bbox": [ + 206, + 480, + 216, + 488 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 478, + 356, + 491 + ], + "score": 1.0, + "content": ". We make two main observations.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 506, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 203, + 507 + ], + "score": 1.0, + "content": "First, in the example of", + "type": "text" + }, + { + "bbox": [ + 203, + 496, + 234, + 506 + ], + "score": 0.89, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 496, + 506, + 507 + ], + "score": 1.0, + "content": "in Figure 2 (left), we can see that MIMO can learn a diverse set of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "predictors. Second, the diverse predictors obtained by MIMO translate into improved performance, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 420, + 530 + ], + "score": 1.0, + "content": "seen in Figure 2 (right). Moreover, in the regression setting, we can decompose", + "type": "text" + }, + { + "bbox": [ + 420, + 517, + 435, + 528 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "into its (squared)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 527, + 460, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 460, + 541 + ], + "score": 1.0, + "content": "bias and variance components (Sec. 2.5 in Hastie et al. (2009)). More formally, we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 548, + 500, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 500, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 500, + 575 + ], + "score": 0.36, + "content": "\\mathcal { E } _ { M } = \\underbrace { \\mathbb { E } } _ { ( \\mathbf { x } ^ { \\prime } , y ^ { \\prime } ) \\in \\mathrm { X } _ { \\mathrm { e x t } } } \\left[ ( \\widetilde { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) - y ^ { \\prime } ) ^ { 2 } \\right] + \\underbrace { \\mathbb { E } } _ { ( \\mathbf { x } ^ { \\prime } , y ^ { \\prime } ) \\in \\mathbb { X } _ { \\mathrm { e x t } } } \\left[ \\mathbb { E } \\left[ ( \\widetilde { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) - \\hat { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) ) ^ { 2 } \\right] \\right]", + "type": "interline_equation", + "image_path": "a3b0a1fe094346eb892c51e369ea85d25eeca19035fb8d4c22d8ce903db10ce7.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 106, + 548, + 500, + 557.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 106, + 557.0, + 500, + 566.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 106, + 566.0, + 500, + 575.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 507, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 134, + 612 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 596, + 197, + 610 + ], + "score": 0.94, + "content": "\\bar { f } _ { M } = \\mathbb { E } _ { \\mathbb { X } } [ \\hat { f } _ { M } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 596, + 507, + 612 + ], + "score": 1.0, + "content": ". The bias-variance decomposition nicely captures the strength of MIMO.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 258, + 622 + ], + "score": 1.0, + "content": "While learning a neural network over", + "type": "text" + }, + { + "bbox": [ + 258, + 609, + 270, + 619 + ], + "score": 0.76, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "-tuples induces a slight bias compared to a standard model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 127, + 632 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 620, + 160, + 630 + ], + "score": 0.9, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 620, + 505, + 632 + ], + "score": 1.0, + "content": ", i.e. the individual members perform slightly worse (Figure 2, middle-left), this is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 631, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 643 + ], + "score": 1.0, + "content": "compensated by the diverse predictions of the ensemble members that lead to lower variance (Figure 2,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 474, + 654 + ], + "score": 1.0, + "content": "middle-right). MIMO yields an improvement when the model has sufficient capacity to fit", + "type": "text" + }, + { + "bbox": [ + 475, + 642, + 505, + 652 + ], + "score": 0.89, + "content": "M > 1", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 653, + 289, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 289, + 665 + ], + "score": 1.0, + "content": "diverse, well-performing ensemble members.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 679, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 677, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 118, + 677, + 505, + 692 + ], + "score": 1.0, + "content": "1We use bold font to denote random variables and italic to denote their instantiations, for example, random", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 689, + 227, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 137, + 701 + ], + "score": 1.0, + "content": "variable", + "type": "text" + }, + { + "bbox": [ + 137, + 691, + 144, + 699 + ], + "score": 0.51, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 689, + 216, + 701 + ], + "score": 1.0, + "content": "and its instantiation", + "type": "text" + }, + { + "bbox": [ + 217, + 691, + 223, + 699 + ], + "score": 0.73, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 689, + 227, + 701 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 119, + 700, + 131, + 710 + ], + "score": 0.68, + "content": "^ 2 \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "standard ResNet28-10 has 36.479M parameters and it takes 10.559G FLOPs to evaluate, while in the", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 709, + 425, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 425, + 722 + ], + "score": 1.0, + "content": "MIMO configuration, it has 36.492M parameters and takes 10.561G FLOPs to evaluate.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 719, + 379, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 379, + 734 + ], + "score": 1.0, + "content": "3We provide an interactive notebook that reproduces these experiments.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 81, + 500, + 275 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 81, + 500, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 81, + 500, + 275 + ], + "spans": [ + { + "bbox": [ + 109, + 81, + 500, + 275 + ], + "score": 0.977, + "type": "image", + "image_path": "ae5a07d29a346ba35da6faa7d3f8fdfd07778953474a355fcadfbdd21235922b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 81, + 500, + 145.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 145.66666666666669, + 500, + 210.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 210.33333333333337, + 500, + 275.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 291, + 506, + 361 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "Figure 3: Accuracy landscape and function space landscape comparison of individual subnetworks for MIMO", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "(top row) and the naive multiheaded architecture (bottom row). (left): The test accuracy in the weight space", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 177, + 323 + ], + "score": 1.0, + "content": "section containing", + "type": "text" + }, + { + "bbox": [ + 177, + 311, + 207, + 321 + ], + "score": 0.9, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 310, + 506, + 323 + ], + "score": 1.0, + "content": "trained subnetworks and the origin. For the MIMO architecture, the individual", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "subnetworks converge to three distinct low-loss basins, while naive multihead leads to the same mode. (middle-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "left to right): The blue, red and green panels show the disagreement between the three trained subnetworks for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "the same section of the weight space. For the MIMO architecture, the subnetworks often disagree, while for the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 351, + 334, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 334, + 363 + ], + "score": 1.0, + "content": "naive multihead architecture they are all essentially equivalent.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 417, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 418, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 418, + 388 + ], + "score": 1.0, + "content": "2.1 ILLUSTRATION OF MIMO ON A SYNTHETIC REGRESSION EXAMPLE", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 403 + ], + "score": 1.0, + "content": "Before applying MIMO to large-scale vision models, we first illustrate its behavior on a simple", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "one-dimensional regression problem. We consider the noisy function from Blundell et al. (2015) (see", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 284, + 426 + ], + "score": 1.0, + "content": "Figure 2, left), with a training and test set of", + "type": "text" + }, + { + "bbox": [ + 284, + 414, + 317, + 424 + ], + "score": 0.9, + "content": "N = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "and 3000 observations respectively. We train a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 425, + 503, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 503, + 437 + ], + "score": 1.0, + "content": "multilayer perceptron with two hidden-layers, composed of (32, 128) units and ReLU activations. 3", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 392, + 506, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 220, + 454 + ], + "score": 1.0, + "content": "For different ensemble sizes", + "type": "text" + }, + { + "bbox": [ + 221, + 442, + 286, + 454 + ], + "score": 0.91, + "content": "M \\in \\{ 1 , \\ldots , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 440, + 506, + 454 + ], + "score": 1.0, + "content": ", we evaluate the resulting models in terms of expected", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 183, + 467 + ], + "score": 1.0, + "content": "mean squared error", + "type": "text" + }, + { + "bbox": [ + 184, + 455, + 199, + 466 + ], + "score": 0.87, + "content": "\\mathcal { E } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 454, + 266, + 467 + ], + "score": 1.0, + "content": ". If we denote by", + "type": "text" + }, + { + "bbox": [ + 267, + 453, + 281, + 466 + ], + "score": 0.88, + "content": "\\hat { f } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 454, + 353, + 467 + ], + "score": 1.0, + "content": "the regressor with", + "type": "text" + }, + { + "bbox": [ + 354, + 455, + 366, + 465 + ], + "score": 0.68, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 454, + 494, + 467 + ], + "score": 1.0, + "content": "ensemble members learned over", + "type": "text" + }, + { + "bbox": [ + 494, + 455, + 502, + 465 + ], + "score": 0.29, + "content": "\\mathbb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 454, + 506, + 467 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 162, + 482 + ], + "score": 1.0, + "content": "we recall that", + "type": "text" + }, + { + "bbox": [ + 162, + 466, + 357, + 480 + ], + "score": 0.93, + "content": "\\mathcal { E } _ { M } = \\mathbb { E } ( \\pmb { x } ^ { \\prime } , \\pmb { y } ^ { \\prime } ) \\in \\mathbb { X } _ { \\mathrm { t e s t } } [ \\mathbb { E } \\pmb { \\mathbb { X } } [ ( \\hat { f } _ { M } ( \\pmb { x } ^ { \\prime } , \\dots , \\pmb { x } ^ { \\prime } ) - \\pmb { y } ^ { \\prime } ) ^ { 2 } ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 466, + 387, + 482 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 387, + 467, + 409, + 479 + ], + "score": 0.9, + "content": "\\mathbb { E } \\mathbb { x } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "denotes the expectation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 478, + 356, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 206, + 491 + ], + "score": 1.0, + "content": "over training sets of size", + "type": "text" + }, + { + "bbox": [ + 206, + 480, + 216, + 488 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 478, + 356, + 491 + ], + "score": 1.0, + "content": ". We make two main observations.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 440, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 506, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 203, + 507 + ], + "score": 1.0, + "content": "First, in the example of", + "type": "text" + }, + { + "bbox": [ + 203, + 496, + 234, + 506 + ], + "score": 0.89, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 496, + 506, + 507 + ], + "score": 1.0, + "content": "in Figure 2 (left), we can see that MIMO can learn a diverse set of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "predictors. Second, the diverse predictors obtained by MIMO translate into improved performance, as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 420, + 530 + ], + "score": 1.0, + "content": "seen in Figure 2 (right). Moreover, in the regression setting, we can decompose", + "type": "text" + }, + { + "bbox": [ + 420, + 517, + 435, + 528 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "into its (squared)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 527, + 460, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 460, + 541 + ], + "score": 1.0, + "content": "bias and variance components (Sec. 2.5 in Hastie et al. (2009)). More formally, we have", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 496, + 506, + 541 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 548, + 500, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 500, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 500, + 575 + ], + "score": 0.36, + "content": "\\mathcal { E } _ { M } = \\underbrace { \\mathbb { E } } _ { ( \\mathbf { x } ^ { \\prime } , y ^ { \\prime } ) \\in \\mathrm { X } _ { \\mathrm { e x t } } } \\left[ ( \\widetilde { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) - y ^ { \\prime } ) ^ { 2 } \\right] + \\underbrace { \\mathbb { E } } _ { ( \\mathbf { x } ^ { \\prime } , y ^ { \\prime } ) \\in \\mathbb { X } _ { \\mathrm { e x t } } } \\left[ \\mathbb { E } \\left[ ( \\widetilde { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) - \\hat { f } _ { M } ( \\mathbf { x } ^ { \\prime } , \\dots , \\mathbf { x } ^ { \\prime } ) ) ^ { 2 } \\right] \\right]", + "type": "interline_equation", + "image_path": "a3b0a1fe094346eb892c51e369ea85d25eeca19035fb8d4c22d8ce903db10ce7.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 106, + 548, + 500, + 557.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 106, + 557.0, + 500, + 566.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 106, + 566.0, + 500, + 575.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 507, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 134, + 612 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 596, + 197, + 610 + ], + "score": 0.94, + "content": "\\bar { f } _ { M } = \\mathbb { E } _ { \\mathbb { X } } [ \\hat { f } _ { M } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 596, + 507, + 612 + ], + "score": 1.0, + "content": ". The bias-variance decomposition nicely captures the strength of MIMO.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 258, + 622 + ], + "score": 1.0, + "content": "While learning a neural network over", + "type": "text" + }, + { + "bbox": [ + 258, + 609, + 270, + 619 + ], + "score": 0.76, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "-tuples induces a slight bias compared to a standard model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 127, + 632 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 620, + 160, + 630 + ], + "score": 0.9, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 620, + 505, + 632 + ], + "score": 1.0, + "content": ", i.e. the individual members perform slightly worse (Figure 2, middle-left), this is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 631, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 643 + ], + "score": 1.0, + "content": "compensated by the diverse predictions of the ensemble members that lead to lower variance (Figure 2,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 474, + 654 + ], + "score": 1.0, + "content": "middle-right). MIMO yields an improvement when the model has sufficient capacity to fit", + "type": "text" + }, + { + "bbox": [ + 475, + 642, + 505, + 652 + ], + "score": 0.89, + "content": "M > 1", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 653, + 289, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 289, + 665 + ], + "score": 1.0, + "content": "diverse, well-performing ensemble members.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 596, + 507, + 665 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 399, + 101, + 502, + 143 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 399, + 101, + 502, + 143 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 399, + 101, + 502, + 143 + ], + "spans": [ + { + "bbox": [ + 399, + 101, + 502, + 143 + ], + "score": 0.513, + "html": "
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(left): Histogram of the conditional variances", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 251, + 187 + ], + "score": 1.0, + "content": "of the pre-activations w.r.t. each input", + "type": "text" + }, + { + "bbox": [ + 252, + 176, + 280, + 186 + ], + "score": 0.85, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 177, + 505, + 187 + ], + "score": 1.0, + "content": ", ResNet28-10). (middle-left): Scatter plot of the conditional", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 198 + ], + "score": 1.0, + "content": "variances of the pre-activations w.r.t. each input. Almost all the pre-activations only have variance with respect", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 322, + 209 + ], + "score": 1.0, + "content": "to one of the inputs: the subnetwork they that are part of", + "type": "text" + }, + { + "bbox": [ + 322, + 196, + 352, + 206 + ], + "score": 0.87, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 195, + 506, + 209 + ], + "score": 1.0, + "content": ", ResNet28-10). (middle-right): Training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 424, + 217 + ], + "score": 1.0, + "content": "trajectories of the subnetworks. The subnetworks converge to different local optima", + "type": "text" + }, + { + "bbox": [ + 424, + 207, + 454, + 216 + ], + "score": 0.87, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 207, + 506, + 217 + ], + "score": 1.0, + "content": ", SmallCNN).", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 448, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 228, + 227 + ], + "score": 1.0, + "content": "(right): Diversity of the members", + "type": "text" + }, + { + "bbox": [ + 229, + 217, + 248, + 226 + ], + "score": 0.79, + "content": "( \\mathcal { D } _ { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 216, + 448, + 227 + ], + "score": 1.0, + "content": "in different efficient ensemble models (ResNet 28-10).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + } + ], + "index": 4.25 + }, + { + "type": "title", + "bbox": [ + 108, + 234, + 322, + 246 + ], + "lines": [ + { + "bbox": [ + 104, + 232, + 323, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 232, + 323, + 250 + ], + "score": 1.0, + "content": "3 UNDERSTANDING THE SUBNETWORKS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "The mapping of each input-output pair in a MIMO configuration is referred to as a subnetwork. In", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "this section, we show that the subnetworks converge to distinct local optima and they functionally", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 276, + 306, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 306, + 286 + ], + "score": 1.0, + "content": "behave as independently trained neural networks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 296, + 254, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 255, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 255, + 308 + ], + "score": 1.0, + "content": "3.1 LOSS-LANDSCAPE ANALYSIS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "score": 1.0, + "content": "Using multiple inputs is the key to training diverse subnetworks. The subnetworks learn independently,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "since features derived from each input are only useful for the corresponding output. In contrast, in a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 443, + 347 + ], + "score": 1.0, + "content": "naive multiheaded architecture, where the input is shared, but the model has separate", + "type": "text" + }, + { + "bbox": [ + 443, + 335, + 455, + 345 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "outputs, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 436, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 436, + 359 + ], + "score": 1.0, + "content": "outputs rely on the same features for prediction, which leads to very low diversity.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "To showcase this, we replicate a study from Fort et al. (2019). We look at the SmallCNN model (3", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "convolutional layers with 16, 32, 32 filters respectively) trained on CIFAR-10, and linearly interpolate", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "between the three subnetworks in weight space. In the case of MIMO, we interpolate the input and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "score": 1.0, + "content": "output layers, since the body of the network is shared, and for the naive multiheaded model, we only", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "interpolate the output layers, since the input and the body of the network is shared. Analogously to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "Deep Ensembles, the subnetworks trained using MIMO converge to disconnected modes in weight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "space due to differences in initialization, while in the case of the naive multiheaded model, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "subnetworks end up in the same mode (Figure 3, left). Figure 3 (right) shows the disagreement i.e.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "the probability that the subnetworks disagree on the predicted class. MIMO’s disconnected modes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "yield diverse predictions, while the predictions of the naive multiheaded model are highly correlated.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 483, + 250, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 252, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 252, + 495 + ], + "score": 1.0, + "content": "3.2 FUNCTION SPACE ANALYSIS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 499, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "We visualize the training trajectories of the subnetworks in MIMO in function space (similarly to Fort", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 324, + 522 + ], + "score": 1.0, + "content": "et al. (2019)). As we train the SmallCNN architecture (", + "type": "text" + }, + { + "bbox": [ + 324, + 511, + 355, + 522 + ], + "score": 0.86, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 511, + 505, + 522 + ], + "score": 1.0, + "content": "), we periodically save the predictions", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "on the test set. Once training is finished, we plot the t-SNE projection (Maaten & Hinton, 2008) of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 476, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 476, + 545 + ], + "score": 1.0, + "content": "the predictions. We observe that the trajectories converge to distinct local optima (Figure 4).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 562 + ], + "score": 1.0, + "content": "For a quantitative measurement of diversity in large scale networks, we look at the average pairwise", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "similarity of the subnetwork’s predictions at test time, and compare against other efficient ensemble", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 284, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 284, + 584 + ], + "score": 1.0, + "content": "methods. The average pairwise similarity is", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 587, + 415, + 600 + ], + "lines": [ + { + "bbox": [ + 194, + 587, + 415, + 600 + ], + "spans": [ + { + "bbox": [ + 194, + 587, + 415, + 600 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { D } = \\mathbb { E } \\left[ D \\left( p _ { \\theta } ( \\mathbf { y } _ { 1 } | \\mathbf { x } , \\mathbf { x } \\ldots \\mathbf { x } ) , p _ { \\theta } ( \\mathbf { y } _ { 2 } | \\mathbf { x } , \\mathbf { x } \\ldots \\ldots \\mathbf { x } ) \\right) \\right] ,", + "type": "interline_equation", + "image_path": "ee92d6152d59576cf3ddca017e34afa5e1a0d5423d7b202ec13919f7fd05a8aa.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 194, + 587, + 415, + 600 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 134, + 615 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 604, + 144, + 614 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 604, + 380, + 615 + ], + "score": 1.0, + "content": "is a distance metric between predictive distributions and", + "type": "text" + }, + { + "bbox": [ + 381, + 604, + 428, + 616 + ], + "score": 0.91, + "content": "( \\mathbf { x } , \\mathbf { y } ) \\in \\mathbb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 604, + 505, + 615 + ], + "score": 1.0, + "content": ". We consider two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 613, + 503, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 409, + 628 + ], + "score": 1.0, + "content": "distance metrics. Disagreement: whether the predicted classes agree,", + "type": "text" + }, + { + "bbox": [ + 409, + 614, + 503, + 626 + ], + "score": 0.86, + "content": "D _ { \\mathrm { d i s a g r e e m e n t } } ( P _ { 1 } , P _ { 2 } ) \\ =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 625, + 503, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 273, + 638 + ], + "score": 0.86, + "content": "\\begin{array} { r c l } { \\operatorname { I } ( \\arg \\operatorname* { m a x } _ { \\hat { y } } P _ { 1 } ( \\hat { y } ) } & { = } & { \\arg \\operatorname* { m a x } _ { \\hat { y } } P _ { 2 } ( \\hat { y } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 625, + 432, + 639 + ], + "score": 1.0, + "content": "and Kullback–Leibler divergence:", + "type": "text" + }, + { + "bbox": [ + 432, + 626, + 503, + 638 + ], + "score": 0.88, + "content": "\\begin{array} { r l } { D _ { \\mathrm { K L } } ( P _ { 1 } , P _ { 2 } ) } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 219, + 650 + ], + "score": 0.87, + "content": "\\mathbb { E } _ { P _ { 1 } } \\left[ \\log P _ { 1 } ( y ) - \\log P _ { 2 } ( y ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 637, + 506, + 651 + ], + "score": 1.0, + "content": ". When the ensemble members give the same prediction at all test", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 335, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 335, + 661 + ], + "score": 1.0, + "content": "points, both their disagreement and KL divergence are 0.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "The first efficient ensemble approach we compare against is the aforementioned naive multiheaded", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "model, where the input and the body of the neural network is shared by the ensemble members, but", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "each member has its own output layer. 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(left): Histogram of the conditional variances", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 251, + 187 + ], + "score": 1.0, + "content": "of the pre-activations w.r.t. each input", + "type": "text" + }, + { + "bbox": [ + 252, + 176, + 280, + 186 + ], + "score": 0.85, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 177, + 505, + 187 + ], + "score": 1.0, + "content": ", ResNet28-10). (middle-left): Scatter plot of the conditional", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 198 + ], + "score": 1.0, + "content": "variances of the pre-activations w.r.t. each input. Almost all the pre-activations only have variance with respect", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 322, + 209 + ], + "score": 1.0, + "content": "to one of the inputs: the subnetwork they that are part of", + "type": "text" + }, + { + "bbox": [ + 322, + 196, + 352, + 206 + ], + "score": 0.87, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 195, + 506, + 209 + ], + "score": 1.0, + "content": ", ResNet28-10). (middle-right): Training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 424, + 217 + ], + "score": 1.0, + "content": "trajectories of the subnetworks. The subnetworks converge to different local optima", + "type": "text" + }, + { + "bbox": [ + 424, + 207, + 454, + 216 + ], + "score": 0.87, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 207, + 506, + 217 + ], + "score": 1.0, + "content": ", SmallCNN).", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 448, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 228, + 227 + ], + "score": 1.0, + "content": "(right): Diversity of the members", + "type": "text" + }, + { + "bbox": [ + 229, + 217, + 248, + 226 + ], + "score": 0.79, + "content": "( \\mathcal { D } _ { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 216, + 448, + 227 + ], + "score": 1.0, + "content": "in different efficient ensemble models (ResNet 28-10).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + } + ], + "index": 4.25 + }, + { + "type": "title", + "bbox": [ + 108, + 234, + 322, + 246 + ], + "lines": [ + { + "bbox": [ + 104, + 232, + 323, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 232, + 323, + 250 + ], + "score": 1.0, + "content": "3 UNDERSTANDING THE SUBNETWORKS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "The mapping of each input-output pair in a MIMO configuration is referred to as a subnetwork. In", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "this section, we show that the subnetworks converge to distinct local optima and they functionally", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 276, + 306, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 306, + 286 + ], + "score": 1.0, + "content": "behave as independently trained neural networks.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 253, + 505, + 286 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 296, + 254, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 255, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 255, + 308 + ], + "score": 1.0, + "content": "3.1 LOSS-LANDSCAPE ANALYSIS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 507, + 326 + ], + "score": 1.0, + "content": "Using multiple inputs is the key to training diverse subnetworks. The subnetworks learn independently,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "since features derived from each input are only useful for the corresponding output. In contrast, in a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 443, + 347 + ], + "score": 1.0, + "content": "naive multiheaded architecture, where the input is shared, but the model has separate", + "type": "text" + }, + { + "bbox": [ + 443, + 335, + 455, + 345 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "outputs, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 346, + 436, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 436, + 359 + ], + "score": 1.0, + "content": "outputs rely on the same features for prediction, which leads to very low diversity.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 312, + 507, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "To showcase this, we replicate a study from Fort et al. (2019). We look at the SmallCNN model (3", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "convolutional layers with 16, 32, 32 filters respectively) trained on CIFAR-10, and linearly interpolate", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "between the three subnetworks in weight space. In the case of MIMO, we interpolate the input and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "score": 1.0, + "content": "output layers, since the body of the network is shared, and for the naive multiheaded model, we only", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "interpolate the output layers, since the input and the body of the network is shared. Analogously to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "Deep Ensembles, the subnetworks trained using MIMO converge to disconnected modes in weight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "space due to differences in initialization, while in the case of the naive multiheaded model, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "subnetworks end up in the same mode (Figure 3, left). Figure 3 (right) shows the disagreement i.e.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "the probability that the subnetworks disagree on the predicted class. MIMO’s disconnected modes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "yield diverse predictions, while the predictions of the naive multiheaded model are highly correlated.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 362, + 506, + 474 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 483, + 250, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 252, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 252, + 495 + ], + "score": 1.0, + "content": "3.2 FUNCTION SPACE ANALYSIS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 499, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "We visualize the training trajectories of the subnetworks in MIMO in function space (similarly to Fort", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 324, + 522 + ], + "score": 1.0, + "content": "et al. (2019)). As we train the SmallCNN architecture (", + "type": "text" + }, + { + "bbox": [ + 324, + 511, + 355, + 522 + ], + "score": 0.86, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 511, + 505, + 522 + ], + "score": 1.0, + "content": "), we periodically save the predictions", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "on the test set. Once training is finished, we plot the t-SNE projection (Maaten & Hinton, 2008) of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 476, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 476, + 545 + ], + "score": 1.0, + "content": "the predictions. We observe that the trajectories converge to distinct local optima (Figure 4).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 500, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 562 + ], + "score": 1.0, + "content": "For a quantitative measurement of diversity in large scale networks, we look at the average pairwise", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "similarity of the subnetwork’s predictions at test time, and compare against other efficient ensemble", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 284, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 284, + 584 + ], + "score": 1.0, + "content": "methods. The average pairwise similarity is", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 548, + 505, + 584 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 587, + 415, + 600 + ], + "lines": [ + { + "bbox": [ + 194, + 587, + 415, + 600 + ], + "spans": [ + { + "bbox": [ + 194, + 587, + 415, + 600 + ], + "score": 0.88, + "content": "\\mathcal { D } _ { D } = \\mathbb { E } \\left[ D \\left( p _ { \\theta } ( \\mathbf { y } _ { 1 } | \\mathbf { x } , \\mathbf { x } \\ldots \\mathbf { x } ) , p _ { \\theta } ( \\mathbf { y } _ { 2 } | \\mathbf { x } , \\mathbf { x } \\ldots \\ldots \\mathbf { x } ) \\right) \\right] ,", + "type": "interline_equation", + "image_path": "ee92d6152d59576cf3ddca017e34afa5e1a0d5423d7b202ec13919f7fd05a8aa.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 194, + 587, + 415, + 600 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 134, + 615 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 604, + 144, + 614 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 604, + 380, + 615 + ], + "score": 1.0, + "content": "is a distance metric between predictive distributions and", + "type": "text" + }, + { + "bbox": [ + 381, + 604, + 428, + 616 + ], + "score": 0.91, + "content": "( \\mathbf { x } , \\mathbf { y } ) \\in \\mathbb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 604, + 505, + 615 + ], + "score": 1.0, + "content": ". We consider two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 613, + 503, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 409, + 628 + ], + "score": 1.0, + "content": "distance metrics. Disagreement: whether the predicted classes agree,", + "type": "text" + }, + { + "bbox": [ + 409, + 614, + 503, + 626 + ], + "score": 0.86, + "content": "D _ { \\mathrm { d i s a g r e e m e n t } } ( P _ { 1 } , P _ { 2 } ) \\ =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 625, + 503, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 273, + 638 + ], + "score": 0.86, + "content": "\\begin{array} { r c l } { \\operatorname { I } ( \\arg \\operatorname* { m a x } _ { \\hat { y } } P _ { 1 } ( \\hat { y } ) } & { = } & { \\arg \\operatorname* { m a x } _ { \\hat { y } } P _ { 2 } ( \\hat { y } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 625, + 432, + 639 + ], + "score": 1.0, + "content": "and Kullback–Leibler divergence:", + "type": "text" + }, + { + "bbox": [ + 432, + 626, + 503, + 638 + ], + "score": 0.88, + "content": "\\begin{array} { r l } { D _ { \\mathrm { K L } } ( P _ { 1 } , P _ { 2 } ) } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 219, + 650 + ], + "score": 0.87, + "content": "\\mathbb { E } _ { P _ { 1 } } \\left[ \\log P _ { 1 } ( y ) - \\log P _ { 2 } ( y ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 637, + 506, + 651 + ], + "score": 1.0, + "content": ". When the ensemble members give the same prediction at all test", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 335, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 335, + 661 + ], + "score": 1.0, + "content": "points, both their disagreement and KL divergence are 0.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 604, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "The first efficient ensemble approach we compare against is the aforementioned naive multiheaded", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "model, where the input and the body of the neural network is shared by the ensemble members, but", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "each member has its own output layer. Next, TreeNet (Lee et al., 2015), where the input and the first", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "two residual groups are shared, but the final residual group and the output layer are trained separately", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "for each member. Finally, BatchEnsemble (Wen et al., 2020), where the members share network", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 497, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 497, + 733 + ], + "score": 1.0, + "content": "parameters up to a rank-1 perturbation, which changes information flow through the full network.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 82, + 497, + 182 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 82, + 497, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 82, + 497, + 182 + ], + "spans": [ + { + "bbox": [ + 110, + 82, + 497, + 182 + ], + "score": 0.96, + "type": "image", + "image_path": "4b208d0efc4979d83270a16d261e7758938409355e703f1ad219a174f6c536d8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 82, + 497, + 115.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 115.33333333333334, + 497, + 148.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 148.66666666666669, + 497, + 182.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 187, + 504, + 208 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "Figure 5: The performance of the subnetworks and the ensemble of the subnetworks as the number of subnetworks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 108, + 197, + 393, + 208 + ], + "spans": [ + { + "bbox": [ + 108, + 198, + 123, + 207 + ], + "score": 0.62, + "content": "( M )", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 197, + 150, + 208 + ], + "score": 1.0, + "content": "varies.", + "type": "text" + }, + { + "bbox": [ + 151, + 198, + 178, + 207 + ], + "score": 0.9, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 197, + 393, + 208 + ], + "score": 1.0, + "content": "is equivalent to a standard neural network (ResNet-28-10).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "We find that the naive multiheaded model fails to induce diversity: the predictions of its subnetworks", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "score": 1.0, + "content": "are nearly identical on all test points as shown in Figure 4 (right). TreeNet and BatchEnsemble have", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "more diversity, although there is still significant correlation in the predictions. MIMO has better", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "diversity than prior efficient ensemble approaches and it matches the diversity of independently", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 260, + 205, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 205, + 272 + ], + "score": 1.0, + "content": "trained neural networks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "These results allow us to pinpoint the source of robustness: The robustness of MIMO comes from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "ensembling the diverse predictions made by the subnetworks, thus MIMO faithfully replicates the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 314, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 314, + 311 + ], + "score": 1.0, + "content": "behaviour of a Deep Ensemble within one network.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 320, + 286, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 288, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 288, + 333 + ], + "score": 1.0, + "content": "3.3 SEPARATION OF THE SUBNETWORKS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "To show that the subnetworks utilize separate parts of the network, we look at the activations and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 308, + 360 + ], + "score": 1.0, + "content": "measure how they react to changes in each of the", + "type": "text" + }, + { + "bbox": [ + 308, + 348, + 320, + 358 + ], + "score": 0.73, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "inputs. Namely, we calculate the conditional", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 358, + 497, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 480, + 372 + ], + "score": 1.0, + "content": "variance of each pre-activation in the network with respect to each individual input. For input", + "type": "text" + }, + { + "bbox": [ + 481, + 361, + 492, + 370 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 358, + 497, + 372 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 374, + 506, + 393 + ], + "lines": [ + { + "bbox": [ + 111, + 374, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 111, + 374, + 506, + 393 + ], + "score": 0.84, + "content": "\\mathrm { 7 a r } ( a _ { i } | \\mathbf { x } _ { 2 } ) = \\underset { \\mathbf { x } _ { 2 } } { \\mathbb { E } } [ \\mathrm { V a r } ( a _ { i } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) ] \\quad ( M = 2 ) , \\ \\mathrm { a n d } \\ \\mathrm { V a r } ( a _ { i } | \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } ) = \\underset { \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } } { \\mathbb { E } } [ \\mathrm { V a r } ( a _ { i } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } ) ] \\quad ( M = 3 )", + "type": "interline_equation", + "image_path": "6f60e436b57962dfafeaa0e3562ba12a7bd59ed6c22bb775eabe5b3286c6566e.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 111, + 374, + 506, + 393 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 133, + 409 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 398, + 144, + 407 + ], + "score": 0.85, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 396, + 207, + 409 + ], + "score": 1.0, + "content": "is the value of", + "type": "text" + }, + { + "bbox": [ + 208, + 397, + 212, + 406 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 396, + 375, + 409 + ], + "score": 1.0, + "content": "-th pre-activation in the function of the", + "type": "text" + }, + { + "bbox": [ + 375, + 397, + 387, + 406 + ], + "score": 0.83, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "inputs. For reference, there", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 406, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 492, + 420 + ], + "score": 1.0, + "content": "are 8190 pre-activations in a ResNet28-10. We can estimate the conditional variance by fixing", + "type": "text" + }, + { + "bbox": [ + 493, + 409, + 504, + 418 + ], + "score": 0.8, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 418, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 236, + 431 + ], + "score": 1.0, + "content": "and calculating the variance of", + "type": "text" + }, + { + "bbox": [ + 236, + 419, + 246, + 429 + ], + "score": 0.86, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 418, + 272, + 431 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + }, + { + "bbox": [ + 273, + 418, + 307, + 429 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 2 } \\in \\mathbb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 418, + 337, + 431 + ], + "score": 1.0, + "content": "(when", + "type": "text" + }, + { + "bbox": [ + 337, + 418, + 370, + 429 + ], + "score": 0.81, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 418, + 374, + 431 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 374, + 418, + 423, + 429 + ], + "score": 0.83, + "content": "\\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } \\in \\mathbb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 418, + 450, + 431 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 450, + 418, + 483, + 429 + ], + "score": 0.89, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 418, + 506, + 431 + ], + "score": 1.0, + "content": ") and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 274, + 442 + ], + "score": 1.0, + "content": "finally averaging over the possible values", + "type": "text" + }, + { + "bbox": [ + 275, + 430, + 306, + 440 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 1 } \\in \\mathbb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 428, + 506, + 442 + ], + "score": 1.0, + "content": ". The conditional variance is analogously defined", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 120, + 453 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 442, + 170, + 451 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 2 } , \\ldots , \\mathbf { x } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 440, + 506, + 453 + ], + "score": 1.0, + "content": ". If the conditional variance of an activation is non-zero w.r.t. an input, that means", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "that the activation changes as the input changes and therefore we consider it part of the subnetwork", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "corresponding to the input. If the subnetworks are independent within the network, we expect that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "conditional variance of each activation is non-zero w.r.t. one of the inputs, the subnetwork to which it", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "score": 1.0, + "content": "belongs, and close-to zero w.r.t. all the other inputs.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "When we plot the conditional variances, this is exactly what we see. In Figure 4 (left), we see", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "the histogram of the pre-activations in the network. Each point has two corresponding values: the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "conditional variance w.r.t. the two inputs. As we can see, all activations have non-zero conditional", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "variance w.r.t. one of the inputs and close-to zero w.r.t. the other. Figure 4 (middle-left) shows the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 233, + 556 + ], + "score": 1.0, + "content": "scatterplot of the activations for", + "type": "text" + }, + { + "bbox": [ + 234, + 545, + 263, + 555 + ], + "score": 0.89, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 545, + 365, + 556 + ], + "score": 1.0, + "content": ". We see that, similarly to", + "type": "text" + }, + { + "bbox": [ + 365, + 545, + 395, + 555 + ], + "score": 0.89, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 545, + 505, + 556 + ], + "score": 1.0, + "content": ", almost all activations have", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "non-zero conditional variance w.r.t. one of the inputs and close-to zero conditional variance w.r.t.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "the others. Since almost all activations are part of exactly one of the subnetworks, which we can", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "identify by calculating the conditional variances, we conclude that the subnetworks separate within", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "the network. This implies an extension to Frankle & Carbin (2018): the subnetworks realize separate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 600, + 332, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 332, + 612 + ], + "score": 1.0, + "content": "winning lottery tickets within a single network instance.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 622, + 312, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 313, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 313, + 633 + ], + "score": 1.0, + "content": "3.4 THE OPTIMAL NUMBER OF SUBNETWORKS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 375, + 650 + ], + "score": 1.0, + "content": "A natural question that arises is the ideal number of subnetworks", + "type": "text" + }, + { + "bbox": [ + 376, + 639, + 388, + 648 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "to fit in a network. Too few", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "subnetworks do not fully leverage the benefits of ensembling, while having too many quickly reaches", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "the network capacity, hurting their individual performances. Ideally, we are looking to fit as many", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 672, + 455, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 455, + 683 + ], + "score": 1.0, + "content": "subnetworks as possible without significantly impacting their individual performances.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 464, + 699 + ], + "score": 1.0, + "content": "Figure 5 shows the performance of both the individual subnetworks and the ensemble as", + "type": "text" + }, + { + "bbox": [ + 465, + 688, + 477, + 698 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "varies.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 137, + 709 + ], + "score": 0.89, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "is equivalent to a traditionally trained network i.e. the performance of the subnetwork matches", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 399, + 722 + ], + "score": 1.0, + "content": "the performance of the ensemble since there is only one subnetwork. As", + "type": "text" + }, + { + "bbox": [ + 399, + 710, + 411, + 720 + ], + "score": 0.74, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "grows, we can see that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "the performance of the subnetworks slowly declines as they utilize more and more of the network", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 82, + 497, + 182 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 82, + 497, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 82, + 497, + 182 + ], + "spans": [ + { + "bbox": [ + 110, + 82, + 497, + 182 + ], + "score": 0.96, + "type": "image", + "image_path": "4b208d0efc4979d83270a16d261e7758938409355e703f1ad219a174f6c536d8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 82, + 497, + 115.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 115.33333333333334, + 497, + 148.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 148.66666666666669, + 497, + 182.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 187, + 504, + 208 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "Figure 5: The performance of the subnetworks and the ensemble of the subnetworks as the number of subnetworks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 108, + 197, + 393, + 208 + ], + "spans": [ + { + "bbox": [ + 108, + 198, + 123, + 207 + ], + "score": 0.62, + "content": "( M )", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 197, + 150, + 208 + ], + "score": 1.0, + "content": "varies.", + "type": "text" + }, + { + "bbox": [ + 151, + 198, + 178, + 207 + ], + "score": 0.9, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 197, + 393, + 208 + ], + "score": 1.0, + "content": "is equivalent to a standard neural network (ResNet-28-10).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "We find that the naive multiheaded model fails to induce diversity: the predictions of its subnetworks", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 239 + ], + "score": 1.0, + "content": "are nearly identical on all test points as shown in Figure 4 (right). TreeNet and BatchEnsemble have", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "more diversity, although there is still significant correlation in the predictions. MIMO has better", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "diversity than prior efficient ensemble approaches and it matches the diversity of independently", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 260, + 205, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 205, + 272 + ], + "score": 1.0, + "content": "trained neural networks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 216, + 506, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "These results allow us to pinpoint the source of robustness: The robustness of MIMO comes from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "ensembling the diverse predictions made by the subnetworks, thus MIMO faithfully replicates the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 314, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 314, + 311 + ], + "score": 1.0, + "content": "behaviour of a Deep Ensemble within one network.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 276, + 505, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 320, + 286, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 288, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 288, + 333 + ], + "score": 1.0, + "content": "3.3 SEPARATION OF THE SUBNETWORKS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "To show that the subnetworks utilize separate parts of the network, we look at the activations and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 308, + 360 + ], + "score": 1.0, + "content": "measure how they react to changes in each of the", + "type": "text" + }, + { + "bbox": [ + 308, + 348, + 320, + 358 + ], + "score": 0.73, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "inputs. Namely, we calculate the conditional", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 358, + 497, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 480, + 372 + ], + "score": 1.0, + "content": "variance of each pre-activation in the network with respect to each individual input. For input", + "type": "text" + }, + { + "bbox": [ + 481, + 361, + 492, + 370 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 358, + 497, + 372 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 337, + 505, + 372 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 374, + 506, + 393 + ], + "lines": [ + { + "bbox": [ + 111, + 374, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 111, + 374, + 506, + 393 + ], + "score": 0.84, + "content": "\\mathrm { 7 a r } ( a _ { i } | \\mathbf { x } _ { 2 } ) = \\underset { \\mathbf { x } _ { 2 } } { \\mathbb { E } } [ \\mathrm { V a r } ( a _ { i } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } ) ] \\quad ( M = 2 ) , \\ \\mathrm { a n d } \\ \\mathrm { V a r } ( a _ { i } | \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } ) = \\underset { \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } } { \\mathbb { E } } [ \\mathrm { V a r } ( a _ { i } ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\mathbf { x } _ { 3 } ) ] \\quad ( M = 3 )", + "type": "interline_equation", + "image_path": "6f60e436b57962dfafeaa0e3562ba12a7bd59ed6c22bb775eabe5b3286c6566e.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 111, + 374, + 506, + 393 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 133, + 409 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 398, + 144, + 407 + ], + "score": 0.85, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 396, + 207, + 409 + ], + "score": 1.0, + "content": "is the value of", + "type": "text" + }, + { + "bbox": [ + 208, + 397, + 212, + 406 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 396, + 375, + 409 + ], + "score": 1.0, + "content": "-th pre-activation in the function of the", + "type": "text" + }, + { + "bbox": [ + 375, + 397, + 387, + 406 + ], + "score": 0.83, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "inputs. For reference, there", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 406, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 492, + 420 + ], + "score": 1.0, + "content": "are 8190 pre-activations in a ResNet28-10. 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The conditional variance is analogously defined", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 120, + 453 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 442, + 170, + 451 + ], + "score": 0.9, + "content": "\\mathbf { x } _ { 2 } , \\ldots , \\mathbf { x } _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 440, + 506, + 453 + ], + "score": 1.0, + "content": ". If the conditional variance of an activation is non-zero w.r.t. an input, that means", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "that the activation changes as the input changes and therefore we consider it part of the subnetwork", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "corresponding to the input. If the subnetworks are independent within the network, we expect that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "conditional variance of each activation is non-zero w.r.t. one of the inputs, the subnetwork to which it", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "score": 1.0, + "content": "belongs, and close-to zero w.r.t. all the other inputs.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 396, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "When we plot the conditional variances, this is exactly what we see. In Figure 4 (left), we see", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "the histogram of the pre-activations in the network. Each point has two corresponding values: the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "conditional variance w.r.t. the two inputs. As we can see, all activations have non-zero conditional", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "variance w.r.t. one of the inputs and close-to zero w.r.t. the other. Figure 4 (middle-left) shows the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 233, + 556 + ], + "score": 1.0, + "content": "scatterplot of the activations for", + "type": "text" + }, + { + "bbox": [ + 234, + 545, + 263, + 555 + ], + "score": 0.89, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 545, + 365, + 556 + ], + "score": 1.0, + "content": ". We see that, similarly to", + "type": "text" + }, + { + "bbox": [ + 365, + 545, + 395, + 555 + ], + "score": 0.89, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 545, + 505, + 556 + ], + "score": 1.0, + "content": ", almost all activations have", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "non-zero conditional variance w.r.t. one of the inputs and close-to zero conditional variance w.r.t.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "the others. Since almost all activations are part of exactly one of the subnetworks, which we can", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "identify by calculating the conditional variances, we conclude that the subnetworks separate within", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "the network. This implies an extension to Frankle & Carbin (2018): the subnetworks realize separate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 600, + 332, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 332, + 612 + ], + "score": 1.0, + "content": "winning lottery tickets within a single network instance.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 500, + 506, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 622, + 312, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 313, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 313, + 633 + ], + "score": 1.0, + "content": "3.4 THE OPTIMAL NUMBER OF SUBNETWORKS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 375, + 650 + ], + "score": 1.0, + "content": "A natural question that arises is the ideal number of subnetworks", + "type": "text" + }, + { + "bbox": [ + 376, + 639, + 388, + 648 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "to fit in a network. Too few", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "subnetworks do not fully leverage the benefits of ensembling, while having too many quickly reaches", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "the network capacity, hurting their individual performances. Ideally, we are looking to fit as many", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 672, + 455, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 455, + 683 + ], + "score": 1.0, + "content": "subnetworks as possible without significantly impacting their individual performances.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 638, + 505, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 464, + 699 + ], + "score": 1.0, + "content": "Figure 5 shows the performance of both the individual subnetworks and the ensemble as", + "type": "text" + }, + { + "bbox": [ + 465, + 688, + 477, + 698 + ], + "score": 0.77, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "varies.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 137, + 709 + ], + "score": 0.89, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "is equivalent to a traditionally trained network i.e. the performance of the subnetwork matches", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 399, + 722 + ], + "score": 1.0, + "content": "the performance of the ensemble since there is only one subnetwork. As", + "type": "text" + }, + { + "bbox": [ + 399, + 710, + 411, + 720 + ], + "score": 0.74, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "grows, we can see that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "the performance of the subnetworks slowly declines as they utilize more and more of the network", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 381, + 270 + ], + "score": 1.0, + "content": "capacity. The performance of the ensemble, however, peaks between", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 381, + 258, + 412, + 268 + ], + "score": 0.9, + "content": "M = 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 412, + 258, + 429, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 430, + 258, + 460, + 268 + ], + "score": 0.89, + "content": "M = 4", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 460, + 258, + 505, + 270 + ], + "score": 1.0, + "content": ", where the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "benefits of ensembling outweigh the slight decrease in performance of the individual subnetworks.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "Interestingly, the accuracy peaks earlier than the log-likelihood, which suggests that ensembling is", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 224, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 224, + 302 + ], + "score": 1.0, + "content": "more beneficial for the latter.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 81, + 498, + 194 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 81, + 498, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 81, + 498, + 194 + ], + "spans": [ + { + "bbox": [ + 109, + 81, + 498, + 194 + ], + "score": 0.962, + "type": "image", + "image_path": "ffa34da70b55fac06c97da7044f3b96b71a936e0c540a4edbcc3bb7ef0a7bacf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 81, + 498, + 118.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 118.66666666666666, + 498, + 156.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 156.33333333333331, + 498, + 193.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 198, + 505, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 247, + 210 + ], + "score": 1.0, + "content": "Figure 6: (a) Performance of MIMO", + "type": "text" + }, + { + "bbox": [ + 248, + 198, + 278, + 208 + ], + "score": 0.84, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 198, + 341, + 210 + ], + "score": 1.0, + "content": ") as a function of", + "type": "text" + }, + { + "bbox": [ + 342, + 200, + 348, + 209 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 198, + 414, + 210 + ], + "score": 1.0, + "content": "on ImageNet. At", + "type": "text" + }, + { + "bbox": [ + 415, + 199, + 439, + 209 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 198, + 505, + 210 + ], + "score": 1.0, + "content": ", the subnetworks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 354, + 220 + ], + "score": 1.0, + "content": "are independent and they are limited by the network capacity. With", + "type": "text" + }, + { + "bbox": [ + 354, + 209, + 377, + 219 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 208, + 505, + 220 + ], + "score": 1.0, + "content": ", the subnetworks are able to share", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 338, + 230 + ], + "score": 1.0, + "content": "features and better utilize the network capacity. Wide ResNet has", + "type": "text" + }, + { + "bbox": [ + 338, + 219, + 351, + 228 + ], + "score": 0.86, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "more filters. (b) Repeating examples in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 227, + 379, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 379, + 241 + ], + "score": 1.0, + "content": "same batch improves convergence and yields a slight boost in performance.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 381, + 270 + ], + "score": 1.0, + "content": "capacity. The performance of the ensemble, however, peaks between", + "type": "text" + }, + { + "bbox": [ + 381, + 258, + 412, + 268 + ], + "score": 0.9, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 258, + 429, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 430, + 258, + 460, + 268 + ], + "score": 0.89, + "content": "M = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 258, + 505, + 270 + ], + "score": 1.0, + "content": ", where the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "benefits of ensembling outweigh the slight decrease in performance of the individual subnetworks.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 293 + ], + "score": 1.0, + "content": "Interestingly, the accuracy peaks earlier than the log-likelihood, which suggests that ensembling is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 224, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 224, + 302 + ], + "score": 1.0, + "content": "more beneficial for the latter.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 506, + 320 + ], + "score": 1.0, + "content": "In Appendix C, we further illustrate how MIMO exploits the capacity of the network. In particular,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "we study the performance of MIMO when the regularization increases (both in terms of L1 and L2", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "regularization), i.e. when the capacity of the network is increasingly constrained. In agreement with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "our hypothesis that MIMO better utilizes capacity, we observe that its performance degrades more", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 350, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 347, + 365 + ], + "score": 1.0, + "content": "quickly as the regularization intensifies. Moreover, the larger", + "type": "text" + }, + { + "bbox": [ + 347, + 352, + 358, + 362 + ], + "score": 0.69, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 350, + 506, + 365 + ], + "score": 1.0, + "content": ", the stronger the effect. Interestingly,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "for the L1 regularization, we can relate the performance of MIMO with the sparsity of the network,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 374, + 338, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 338, + 385 + ], + "score": 1.0, + "content": "strengthening the connection to Frankle & Carbin (2018).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 395, + 263, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 265, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 265, + 407 + ], + "score": 1.0, + "content": "3.5 INPUT AND BATCH REPETITION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "MIMO works well by simply adding the multi-input and multi-output configuration to an existing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "baseline, and varying only one additional hyperparameter (Section 3.4’s number of subnetworks", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 107, + 434, + 119, + 444 + ], + "score": 0.59, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "). We found two additional hyperparameters can further improve performance, and they can be", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 445, + 299, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 299, + 457 + ], + "score": 1.0, + "content": "important when the network capacity is limited.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Input repetition Selecting independent examples for the multi-input configuration during training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "forces the subnetworks not to share any features. This is beneficial when the network has sufficient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "capacity, but when the network has limited excess capacity, we found that relaxing independence is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 508 + ], + "score": 1.0, + "content": "beneficial. For example, ResNet50 on ImageNet (He et al., 2016) does not have sufficient capacity to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 397, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 263, + 518 + ], + "score": 1.0, + "content": "support two independent subnetworks", + "type": "text" + }, + { + "bbox": [ + 263, + 506, + 294, + 516 + ], + "score": 0.86, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 505, + 397, + 518 + ], + "score": 1.0, + "content": ") in MIMO configuration.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "Our proposed solution is to relax independence between the inputs. 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As", + "type": "text" + }, + { + "bbox": [ + 308, + 596, + 315, + 606 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "grows, the subnetworks share increasingly more", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 338, + 618 + ], + "score": 1.0, + "content": "features, which improves their performance. However, as", + "type": "text" + }, + { + "bbox": [ + 338, + 607, + 345, + 616 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "approaches 1, the subnetworks become", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "highly correlated and the benefit of ensembling is lost. 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Like the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "choice of batch size which determines the number of unique examples in the SGD step (Shallue", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "et al., 2018), varying the number of repetitions has an implicit regularization effect. Figure 6b shows", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "performance over the number of batch repetitions, where each batch repetition setting indicates a box", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 291, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 81, + 498, + 194 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 81, + 498, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 81, + 498, + 194 + ], + "spans": [ + { + "bbox": [ + 109, + 81, + 498, + 194 + ], + "score": 0.962, + "type": "image", + "image_path": "ffa34da70b55fac06c97da7044f3b96b71a936e0c540a4edbcc3bb7ef0a7bacf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 81, + 498, + 118.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 118.66666666666666, + 498, + 156.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 156.33333333333331, + 498, + 193.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 198, + 505, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 247, + 210 + ], + "score": 1.0, + "content": "Figure 6: (a) Performance of MIMO", + "type": "text" + }, + { + "bbox": [ + 248, + 198, + 278, + 208 + ], + "score": 0.84, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 198, + 341, + 210 + ], + "score": 1.0, + "content": ") as a function of", + "type": "text" + }, + { + "bbox": [ + 342, + 200, + 348, + 209 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 198, + 414, + 210 + ], + "score": 1.0, + "content": "on ImageNet. At", + "type": "text" + }, + { + "bbox": [ + 415, + 199, + 439, + 209 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 198, + 505, + 210 + ], + "score": 1.0, + "content": ", the subnetworks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 354, + 220 + ], + "score": 1.0, + "content": "are independent and they are limited by the network capacity. With", + "type": "text" + }, + { + "bbox": [ + 354, + 209, + 377, + 219 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 208, + 505, + 220 + ], + "score": 1.0, + "content": ", the subnetworks are able to share", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 338, + 230 + ], + "score": 1.0, + "content": "features and better utilize the network capacity. Wide ResNet has", + "type": "text" + }, + { + "bbox": [ + 338, + 219, + 351, + 228 + ], + "score": 0.86, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "more filters. (b) Repeating examples in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 227, + 379, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 379, + 241 + ], + "score": 1.0, + "content": "same batch improves convergence and yields a slight boost in performance.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 302 + ], + "lines": [], + "index": 8.5, + "bbox_fs": [ + 105, + 258, + 506, + 302 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 308, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 506, + 320 + ], + "score": 1.0, + "content": "In Appendix C, we further illustrate how MIMO exploits the capacity of the network. In particular,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "we study the performance of MIMO when the regularization increases (both in terms of L1 and L2", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "regularization), i.e. when the capacity of the network is increasingly constrained. 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Interestingly,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "for the L1 regularization, we can relate the performance of MIMO with the sparsity of the network,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 374, + 338, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 338, + 385 + ], + "score": 1.0, + "content": "strengthening the connection to Frankle & Carbin (2018).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 308, + 506, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 395, + 263, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 265, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 265, + 407 + ], + "score": 1.0, + "content": "3.5 INPUT AND BATCH REPETITION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "MIMO works well by simply adding the multi-input and multi-output configuration to an existing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "baseline, and varying only one additional hyperparameter (Section 3.4’s number of subnetworks", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 107, + 434, + 119, + 444 + ], + "score": 0.59, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "). We found two additional hyperparameters can further improve performance, and they can be", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 445, + 299, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 299, + 457 + ], + "score": 1.0, + "content": "important when the network capacity is limited.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 412, + 505, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Input repetition Selecting independent examples for the multi-input configuration during training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "forces the subnetworks not to share any features. This is beneficial when the network has sufficient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "capacity, but when the network has limited excess capacity, we found that relaxing independence is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 508 + ], + "score": 1.0, + "content": "beneficial. For example, ResNet50 on ImageNet (He et al., 2016) does not have sufficient capacity to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 397, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 263, + 518 + ], + "score": 1.0, + "content": "support two independent subnetworks", + "type": "text" + }, + { + "bbox": [ + 263, + 506, + 294, + 516 + ], + "score": 0.86, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 505, + 397, + 518 + ], + "score": 1.0, + "content": ") in MIMO configuration.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 461, + 506, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "Our proposed solution is to relax independence between the inputs. 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At", + "type": "text" + }, + { + "bbox": [ + 305, + 583, + 329, + 594 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 582, + 506, + 595 + ], + "score": 1.0, + "content": ", the subnetworks are independent but their", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 308, + 607 + ], + "score": 1.0, + "content": "performance is limited by the network capacity. As", + "type": "text" + }, + { + "bbox": [ + 308, + 596, + 315, + 606 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "grows, the subnetworks share increasingly more", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 338, + 618 + ], + "score": 1.0, + "content": "features, which improves their performance. However, as", + "type": "text" + }, + { + "bbox": [ + 338, + 607, + 345, + 616 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "approaches 1, the subnetworks become", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "highly correlated and the benefit of ensembling is lost. Unlike ResNet50, Wide ResNet50 has more", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 627, + 417, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 332, + 640 + ], + "score": 1.0, + "content": "capacity and benefits less from input repetition (roughly", + "type": "text" + }, + { + "bbox": [ + 332, + 627, + 365, + 637 + ], + "score": 0.87, + "content": "78 \\text{‰}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 627, + 377, + 640 + ], + "score": 1.0, + "content": "vs", + "type": "text" + }, + { + "bbox": [ + 378, + 627, + 411, + 638 + ], + "score": 0.83, + "content": "7 4 \\mathrm { - } 7 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 627, + 417, + 640 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 582, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Batch repetition For stochastic models, most notably MC dropout and variational Bayesian neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "nets, drawing multiple approximate posterior samples for each example during training can improve", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "performance as it reduces gradient noise w.r.t. the network’s model uncertainty, e.g., Dusenberry", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "et al. (2020a). We achieve a similar effect by repeating examples in the minibatch: this forms a new", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 201, + 700 + ], + "score": 1.0, + "content": "minibatch size of, e.g.,", + "type": "text" + }, + { + "bbox": [ + 202, + 688, + 231, + 698 + ], + "score": 0.79, + "content": "5 1 2 \\cdot 5", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "(batch size and number of batch repetitions respectively). Like the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "choice of batch size which determines the number of unique examples in the SGD step (Shallue", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "et al., 2018), varying the number of repetitions has an implicit regularization effect. Figure 6b shows", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "performance over the number of batch repetitions, where each batch repetition setting indicates a box", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "plot over a sweep of 12 settings of batch size, learning rate, and ensemble size. Higher repetitions", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 224, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 224, + 106 + ], + "score": 1.0, + "content": "typically yield a slight boost.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "plot over a sweep of 12 settings of batch size, learning rate, and ensemble size. Higher repetitions", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 224, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 224, + 106 + ], + "score": 1.0, + "content": "typically yield a slight boost.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 108, + 117, + 201, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 115, + 201, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 201, + 131 + ], + "score": 1.0, + "content": "4 BENCHMARKS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 135, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "We described and analyzed MIMO. In this section, we compare MIMO on benchmarks building on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 145, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 506, + 160 + ], + "score": 1.0, + "content": "Uncertainty Baselines.4 This framework allows us to benchmark the performance and to compare", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "score": 1.0, + "content": "against high-quality, well-optimized implementations of baseline methods (see framework for further", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 169, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 506, + 181 + ], + "score": 1.0, + "content": "baselines than ones highlighted here). We looked at three model/dataset combinations: ResNet28-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "10/CIFAR10, ResNet28-10/CIFAR100, and ResNet50/ImageNet. MIMO’s code is open-sourced.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 190, + 112, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 112, + 197 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 505, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "Baselines Our baselines include the reference implementations of a deterministic deep neural network", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "(trained with SGD), MC-Dropout, BatchEnsemble, and ensemble models, as well as two related", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "score": 1.0, + "content": "models, Naive multihead and TreeNet. Thin networks use half the number of convolutional filters", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 455, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 455, + 253 + ], + "score": 1.0, + "content": "while wide models use double. See Appendix B for the details on the hyperparameters.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "Metrics To measure robustness, we look at accuracy, negative log-likelihood (NLL), and expected", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "calibration error (ECE) on the IID test set as well as a corrupted test set where the test images are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "perturbed (e.g. added blur, compression artifacts, frost effects) (Hendrycks & Dietterich, 2019).", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "Appendix D includes ImageNet results for 5 additional out-of-distribution datasets. To measure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "computational cost, we look at how long it takes to evaluate the model on a TPUv2 core, measured in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 174, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 174, + 325 + ], + "score": 1.0, + "content": "ms per example.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 108, + 334, + 503, + 416 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 334, + 503, + 416 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 334, + 503, + 416 + ], + "spans": [ + { + "bbox": [ + 108, + 334, + 503, + 416 + ], + "score": 0.977, + "html": "
NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic960.1590.02376.11.0500.1530.6321
Dropout95.90.1600.02468.81.2700.1660.6561
Naive mutlihead (M= 3)95.90.1610.02276.60.9690.1440.6361
MIMO (M= 3)(This work)96.40.1230.01076.60.9270.1120.6391
TreeNet (M= 3)95.90.1580.01875.50.9690.1370.9611.5
BatchEnsemble (M=4)96.20.1430.02177.51.0200.1292.5524
Thin Ensemble (M=4)96.30.1150.00877.20.8400.0890.8234
Ensemble (M=4)96.60.1140.01077.90.8100.0872.5364
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NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic79.80.8750.08651.42.7000.2390.6321
Monte Carlo Dropout79.60.8300.05042.62.9000.2020.6561
Naive mutlihead (M= 3)79.50.8340.04852.12.3390.1560.6361
MIMO (M= 3)(This work)82.00.6900.02253.72.2840.1290.6391
TreeNet (M= 3)80.80.7770.04753.52.2950.1760.9611.5
BatchEnsemble (M= 4)81.50.7400.05654.12.4900.1912.5524
Thin Ensemble (M= 4)81.50.6940.01753.72.1900.1110.8234
Ensemble (M=4)82.70.6660.02154.12.2700.1382.5364
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NameAccuracy (↑)NLL (↓)ECE(↓)cAcc (↑)cNLL (↓)cECE(↓)Prediction time (↓)#Forward passes (↓)
Deterministic76.1000.9430.03940.5003.2000.1050.6401
Naive mutlihead (M = 3)76.6110.9290.04340.6163.2500.1220.6381
MIMO (M=2)(p=0.6)(This work)77.5000.8870.03743.3003.0300.1060.6351
TreeNet(M=2)78.1390.8520.01742.4203.0520.0730.8481.5
BatchEnsemble (M =4)76.7000.9440.04941.8003.1800.1102.5924
Ensemble (M=4)77.5000.8770.03142.1002.9900.0512.6244
WideDeterministic77.8850.9380.07245.0003.1000.1501.6741
Wide MIMO (M=2)(p= 0.6)(This work)79.3000.8430.06145.7913.0480.1471.7061
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In this section, we compare MIMO on benchmarks building on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 145, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 506, + 160 + ], + "score": 1.0, + "content": "Uncertainty Baselines.4 This framework allows us to benchmark the performance and to compare", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "score": 1.0, + "content": "against high-quality, well-optimized implementations of baseline methods (see framework for further", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 169, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 506, + 181 + ], + "score": 1.0, + "content": "baselines than ones highlighted here). We looked at three model/dataset combinations: ResNet28-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 192 + ], + "score": 1.0, + "content": "10/CIFAR10, ResNet28-10/CIFAR100, and ResNet50/ImageNet. MIMO’s code is open-sourced.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 190, + 112, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 112, + 197 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 135, + 506, + 197 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 207, + 505, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "Baselines Our baselines include the reference implementations of a deterministic deep neural network", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "(trained with SGD), MC-Dropout, BatchEnsemble, and ensemble models, as well as two related", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 241 + ], + "score": 1.0, + "content": "models, Naive multihead and TreeNet. 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See Appendix B for the details on the hyperparameters.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 208, + 505, + 253 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "Metrics To measure robustness, we look at accuracy, negative log-likelihood (NLL), and expected", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "calibration error (ECE) on the IID test set as well as a corrupted test set where the test images are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "perturbed (e.g. added blur, compression artifacts, frost effects) (Hendrycks & Dietterich, 2019).", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 303 + ], + "score": 1.0, + "content": "Appendix D includes ImageNet results for 5 additional out-of-distribution datasets. 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NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic960.1590.02376.11.0500.1530.6321
Dropout95.90.1600.02468.81.2700.1660.6561
Naive mutlihead (M= 3)95.90.1610.02276.60.9690.1440.6361
MIMO (M= 3)(This work)96.40.1230.01076.60.9270.1120.6391
TreeNet (M= 3)95.90.1580.01875.50.9690.1370.9611.5
BatchEnsemble (M=4)96.20.1430.02177.51.0200.1292.5524
Thin Ensemble (M=4)96.30.1150.00877.20.8400.0890.8234
Ensemble (M=4)96.60.1140.01077.90.8100.0872.5364
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NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic79.80.8750.08651.42.7000.2390.6321
Monte Carlo Dropout79.60.8300.05042.62.9000.2020.6561
Naive mutlihead (M= 3)79.50.8340.04852.12.3390.1560.6361
MIMO (M= 3)(This work)82.00.6900.02253.72.2840.1290.6391
TreeNet (M= 3)80.80.7770.04753.52.2950.1760.9611.5
BatchEnsemble (M= 4)81.50.7400.05654.12.4900.1912.5524
Thin Ensemble (M= 4)81.50.6940.01753.72.1900.1110.8234
Ensemble (M=4)82.70.6660.02154.12.2700.1382.5364
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NameAccuracy (↑)NLL (↓)ECE(↓)cAcc (↑)cNLL (↓)cECE(↓)Prediction time (↓)#Forward passes (↓)
Deterministic76.1000.9430.03940.5003.2000.1050.6401
Naive mutlihead (M = 3)76.6110.9290.04340.6163.2500.1220.6381
MIMO (M=2)(p=0.6)(This work)77.5000.8870.03743.3003.0300.1060.6351
TreeNet(M=2)78.1390.8520.01742.4203.0520.0730.8481.5
BatchEnsemble (M =4)76.7000.9440.04941.8003.1800.1102.5924
Ensemble (M=4)77.5000.8770.03142.1002.9900.0512.6244
WideDeterministic77.8850.9380.07245.0003.1000.1501.6741
Wide MIMO (M=2)(p= 0.6)(This work)79.3000.8430.06145.7913.0480.1471.7061
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It approaches the robustness of a Deep Ensemble, without increasing the computational", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 688, + 131, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 131, + 699 + ], + "score": 1.0, + "content": "costs.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 665, + 505, + 699 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 209, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 210, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 210, + 96 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "Multi-headed networks have been previously studied by Lee et al. (2015); Osband et al. (2016);", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 506, + 124 + ], + "score": 1.0, + "content": "Tran et al. (2020). In this approach to efficient ensembles, the input and part of the network are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "shared by the members while the final few layers and the outputs are separate. The advantage of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "score": 1.0, + "content": "the approach is that the computational cost is reduced compared to typical ensembles, since many", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 157 + ], + "score": 1.0, + "content": "layers are shared, but the ensemble diversity (and resulting performance) is quite lacking. MIMO’s", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 168 + ], + "score": 1.0, + "content": "multi-input configuration makes a significant impact as each ensemble member may take different", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 178 + ], + "score": 1.0, + "content": "paths throughout the full network. Further, MIMO has lower computational cost than multi-headed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 383, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 383, + 189 + ], + "score": 1.0, + "content": "approaches, since all of the layers except the first and last are shared.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 194, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "Related efficient ensemble approaches include BatchEnsemble, Rank-1 BNNs, and hyper batch", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "ensembles (Wen et al., 2020; Dusenberry et al., 2020a; Wenzel et al., 2020). In these methods, most", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "of the model parameters are shared among the members, which reduces the memory requirement of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "the model, but the evaluation cost still requires multiple forward passes. Interestingly, like MIMO,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 239, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 250 + ], + "score": 1.0, + "content": "these methods also apply a multi-input multi-output configuration, treating an ensemble of networks", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 261 + ], + "score": 1.0, + "content": "as a single bigger network; however, MIMO still outperforms BatchEnsemble. We believe important", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 260, + 441, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 441, + 272 + ], + "score": 1.0, + "content": "insights such as Section 3.5’s input independence may also improve these methods.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "Finally, there are simple heuristics which also retain efficient compute such as data augmentation,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "temperature scaling, label smoothing, contrastive training. 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The key benefits of MIMO are its simplicity, since it does not require", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "significant modifications to the network architecture and it has few hyperparameters, and also its", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "computational efficiency, since it can be evaluated in a single forward pass. Our empirical results", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "confirm that MIMO improves performance and robustness with minor changes to the number of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 242, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 242, + 419 + ], + "score": 1.0, + "content": "parameters and the compute cost.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 431, + 225, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 226, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 226, + 445 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 441, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 443, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 443, + 464 + ], + "score": 1.0, + "content": "Marton Havasi is funded by EPSRC. We thank Ellen Jiang for helping with writing.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 175, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 176, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 176, + 486 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 504, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 116, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 116, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "neural networks. 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The lottery ticket hypothesis: Finding sparse, trainable neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 115, + 721, + 321, + 732 + ], + "spans": [ + { + "bbox": [ + 115, + 721, + 321, + 732 + ], + "score": 1.0, + "content": "networks. arXiv preprint arXiv:1803.03635, 2018.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 209, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 210, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 210, + 96 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "Multi-headed networks have been previously studied by Lee et al. (2015); Osband et al. (2016);", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 506, + 124 + ], + "score": 1.0, + "content": "Tran et al. (2020). In this approach to efficient ensembles, the input and part of the network are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "shared by the members while the final few layers and the outputs are separate. The advantage of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "score": 1.0, + "content": "the approach is that the computational cost is reduced compared to typical ensembles, since many", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 157 + ], + "score": 1.0, + "content": "layers are shared, but the ensemble diversity (and resulting performance) is quite lacking. MIMO’s", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 168 + ], + "score": 1.0, + "content": "multi-input configuration makes a significant impact as each ensemble member may take different", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 178 + ], + "score": 1.0, + "content": "paths throughout the full network. Further, MIMO has lower computational cost than multi-headed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 383, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 383, + 189 + ], + "score": 1.0, + "content": "approaches, since all of the layers except the first and last are shared.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 100, + 506, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 194, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "Related efficient ensemble approaches include BatchEnsemble, Rank-1 BNNs, and hyper batch", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "ensembles (Wen et al., 2020; Dusenberry et al., 2020a; Wenzel et al., 2020). In these methods, most", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "of the model parameters are shared among the members, which reduces the memory requirement of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "the model, but the evaluation cost still requires multiple forward passes. Interestingly, like MIMO,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 239, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 250 + ], + "score": 1.0, + "content": "these methods also apply a multi-input multi-output configuration, treating an ensemble of networks", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 261 + ], + "score": 1.0, + "content": "as a single bigger network; however, MIMO still outperforms BatchEnsemble. We believe important", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 260, + 441, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 441, + 272 + ], + "score": 1.0, + "content": "insights such as Section 3.5’s input independence may also improve these methods.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 194, + 506, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "Finally, there are simple heuristics which also retain efficient compute such as data augmentation,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "temperature scaling, label smoothing, contrastive training. These methods are orthogonal to MIMO", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 299, + 450, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 450, + 311 + ], + "score": 1.0, + "content": "and they can provide a performance boost, without increasing the computational cost.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 276, + 506, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 321, + 202, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 203, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 203, + 337 + ], + "score": 1.0, + "content": "6 CONCLUSIONS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 341, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "We propose MIMO, a novel approach for training multiple independent subnetworks within a network.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "We show that the subnetworks separate within the model and that they behave as independently", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "trained neural networks. The key benefits of MIMO are its simplicity, since it does not require", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "significant modifications to the network architecture and it has few hyperparameters, and also its", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "computational efficiency, since it can be evaluated in a single forward pass. 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We thank Ellen Jiang for helping with writing.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 447, + 443, + 464 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 175, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 176, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 176, + 486 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 504, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 116, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 116, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "neural networks. 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1:for t=1...Niter do
2:(x1:M,Y1:M) ~ U(X)
3:Pe(yi|x1:M)...Pe(ym|x1:m) ← MIMO(x1:M)
4:
5:0←0-∈VLM(0)
6: end for
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For ResNet50/ImageNet the model has lower capacity so we used", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 514, + 194, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 514, + 137, + 524 + ], + "score": 0.89, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 514, + 158, + 527 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 158, + 514, + 191, + 525 + ], + "score": 0.91, + "content": "\\rho = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 514, + 194, + 527 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 477, + 563 + ], + "lines": [ + { + "bbox": [ + 107, + 537, + 477, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 120, + 549 + ], + "score": 1.0, + "content": "C", + "type": "text" + }, + { + "bbox": [ + 123, + 537, + 477, + 550 + ], + "score": 1.0, + "content": "MIMO BETTER EXPLOITS THE NETWORK CAPACITY: PERFORMANCE", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 127, + 553, + 322, + 565 + ], + "spans": [ + { + "bbox": [ + 127, + 553, + 322, + 565 + ], + "score": 1.0, + "content": "VERSUS REGULARIZATION STRENGTH", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 108, + 569, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 583 + ], + "score": 1.0, + "content": "In this section, we further illustrate how MIMO better exploits the capacity of the network, through", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 580, + 276, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 276, + 594 + ], + "score": 1.0, + "content": "the lens of its sensitivity to regularization.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 358, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 358, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 358, + 611 + ], + "score": 1.0, + "content": "Our experimental protocol is guided by the following rationale", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 132, + 617, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 132, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 132, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "• Regularization controls the capacity of the network: the higher the regularization, the more", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 628, + 242, + 641 + ], + "spans": [ + { + "bbox": [ + 142, + 628, + 242, + 641 + ], + "score": 1.0, + "content": "constrained the capacity.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 137, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 137, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "MIMO makes better use of the capacity of the network: the more ensemble members, the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 654, + 252, + 666 + ], + "spans": [ + { + "bbox": [ + 141, + 654, + 252, + 666 + ], + "score": 1.0, + "content": "more capacity is exploited.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 134, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 134, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "• As a result, MIMO should be more sensitive to the constraining of the capacity of the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 141, + 679, + 371, + 691 + ], + "score": 1.0, + "content": "network. 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1:for t=1...Niter do
2:(x1:M,Y1:M) ~ U(X)
3:Pe(yi|x1:M)...Pe(ym|x1:m) ← MIMO(x1:M)
4:
5:0←0-∈VLM(0)
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For", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "the ResNet50/ImageNet models, we use a batch-size of 4096 with decaying learning rate of 0.1", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 484 + ], + "score": 1.0, + "content": "(decay rate 0.1), L2 regularization 1e-4, 150 training epochs, and batch repetition of 2. This makes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "score": 1.0, + "content": "the training cost of MIMO comparable to that of BatchEnsemble and Ensemble models. For the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 281, + 505 + ], + "score": 1.0, + "content": "ResNet28-10/CIFAR experiments, we use", + "type": "text" + }, + { + "bbox": [ + 282, + 492, + 314, + 502 + ], + "score": 0.9, + "content": "M = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "subnetworks because it performs well in both", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "accuracy and log-likelihood. For ResNet50/ImageNet the model has lower capacity so we used", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 514, + 194, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 514, + 137, + 524 + ], + "score": 0.89, + "content": "M = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 514, + 158, + 527 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 158, + 514, + 191, + 525 + ], + "score": 0.91, + "content": "\\rho = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 514, + 194, + 527 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 425, + 506, + 527 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 477, + 563 + ], + "lines": [ + { + "bbox": [ + 107, + 537, + 477, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 120, + 549 + ], + "score": 1.0, + "content": "C", + "type": "text" + }, + { + "bbox": [ + 123, + 537, + 477, + 550 + ], + "score": 1.0, + "content": "MIMO BETTER EXPLOITS THE NETWORK CAPACITY: PERFORMANCE", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 127, + 553, + 322, + 565 + ], + "spans": [ + { + "bbox": [ + 127, + 553, + 322, + 565 + ], + "score": 1.0, + "content": "VERSUS REGULARIZATION STRENGTH", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 108, + 569, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 583 + ], + "score": 1.0, + "content": "In this section, we further illustrate how MIMO better exploits the capacity of the network, through", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 580, + 276, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 276, + 594 + ], + "score": 1.0, + "content": "the lens of its sensitivity to regularization.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 568, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 358, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 358, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 358, + 611 + ], + "score": 1.0, + "content": "Our experimental protocol is guided by the following rationale", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 597, + 358, + 611 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 617, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 132, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 132, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "• Regularization controls the capacity of the network: the higher the regularization, the more", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 628, + 242, + 641 + ], + "spans": [ + { + "bbox": [ + 142, + 628, + 242, + 641 + ], + "score": 1.0, + "content": "constrained the capacity.", + "type": "text" + } + ], + "index": 37, + "is_list_end_line": true + }, + { + "bbox": [ + 137, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 137, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "MIMO makes better use of the capacity of the network: the more ensemble members, the", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 654, + 252, + 666 + ], + "spans": [ + { + "bbox": [ + 141, + 654, + 252, + 666 + ], + "score": 1.0, + "content": "more capacity is exploited.", + "type": "text" + } + ], + "index": 39, + "is_list_end_line": true + }, + { + "bbox": [ + 134, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 134, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "• As a result, MIMO should be more sensitive to the constraining of the capacity of the", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 141, + 679, + 371, + 691 + ], + "score": 1.0, + "content": "network. And the more ensemble members (i.e., the larger", + "type": "text" + }, + { + "bbox": [ + 371, + 680, + 383, + 689 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "), the stronger the effect should", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 690, + 158, + 702 + ], + "spans": [ + { + "bbox": [ + 141, + 690, + 158, + 702 + ], + "score": 1.0, + "content": "be.", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + } + ], + "index": 39, + "bbox_fs": [ + 132, + 617, + 505, + 702 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We consider the ResNet28-10/CIFAR10 and ResNet28-10/CIFAR100 settings used in the main", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "paper where we additionally vary the L1 (respectively L2) regularization while keeping the other L2", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 366, + 504, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 504, + 380 + ], + "score": 1.0, + "content": "(respectively L1) term equal to zero. 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We display in Figures 7-8 the accuracy and log-likelihood over", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 378, + 421, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 421, + 390 + ], + "score": 1.0, + "content": "those regularization paths (averaged over three repetitions of the experiments).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "score": 1.0, + "content": "As previously hypothesised, we observe that MIMO is indeed more sensitive to the constraining", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 418 + ], + "score": 1.0, + "content": "of the capacity of the network as the regularization increases. 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We report those results in Figure 9", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 438, + 290, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 290, + 451 + ], + "score": 1.0, + "content": "where we can observe the same phenomenon.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 336, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 337, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 337, + 476 + ], + "score": 1.0, + "content": "D ADDITIONAL IMAGENET OOD RESULTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "In the following table, we evaluate trained ResNet-50 models on 7 datasets. ImageNet, ImageNet-C,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "ImageNet-A, and ImageNetV2 each display three metrics: negative log-likelihood, accuracy, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "expected calibration error respectively. ImageNet-C further includes mCE (mean corruption error) in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "parentheses. ImageNet-Vid-Robust, YTTB-Robust, and ObjectNet use their own pre-defined stability", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 525, + 141, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 141, + 537 + ], + "score": 1.0, + "content": "metrics.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "These experiments were expensive to run, so we were only able to obtain them for a smaller set", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "of methods. We find these results are consistent with the main text’s benchmarks, showing MIMO", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 564, + 502, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 502, + 576 + ], + "score": 1.0, + "content": "consistently outperforms methods not only on corrupted images, but also across distribution shifts.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "table", + "bbox": [ + 108, + 582, + 503, + 664 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 582, + 503, + 664 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 582, + 503, + 664 + ], + "spans": [ + { + "bbox": [ + 108, + 582, + 503, + 664 + ], + "score": 0.981, + "html": "
NameImageNetImageNet-CImageNet-A
Deterministic0.939/76.2%/0.0323.21/40.5%/0.103 (75.4%)8.09/ 0.7% /0.425
MIMO (M= 2,ρ = 0.6) Wide MIMO (M=2,ρ = 0.6)0.887 / 77.5% /0.037 0.843 /79.3% /0.0613.03 /43.3% / 0.106 (71.7%) 3.1/45.0% /0.150 (69.6%)7.76 /1.4% /0.432 7.52 /3.3% /0.46
NameImageNetV2YTTB-Robust ObjectNet
Deterministic1.58 / 64.4% / 0.074ImageNet-Vid-Robust 29.9%21.7%
1.51/ 65.7% / 0.08425.9%
MIMO (M=2,𝜌= 0.6)31.8%22.2% 28.1%
Wide MIMO (M= 2,p= 0.6)1.49 / 67.9% / 0.10935.3%22.9% 29.5%
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We report those results in Figure 9", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 438, + 290, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 290, + 451 + ], + "score": 1.0, + "content": "where we can observe the same phenomenon.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 393, + 506, + 451 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 336, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 337, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 337, + 476 + ], + "score": 1.0, + "content": "D ADDITIONAL IMAGENET OOD RESULTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "In the following table, we evaluate trained ResNet-50 models on 7 datasets. ImageNet, ImageNet-C,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "ImageNet-A, and ImageNetV2 each display three metrics: negative log-likelihood, accuracy, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "expected calibration error respectively. ImageNet-C further includes mCE (mean corruption error) in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "parentheses. ImageNet-Vid-Robust, YTTB-Robust, and ObjectNet use their own pre-defined stability", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 525, + 141, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 141, + 537 + ], + "score": 1.0, + "content": "metrics.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 481, + 506, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "These experiments were expensive to run, so we were only able to obtain them for a smaller set", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "of methods. We find these results are consistent with the main text’s benchmarks, showing MIMO", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 564, + 502, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 502, + 576 + ], + "score": 1.0, + "content": "consistently outperforms methods not only on corrupted images, but also across distribution shifts.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 542, + 505, + 576 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 582, + 503, + 664 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 582, + 503, + 664 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 582, + 503, + 664 + ], + "spans": [ + { + "bbox": [ + 108, + 582, + 503, + 664 + ], + "score": 0.981, + "html": "
NameImageNetImageNet-CImageNet-A
Deterministic0.939/76.2%/0.0323.21/40.5%/0.103 (75.4%)8.09/ 0.7% /0.425
MIMO (M= 2,ρ = 0.6) Wide MIMO (M=2,ρ = 0.6)0.887 / 77.5% /0.037 0.843 /79.3% /0.0613.03 /43.3% / 0.106 (71.7%) 3.1/45.0% /0.150 (69.6%)7.76 /1.4% /0.432 7.52 /3.3% /0.46
NameImageNetV2YTTB-Robust ObjectNet
Deterministic1.58 / 64.4% / 0.074ImageNet-Vid-Robust 29.9%21.7%
1.51/ 65.7% / 0.08425.9%
MIMO (M=2,𝜌= 0.6)31.8%22.2% 28.1%
Wide MIMO (M= 2,p= 0.6)1.49 / 67.9% / 0.10935.3%22.9% 29.5%
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NameAccuracy (↑)NLL (↓)ECE(↓)cAcc (↑)cNLL (↓)cECE(↓)Prediction time (↓)#Forward passes (↓)
Deterministic76.1000.9430.03940.5003.2000.1050.6401
Naive mutlihead (M = 3)76.6110.9290.04340.6163.2500.1220.6381
MIMO (M=2)(p=0.6)(This work)77.5000.8870.03743.3003.0300.1060.6351
TreeNet(M=2)78.1390.8520.01742.4203.0520.0730.8481.5
BatchEnsemble (M =4)76.7000.9440.04941.8003.1800.1102.5924
Ensemble (M=4)77.5000.8770.03142.1002.9900.0512.6244
WideDeterministic77.8850.9380.07245.0003.1000.1501.6741
Wide MIMO (M=2)(p= 0.6)(This work)79.3000.8430.06145.7913.0480.1471.7061
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NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic79.80.8750.08651.42.7000.2390.6321
Monte Carlo Dropout79.60.8300.05042.62.9000.2020.6561
Naive mutlihead (M= 3)79.50.8340.04852.12.3390.1560.6361
MIMO (M= 3)(This work)82.00.6900.02253.72.2840.1290.6391
TreeNet (M= 3)80.80.7770.04753.52.2950.1760.9611.5
BatchEnsemble (M= 4)81.50.7400.05654.12.4900.1912.5524
Thin Ensemble (M= 4)81.50.6940.01753.72.1900.1110.8234
Ensemble (M=4)82.70.6660.02154.12.2700.1382.5364
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NameAccuracy (↑)NLL (↓)ECE (↓)cAcc (↑)cNLL (↓)cECE (↓)Prediction time (↓)#Forward passes (↓)
Deterministic960.1590.02376.11.0500.1530.6321
Dropout95.90.1600.02468.81.2700.1660.6561
Naive mutlihead (M= 3)95.90.1610.02276.60.9690.1440.6361
MIMO (M= 3)(This work)96.40.1230.01076.60.9270.1120.6391
TreeNet (M= 3)95.90.1580.01875.50.9690.1370.9611.5
BatchEnsemble (M=4)96.20.1430.02177.51.0200.1292.5524
Thin Ensemble (M=4)96.30.1150.00877.20.8400.0890.8234
Ensemble (M=4)96.60.1140.01077.90.8100.0872.5364
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1:for t=1...Niter do
2:(x1:M,Y1:M) ~ U(X)
3:Pe(yi|x1:M)...Pe(ym|x1:m) ← MIMO(x1:M)
4:
5:0←0-∈VLM(0)
6: end for
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NameImageNetImageNet-CImageNet-A
Deterministic0.939/76.2%/0.0323.21/40.5%/0.103 (75.4%)8.09/ 0.7% /0.425
MIMO (M= 2,ρ = 0.6) Wide MIMO (M=2,ρ = 0.6)0.887 / 77.5% /0.037 0.843 /79.3% /0.0613.03 /43.3% / 0.106 (71.7%) 3.1/45.0% /0.150 (69.6%)7.76 /1.4% /0.432 7.52 /3.3% /0.46
NameImageNetV2YTTB-Robust ObjectNet
Deterministic1.58 / 64.4% / 0.074ImageNet-Vid-Robust 29.9%21.7%
1.51/ 65.7% / 0.08425.9%
MIMO (M=2,𝜌= 0.6)31.8%22.2% 28.1%
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Unlike previous works that consider model pruning and quantization separately, we seek to optimize them jointly. To deal with the resultant large designing space, we propose a novel super-bit model, a single-path method, to encode all candidate compression configurations, rather than maintaining separate paths for each configuration. Specifically, we first propose a novel decomposition of quantization that encapsulates all the candidate bitwidths in the search space. Starting from a low bitwidth, we sequentially consider higher bitwidths by recursively adding reassignment offsets. We then introduce learnable binary gates to encode the choice of bitwidth, including filter-wise 0-bit for pruning. By jointly training the binary gates in conjunction with network parameters, the compression configurations of each layer can be automatically determined. Our ABS brings two benefits for model compression: 1) It avoids the combinatorially large design space, with a reduced number of trainable parameters and search costs. 2) It also averts directly fitting an extremely low bit quantizer to the data, hence greatly reducing the optimization difficulty due to the non-differentiable quantization. Experiments on CIFAR-100 and ImageNet show that our methods achieve significant computational cost reduction while preserving promising performance. + +# 1 INTRODUCTION + +Deep neural networks (DNNs) have achieved great success in many challenging computer vision tasks, including image classification (Krizhevsky et al., 2012; He et al., 2016) and object detection (Lin et al., 2017a;b). However, a deep model usually has a large number of parameters and consumes huge amounts of computational resources, which remains great obstacles for many applications, especially on resource-limited devices with limited memory and computational resources, such as smartphones. To reduce the number of parameters and computational overhead, many methods (He et al., 2019; Zhou et al., 2016) have been proposed to conduct model compression by removing the redundancy while maintaining the performance. + +In the last decades, we have witnessed a lot of model compression methods, such as network pruning (He et al., 2017; 2019) and quantization (Zhou et al., 2016; Hubara et al., 2016). Specifically, network pruning reduces the model size and computational costs by removing redundant modules while network quantization maps the full-precision values to low-precision ones. It has been shown that sequentially perform network pruning and quantization is able to get a compressed network with small model size and lower computational overhead (Han et al., 2016). However, performing pruning and quantization in a separate step may lead to sub-optimal results. For example, the best quantization strategy for the uncompressed network is not necessarily the optimal one after network pruning. Therefore, we need to consider performing pruning and quantization simultaneously. + +Recently, many attempts have been made to automatically determine the compression configurations of each layer (i.e., pruning ratios, and/or bitwidths), either based on reinforcement learning (RL) (Wang et al., 2019), evolutionary search (ES) (Wang et al., 2020), Bayesian optimization (BO) (Tung & Mori, 2018) or differentiable methods (Wu et al., 2018; Dong & Yang, 2019). In particular, previous differentiable methods formulate model compression as a differentiable searching problem to explore the search space using gradient-based optimization. As shown in Figure 1(a), each candidate operation is maintained as a separate path, which leads to a huge number of trainable parameters and high computational overhead when the search space becomes combinatorially large. Moreover, due to the non-differentiable quantizer and pruning process, the optimization of heavily compressed candidate networks can be more challenging than that in the conventional search problem. + +![](images/d661d86f6402cd19d79021a39a0281ea92dacd0285f0d8f2e32e077e497f6529.jpg) +Figure 1: Multi-path v.s. single-path compression scheme. (a) Multi-path search scheme (Wu et al., 2018): represents each candidate configuration as a separate path and formulates the compression problem as a path selection problem, which gives rise to huge numbers of trainable parameters and high computational overhead when the search space becomes combinatorially large. Here, $z _ { k }$ is the $k$ -bit quantized version of $z$ and $\alpha _ { k } ^ { q }$ is the architecture parameters corresponding to the path of $k$ -bit quantization. (b) Single-path search scheme (Ours): represents each candidate configuration as a subset of a “super-bit” and formulates the compression problem as a subset selection problem, which greatly reduces the computational costs and optimization difficulty from the discontinuity of quantization. Here, the super-bit denotes the highest bitwidth in the search space, $g _ { k } ^ { q }$ is a binary gate that controls the decision of bitwidth, and $\epsilon _ { k }$ is the re-assignment offset (quantized residual error). + +In this paper, we propose a simple yet effective model compression method named Automatic Bit Sharing (ABS) to reduce the search cost and ease the optimization for the compressed candidates. Inspired by recent single-path neural architecture search (NAS) methods (Stamoulis et al., 2019; Guo et al., 2020), the proposed ABS introduces a novel single-path super-bit to encode all effective bitwidths in the search space instead of formulating each candidate operation as a separate path, as shown in Figure 1(b). Specifically, we build upon the observation that the quantized values of a high bitwidth can share the ones of low bitwidths under some conditions. Therefore, we are able to decompose the quantized representation into the sum of the lowest bit quantization and a series of re-assignment offsets. We then introduce learnable binary gates to encode the choice of bitwidth, including filter-wise 0-bit for pruning. By jointly training the binary gates and network parameters, the compression ratio of each layer can be automatically determined. The proposed scheme has several advantages. First, we only need to solve the search problem as finding which subset of the super-bit to use for each layer’s weights and activations rather than selecting from different paths. Second, we enforce the candidate bitwidths to share the common quantized values. Hence, we are able to optimize them jointly instead of separately, which greatly reduces the optimization difficulty from the discontinuity of discretization. + +Our main contributions are summarized as follows: + +• We devise a novel super-bit scheme that encapsulates multiple compression configurations in a unified single-path framework. Relying on the super-bit scheme, we further introduce learnable binary gates to determine the optimal bitwidths (including filter-wise 0-bit for +pruning). The proposed ABS casts the search problem as subset selection problem, hence significantly reducing the search cost. +• We formulate the quantized representation as a gated combination of the lowest bitwidth quantization and a series of re-assignment offsets, in which we explicitly share the quantized values between different bitwidths. In this way, we enable the candidate operations to learn jointly rather than separately, hence greatly easing the optimization, especially in the non-differentiable quantization scenario. + +• We evaluate our ABS on CIFAR-100 and ImageNet over various network architectures. Extensive experiments show that the proposed method achieves the state-of-the-art performance. For example, on ImageNet, our ABS compressed MobileNetV2 achieves $2 8 . 5 \times$ Bit-Operation (BOP) reduction with only $0 . 2 \%$ performance drop on the Top-1 accuracy. + +# 2 RELATED WORK + +Network quantization. Network quantization represents the weights, activations and even gradients in low-precision to yield compact DNNs. With low-precision integers or power-of-two representations, the heavy matrix multiplications can be replaced by efficient bitwise operations, leading to much faster test-time inference and lower power consumption. To improve the quantization performance, current methods either focus on designing accurate quantizers by fitting the quantizer to the data (Jung et al., 2019; Zhang et al., 2018; Choi et al., 2018; Cai et al., 2017), or seek to approximate the gradients due to the non-differentiable discretization (Ding et al., 2019; Louizos et al., 2019; Zhuang et al., 2020). Moreover, most previous works assign the same bitwidth for all layers (Zhou et al., 2016; Zhuang et al., 2018a; 2019; Jung et al., 2019; Jin et al., 2019; Li et al., 2020; Esser et al., 2020). Though attractive for simplicity, setting a uniform precision places no guarantee on optimizing network performance since different layers have different redundancy and arithmetic intensity. Therefore, several studies proposed mixed-precision quantization (Wang et al., 2019; Dong et al., 2019; Wu et al., 2018; Uhlich et al., 2020) to set different bitwidths according to the redundancy of each layer. In this paper, based on the proposed quantization decomposition, we devise an approach that can effectively learn appropriate bitwidths for each layer through gradient-based optimization. + +NAS and pruning. Neural architecture search (NAS) aims to automatically design efficient architectures with low model size and computational costs, either based on reinforcement learning (Pham et al., 2018; Guo et al., 2019), evolutionary search (Real et al., 2019) or gradient-based methods (Liu et al., 2019a). In particular, gradient-based NAS has gained increased popularity, where the search space can be divided into the multi-path design (Liu et al., 2019a; Cai et al., 2019) and single-path formulation (Stamoulis et al., 2019; Guo et al., 2020), depending on whether adding each operation as a separate path or not. While prevailing NAS methods optimize the network topology, the focus of this paper is to search optimal compression ratios for a given architecture. Moreover, network pruning can be treated as fine-grained NAS, which aims at removing redundant modules to accelerate the run-time inference speed, giving rise to methods based on unstructured weight pruning (Han et al., 2016; Guo et al., 2016) or structured channel pruning (He et al., 2017; Zhuang et al., 2018b; Luo et al., 2017). Based on channel pruning, our paper further takes quantization into consideration to generate more compact networks. + +AutoML for model compression. Recently, much effort has been put into automatically determining either the optimal pruning rate (Tung & Mori, 2018; Dong & Yang, 2019; He et al., 2018), or the bitwidth (Lou et al., 2019; Cai & Vasconcelos, 2020) of each layer via hyper-parameter search, without relying on heuristics. In particular, HAQ (Wang et al., 2019) employs reinforcement learning to search bitwidth strategies with the hardware accelerator’s feedback. Meta-pruning (Liu et al., 2019b) uses meta-learning to generate the weight parameters of the pruned networks and then adopts an evolutionary search algorithm to find the layer-wise sparsity for channel pruning. More recently, several studies (Wu et al., 2018; Cai & Vasconcelos, 2020) have focused on using differentiable schemes via gradient-based optimization. + +Closely related methods. To further improve the compression ratio, several methods propose to jointly optimize pruning and quantization strategies. In particular, some works only support weight quantization (Tung & Mori, 2018; Ye et al., 2019) or use fine-grained pruning (Yang et al., 2020). However, the resultant networks cannot be implemented efficiently on edge devices. Recently, several methods (Wu et al., 2018; Wang et al., 2020; Ying et al., 2020) have been proposed to consider filter pruning, weight quantization, and activation quantization jointly. In contrast to these methods, we carefully design the compression search space by sharing the quantized values between different candidate configurations, which significantly reduces the search cost and eases the optimization. Compared with those methods that share the similarities of using quantized residual errors (Chen et al., 2010; Gong et al., 2014; Li et al., 2017b; van Baalen et al., 2020), our proposed method recursively uses quantized residual errors to decompose a quantized representation as a set of candidate bitwidths and parameterize the selection of optimal bitwidth via binary gates. + +Our proposed ABS and Bayesian Bits (van Baalen et al., 2020) are developed concurrently that share a similar idea of quantization decomposition. Critically, our ABS differs from Bayesian Bits in several aspects: 1) The quantization decomposition in our methods can be extended to non-powerof-two bit widths (i.e., $b _ { 1 }$ can be set to arbitrary appropriate integer values), which is a general case of the one in Bayesian Bits. 2) The optimization problems are different. Specifically, we formulate model compression as a single-path subset selection problem while Bayesian Bits casts the optimization of the binary gates to a variational inference problem that requires more relaxations and hyperparameters. 3) Our compressed models with less or comparable BOPs outperform those of Bayesian Bits by a large margin on ImageNet (See Table 2). + +# 3 PROPOSED METHOD + +# 3.1 PRELIMINARY: NORMALIZATION AND QUANTIZATION FUNCTION + +Without loss of generality, given a convolutional layer, let $x$ and $w$ be the activations of the last layer and its weight parameters, respectively. First, for convenience, following (Choi et al., 2018; Bai et al., 2019), we can normalize $x$ and $w$ into scale [0, 1] by $T _ { x }$ and $T _ { w }$ , respectively: + +$$ +\begin{array} { l } { \displaystyle { z _ { x } = T _ { x } ( x ) = \mathrm { c l i p } \left( \frac { x } { v _ { x } } , 0 , 1 \right) , } } \\ { \displaystyle { z _ { w } = T _ { w } ( w ) = \frac { 1 } { 2 } \left( \mathrm { c l i p } \left( \frac { w } { v _ { w } } , - 1 , 1 \right) + 1 \right) , } } \end{array} +$$ + +where the function $\mathrm { c l i p } \left( v , v _ { \mathrm { l o w } } , v _ { \mathrm { u p } } \right) = \mathrm { m i n } ( \mathrm { m a x } ( v , v _ { \mathrm { l o w } } ) , v _ { \mathrm { u p } } )$ clips any number $v$ into the range $[ v _ { \mathrm { l o w } } , v _ { \mathrm { u p } } ]$ , and $v _ { x }$ and $v _ { w }$ are trainable quantization intervals which indicate the range of weights and activations to be quantized. Then, we can apply the following function to quantize the normalized activations and parameters, namely $z _ { x } \in [ 0 , 1 ]$ and $z _ { w } \in [ 0 , 1 ]$ , to discretized ones: + +$$ +D ( z , s ) = s \cdot { \mathrm { r o u n d } } \left( { \frac { z } { s } } \right) , +$$ + +where round $( \cdot )$ returns the nearest integer of a given value and $s$ denotes the normalized step size. Typically, for $k$ -bit quantization, the normalized step size $s$ can be computed by + +$$ +s = { \frac { 1 } { 2 ^ { k } - 1 } } . +$$ + +After doing the $k$ -bit quantization, we shall have $2 ^ { k } - 1$ quantized values. Specifically, we obtain the quantization $Q ( w )$ and $Q ( x )$ by + +$$ +\begin{array} { r l } & { Q ( w ) = { \cal T } _ { w } ^ { - 1 } ( D ( z _ { w } , s ) ) = v _ { w } \cdot ( 2 \cdot D ( z _ { w } , s ) - 1 ) , } \\ & { ~ Q ( x ) = { \cal T } _ { x } ^ { - 1 } ( D ( z _ { x } , s ) ) = v _ { x } \cdot D ( z _ { x } , s ) , } \end{array} +$$ + +where $T _ { w } ^ { - 1 }$ and $T _ { x } ^ { - 1 }$ denote the inverse functions of $T _ { w }$ and $T _ { x }$ , respectively. + +# 3.2 BIT SHARING DECOMPOSITION + +Previous methods consider different compression configurations as different paths and reformulate model compression as a path selection problem, which gives rise to a huge number of trainable parameters and high computational costs. In this paper, we seek to conduct filter pruning and quantization simultaneously by solving the following problem: + +$$ +\begin{array} { r } { \underset { \mathbf { W } , \alpha ^ { p } , \alpha ^ { q } } { \operatorname* { m i n } } \mathcal { L } \left( \mathbf { W } , \alpha ^ { p } , \alpha ^ { q } \right) , } \end{array} +$$ + +where $\mathcal { L } ( \cdot )$ denotes some losses, and $\mathbf { W }$ is the parameters of the network. $\alpha ^ { p }$ and $\alpha ^ { q }$ are the pruning and quantization configurations, respectively. As shown in Eq. (7), we propose to encode all compression configurations in a single-path super-bit model (See Figure 1(b)). In the following, we first introduce the bit sharing decomposition and then describe how to learn for compression. + +To illustrate the bit sharing decomposition, we begin with an example of 2-bit quantization for $z \in \{ z _ { x } , z _ { w } \}$ . Specifically, we consider using the following equation to quantize $z$ to 2-bit: + +$$ +z _ { 2 } = D ( z , s _ { 2 } ) , \quad s _ { 2 } = \frac { 1 } { 2 ^ { 2 } - 1 } , +$$ + +where $z _ { 2 }$ and $s _ { 2 }$ are the quantized value and the step size of 2-bit quantization, respectively. Due to the large step size, the residual error $z - z _ { 2 } \in [ - s _ { 2 } / 2 , s _ { 2 } / 2 ]$ may be big and result in a significant performance drop. To reduce the residual error, an intuitive way is to use a smaller step size, which indicates that we quantize $z$ to a higher bitwidth. Since the step size $s _ { 4 } = 1 / ( 2 ^ { 4 } - 1 )$ in 4-bit quantization is a divisor of the step size $s _ { 2 }$ in 2-bit quantization, the quantized values of 2-bit quantization are among the ones of 4-bit quantization. In fact, based on 2-bit quantization, the 4-bit counterpart introduces additional unshared quantized values. In particular, if $z _ { 2 }$ has zero residual error, then 4-bit quantization maps $z$ to the shared quantized values (i.e., $z _ { 2 }$ ). In contrast, if $z _ { 2 }$ is with non-zero residual error, 4-bit quantization is likely to map $z$ to the unshared quantized values. In this case, 4-bit quantization can be regarded as performing quantized value re-assignment based on $z _ { 2 }$ . Such a re-assignment process can be formulated as follows: + +$$ +z _ { 4 } = z _ { 2 } + \epsilon _ { 4 } , +$$ + +where $z _ { 4 }$ is the 4-bit quantized value and $\epsilon _ { 4 }$ is the re-assignment offset based on $z _ { 2 }$ . To ensure that the results of re-assignment fall into the unshared quantized values, the re-assignment offset $\epsilon _ { 4 }$ must be an integer multiplying of the 4-bit step size $s _ { 4 }$ . Formally, $\epsilon _ { 4 }$ can be computed by performing 4-bit quantization on the residual error of $z _ { 2 }$ : + +$$ +\epsilon _ { 4 } = D ( z - z _ { 2 } , s _ { 4 } ) , ~ s _ { 4 } = \frac { s _ { 2 } } { 2 ^ { 2 } + 1 } = \frac { 1 } { 2 ^ { 4 } - 1 } . +$$ + +Therefore, according to Eq. (9), a 4-bit quantized value can be decomposed into the 2-bit representation and its re-assignment offset. Similarly, an 8-bit quantized value can also be decomposed into the 4-bit representation and its corresponding re-assignment offset. In this way, we can generalize the idea of decomposition to arbitrary effective bitwidths as follows. + +Definition 1 (Quantization decomposition) Let $z \in [ 0 , 1 ]$ be a normalized full-precision input, $\big \{ b _ { 1 } , . . . , b _ { K } \big \}$ be a sequence of candidate bitwidths, and $b _ { 1 } ~ < ~ b _ { 2 } , . . . , < ~ b _ { K - 1 } < b _ { K }$ . We use the following quantized $\widehat { z }$ to approximate $z$ : + +$$ +{ \widehat { z } } = z _ { b _ { 1 } } + \sum _ { j = 2 } ^ { K } \epsilon _ { b _ { j } } , \quad { \mathrm { w h e r e ~ } } \epsilon _ { b _ { j } } = D ( z - z _ { b _ { j - 1 } } , s _ { b _ { j } } ) , \quad s _ { b _ { j } } = { \frac { s _ { b _ { j - 1 } } } { 2 ^ { b _ { j - 1 } } + 1 } } = { \frac { 1 } { 2 ^ { b _ { j } } - 1 } } . +$$ + +In other words, the quantized approximation $\widehat { z }$ can be decomposed into the sum of the lowest bit bquantization and a series of recursive re-assignment offsets. In Definition (1), to enable quantized value re-assignment, we need to constrain that $s _ { b _ { j - 1 } }$ is divisible by $s _ { b _ { j } }$ , which requires the bitwidths $b _ { j } ( j > 1 )$ to satisfy the following relation: + +$$ +b _ { j } = 2 ^ { j - 1 } \cdot b _ { 1 } . +$$ + +In fact, the bitwidth $b _ { 1 }$ can be set to arbitrary appropriate integer values (e.g., 1, 2, 3, etc.). To get a hardware-friendly compressed network1, we set $b _ { 1 }$ to 2, which ensures that all the decomposition bitwidths are power-of-two. Moreover, since 8-bit quantization achieves lossless performance compared with the full-precision counterpart (Zhou et al., 2016), we only consider those candidate bitwidths that are not greater than 8-bit. In other words, we constrain the value of $j$ to [1, 3]. + +Remark 1 The proposed bit sharing decomposition has several advantages. First, the proposed method only needs to maintain a small number of trainable parameters, which greatly reduces the computational costs during search. Second, we are able to directly extract a low-precision representation from its higher precision, which allows optimizing different bitwidths jointly and ease the discontinuous optimization due to quantization. + +# 3.3 LEARNING FOR COMPRESSION + +Note that different layers have different levels of redundancy, which indicates that different layers may choose different subsets of the quantized values. To learn the quantized approximation for each layer, we introduce a layer-wise binary quantization gate $g _ { b _ { j } } ^ { q } \in \{ 0 , 1 \}$ on each of the re-assignment offsets in Eq. (11) to encode the choice of the quantization bitwidth, which can be formulated as + +$$ +\begin{array} { c } { g _ { b _ { j } } ^ { q } = \mathbb { 1 } \left( | | z - z _ { b _ { j - 1 } } | | - \alpha _ { b _ { j } } ^ { q } > 0 \right) , } \\ { \widehat { z } = z _ { b _ { 1 } } + g _ { b _ { 2 } } ^ { q } \big ( \epsilon _ { b _ { 2 } } + \cdot \cdot \cdot + g _ { b _ { K - 1 } } ^ { q } \big ( \epsilon _ { b _ { K - 1 } } + g _ { b _ { K } } ^ { q } \epsilon _ { b _ { K } } \big ) \big ) , } \end{array} +$$ + +where $\mathbb { 1 } ( \cdot )$ is the indicator function and $\alpha _ { b _ { j } } ^ { q }$ is a layer-wise threshold that controls the choice of bitwidth. Specifically, if the quantization error $\left| \left| z \textrm { -- } z _ { b _ { j - 1 } } \right| \right|$ is greater than the threshold $\alpha _ { b _ { j } } ^ { q }$ , we activate the corresponding quantization gate to increase the bitwidth so that the residual error can be reduced, and vice versa. + +Note that from Eq. (13), we can consider the filter pruning as 0-bit filter-wise quantization. To avoid the prohibitively large filter-wise search space, we propose to divide the filters into groups based on indexes and consider the group-wise sparsity instead. To be specific, we introduce a binary gate $g _ { c } ^ { p }$ for each group to encode the choice of pruning, which can be formulated as follows: + +$$ +\begin{array} { c } { g _ { c } ^ { p } = \mathbb { 1 } ( | | w _ { c } | | - \alpha ^ { p } > 0 ) , } \\ { \widehat { z } _ { c } = g _ { c } ^ { p } \cdot \big ( z _ { c , b _ { 1 } } + g _ { b _ { 2 } } ^ { q } \big ( \epsilon _ { c , b _ { 2 } } + \cdot \cdot \cdot + g _ { b _ { K - 1 } } ^ { q } ( \epsilon _ { c , b _ { K - 1 } } + g _ { b _ { K } } ^ { q } \epsilon _ { c , b _ { K } } ) \big ) \big ) , } \end{array} +$$ + +where $\widehat { z } _ { c }$ is the $c$ -th group of quantized filters and $\epsilon _ { c , b _ { j } }$ is the corresponding re-assignment offset by bquantizing the residual error $z _ { c } - z _ { c , b _ { j - 1 } }$ . Here, $\alpha ^ { p }$ is a layer-wise threshold for filter pruning. Following PFEC (Li et al., 2017a), we use $\ell _ { 1 }$ -norm to evaluate the importance of the filter. Specifically, if a group of filters is important, the corresponding pruning gate will be activated and vice versa. + +Note that both quantization and pruning have their corresponding thresholds. Instead of manually setting the thresholds, we propose to learn them via gradient descent. However, the indicator function in Eq. (13) is non-differentiable. To address this, we use straight-through estimator (STE) (Bengio et al., 2013; Zhou et al., 2016) to approximate the gradient of the indicator function $\mathbb { 1 } ( \cdot )$ using the gradient of the sigmoid function $\sigma ( \cdot )$ , which can be formulated as: + +$$ +\frac { \partial g } { \partial \alpha } = \frac { \partial \mathbb { 1 } \left( A - \alpha \right) } { \partial \alpha } \approx \frac { \partial \sigma \left( A - \alpha \right) } { \partial \alpha } = - \sigma \left( A - \alpha \right) \left( 1 - \sigma ( A - \alpha ) \right) , +$$ + +where $g$ is the output of a binary gate, $\alpha \in \{ \alpha ^ { p } , \alpha ^ { q } \}$ is the corresponding threshold and $A$ denotes some specific metrics (i.e., $\ell _ { 1 }$ -norm of the filter or the quantization error). By jointly training the binary gates and the network parameters, the pruning ratio and bitwidth of each layer can be automatically determined. However, the gradient approximation of the binary gate inevitably introduces noisy signals, which can be even more severe when we quantize both weights and activations. Thus, we propose to train the binary gates of weights and activations in an alternative manner. Specifically, when training the binary gates of weights, we fix the binary gates of activations, and vice versa. + +Search Space for Model Compression. Given an uncompressed network with $L$ layers, we use $C _ { l }$ to denote the number of filters at the $l$ -th layer. To obtain the compressed model, we first divide the filters of each layer into groups and then search for the optimal bitwidths for the considered layer. Let $B$ be the number of filters in a group. For any layer $l$ , there would be $\displaystyle \left\lfloor \frac { C _ { l } } { B } \right\rfloor$ groups in total. Since we quantize both weights and activations, given $K$ candidate bitwidths, there are $K ^ { 2 }$ different quantization configurations for each layer. Thus, for the whole network with $L$ layers, the size of the search space $\Omega$ can be computed by + +$$ +\left| \Omega \right| = \prod _ { l = 1 } ^ { L } \left( K ^ { 2 } \times \left\lfloor \frac { C _ { l } } { B } \right\rfloor \right) . +$$ + +Eq. (16) indicates that the search space is large enough to cover the potentially good configurations. + +Training Objective. To design a hardware-efficient network, the objective function in Eq. (7) should reflect both the accuracy of the compressed network and its computational costs. Following (Cai et al., 2019), we train the network and architecture by minimizing following loss function: + +$$ +{ \mathcal { L } } ( \mathbf { W } , \alpha ^ { p } , \alpha ^ { q } ) = { \mathcal { L } } _ { c e } ( \mathbf { W } , \alpha ^ { p } , \alpha ^ { q } ) + \lambda \log R ( \mathbf { W } , \alpha ^ { p } , \alpha ^ { q } ) , +$$ + +where $\mathcal { L } _ { c e } ( \cdot )$ is the cross-entropy loss, $R ( \cdot )$ is the computational costs of the network and $\lambda$ is a balancing hyper-parameter. Following single-path NAS (Stamoulis et al., 2019), we use a similar formulation of computational costs to preserve the differentiability of the objective function. The details of the differentiable computational loss can be found in Appendix B. Once the training is finished, we can obtain the compressed network by selecting those filters and bitwidths with activated binary gates. Then, we fine-tune the compressed network to compensate the accuracy loss. + +Table 1: Comparisons of different methods on CIFAR-100. “W” and “A” represent the average quantization bitwidth of the weights and activations, respectively. + +
NetworkMethodBOPs (M)BOP comp.ratioWIATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-20Full-precision41798.61.032.0/32.067.590.8
4-bit precision674.662.04.0/4.067.8±0.390.4±0.2
DQ1180.035.45.3/6.167.7±0.690.4±0.5
HAQ653.464.03.7/4.267.7±0.190.4±0.3
DNAS660.062.94.6/3.867.8±0.390.4±0.2
ABS-P(Ours)28586.51.532.0/32.067.9±0.190.7±0.2
ABS-Q(Ours) ABS (Ours)649.564.44.4/4.268.1±0.190.5±0.0
630.666.34.4/4.268.1±0.390.6±0.2
ResNet-56Full-precision128771.71.032.0/32.071.792.2
4-bit precision2033.663.34.0/4.070.9±0.391.2±0.4
DQ2222.957.93.8 /4.670.7±0.291.4±0.4
HAQ2014.963.93.3/4.971.2±0.191.1±0.2
DNAS2035.765.35.3/3.271.2±0.291.3±0.3
ABS-P(Ours)87021.61.532.0/32.071.5±0.191.8±0.2
ABS-Q(Ours)1970.765.34.1/4.071.5±0.291.5±0.2
ABS (Ours)1918.867.14.2/4.171.6±0.191.8±0.4
+ +![](images/6c07349b87f9e6d672e29a547e2de1141f2b382554b25e27f434c7c0ff27512b.jpg) +Figure 2: Results of different compressed networks with different BOPs and memory footprints. We use different methods to compress ResNet-56 and report the results on CIFAR-100. + +# 4 EXPERIMENTS + +Compared methods. To investigate the effectiveness of the proposed method, we consider the following methods for comparisons: ABS: our proposed method with joint pruning and quantization; ABS-Q: ABS with quantization only; ABS-P: ABS with pruning only; and several state-ofthe-art model compression methods including HAQ (Wang et al., 2019), DQ (Uhlich et al., 2020), DJPQ (Ying et al., 2020), Bayesian Bits (van Baalen et al., 2020) and DNAS (Wu et al., 2018). We measure the performance of different methods in terms of the Top-1 and Top-5 accuracy. Following (Guo et al., 2020; Ying et al., 2020), we measure the computational costs by the Bit-Operation (BOP) count. The BOP compression ratio is defined as the ratio between the total BOPs of the uncompressed and compressed models. We can also measure the computational costs with the total weights and activations memory footprints following DQ (Uhlich et al., 2020). Moreover, following (Stamoulis et al., 2019; Liu et al., 2019a), we use the search cost to measure the time of finding an optimal compressed model. + +Implementation details. Following HAQ (Wang et al., 2019), we quantize all the layers, in which the first and the last layers are quantized to 8-bit. Following ThiNet (Luo et al., 2017), we only conduct filter pruning for the first layer in the residual block. For ResNet-20 and ResNet-56 on CIFAR-100 (Krizhevsky et al., 2009), we set $B$ to 4. For ResNet-18 and MobileNetV2 on ImageNet (Russakovsky et al., 2015), $B$ is set to 16 and 8, respectively. We first train the full-precision models and then use the pretrained weights to initialize the compressed models. Following (Li et al., 2020; Esser et al., 2020), we introduce weight normalization during training. We use SGD with nesterov (Nesterov, 1983) for optimization, with a momentum of 0.9. For CIFAR-100, we use the same data augmentation as in (He et al., 2016), including translation and horizontal flipping. For ImageNet, images are resized to $2 5 6 \times 2 5 6$ , and then a $2 2 4 \times 2 2 4$ patch is randomly cropped from an image or its horizontal flip for training. For testing, a $2 2 4 \times 2 2 4$ center cropped is chosen. We first train the uncompressed network for 30 epochs on CIFAR-100 and 10 epochs on ImageNet. + +Table 2: Comparisons on ImageNet. “\*” denotes that we get the results from the figures in (van Baalen et al., 2020) and “–” denotes that the results are not reported. Moreover, “W” and “A” represent the average quantization bitwidth of the weights and activations, respectively. + +
NetworkMethodBOPs (G)BOP comp. ratioW/ATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-18Full-precision1857.61.032.0/32.070.789.8
4-bit precision34.753.54.0/4.071.089.8
DQ40.740.61/168.51
DJPQ35.552.3-/169.1
HAQ34.753.54.0/4.070.289.5
Bayesian Bits*35.951.71/169.5
ABS-Q(Ours)33.156.14.5/3.870.989.7
ABS (Ours)32.357.54.6/4.270.889.6
MobileNetV2Full-precision308.01.032.0/32.071.990.3
6-bit precision11.227.56.0/6.071.890.3
DQ19.61.96.8/8.070.489.7
HAQ10.828.55.61/6.2771.290.0
Bayesian Bits *10.828.5-/170.9一
ABS-Q(Ours)10.928.36.8/6.871.890.4
ABS (Ours)10.828.56.1/7.171.790.3
+ +The learning rate is set to 0.001. We then fine-tune the searched compressed network to recover the performance drop. On CIFAR-100, we train the searched network for 200 epochs with a mini-batch size of 128. The learning rate is initialized to 0.1 and is divided by 10 at 80-th and 120-th epochs. Experiments on CIFAR-100 are repeated for 5 times and we report the mean and standard deviation. For ResNet-18 on ImageNet, we finetune the searched network for 90 epochs with a mini-batch size of 256. For MobileNetV2 on ImageNet, we fine-tune for 150 epochs. For all models on ImageNet, the learning rate starts at 0.01 and decays with cosine annealing (Loshchilov & Hutter, 2017). + +# 4.1 MAIN RESULTS + +We apply the proposed methods to compress ResNet-20, ResNet-56 on CIFAR-100 and ResNet18, MobileNetV2 on ImageNet. We compare the performance of different methods in Table 1 and Table 2. We also show the results of the compressed ResNet-56 with different BOPs and memory footprints in Figure 2. From the results, we can see that 4-bit quantized networks achieve lossless performance. Also, 6-bit MobileNetV2 only leads to a $0 . 1 \%$ performance drop on the Top-1 Accuracy. Compared with fixed-precision quantization, mixed-precision methods are able to reduce the BOPs while preserving the performance. Critically, our proposed ABS-Q outperforms the stateof-the-arts baselines with less computational costs. Specifically, ABS-Q compressed ResNet-18 outperforms the one compressed by HAQ with more BOPs reduction. More critically, our proposed ABS achieves significant improvement in terms of BOPs and memory footprints. For example, in Figure 2(a), our ABS compressed ResNet-56 model yields much fewer BOPs (395.25 vs. 536.24) but achieves comparable performance compared with the fixed-precision counterpart. Moreover, by combing pruning and quantization, ABS achieves nearly lossless performance while further reducing the computational costs of ABS-Q. For example, ABS compressed ResNet-18 reduces the BOPs by $5 7 . 5 \times$ while still outperforming the full-precision network by $0 . 1 \%$ in terms of the Top-1 accuracy on ImageNet. + +# 4.2 FURTHER STUDIES + +Effect of the bit-sharing scheme. To investigate the effect of the bit-sharing scheme, we apply our methods to quantize ResNet-20 and ResNet-56 with and without the bit sharing scheme on CIFAR100. We report the testing accuracy and BOPs in Table 3. We also present the search costs and consumed GPU memory measured on a GPU device (NVIDIA TITAN $\mathrm { X p }$ ). It can be seen from the results that the method with the bit sharing scheme consistently outperforms the ones without the bit sharing scheme while significantly reducing the search costs and GPU memory. + +Effect of the one-stage compression. To investigate the effect of the one-stage compression scheme (perform pruning and quantization jointly), we extend ABS to two-stage optimization, where we sequentially do filter pruning and quantization, denoted as ABS- $\mathbf { \partial } \cdot \mathbf { P } { } \mathbf { A } \mathbf { B } \mathbf { S } { - } \mathbf { Q }$ . The results are shown in Table 4. Compared with the two-stage counterpart, ABS achieves better performance with less computational costs, which shows the superiority of the one-stage optimization. For example, ABS compressed ResNet-56 outperforms the counterpart by $0 . 4 \%$ on the Top-1 accuracy with less computational overhead. + +Table 3: Effect of the bit-sharing scheme. We report the testing accuracy, BOPs, and search costs on CIFAR-100. The search costs are measured on a GPU device (NVIDIA TITAN Xp). + +
NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)Search Cost (GPU hours)GPUMemory (GB)
ResNet-20 w/o bit sharing67.8±0.190.5±0.2664.22.84.4
w/bit sharing68.1±0.190.5±0.0649.50.81.5
ResNet-56w/o bit sharing71.3±0.391.4±0.42001.18.710.9
w/ bit sharing71.5±0.291.5±0.21970.71.93.0
+ +Table 4: Effect of the one-stage compression. We report the results of ResNet-56 on CIFAR-100. + +
NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56ABS-P→ABS-Q70.4±0.190.8±0.21077.7
ABS70.8±0.491.2±0.11042.5
+ +Effect of the alternative training scheme. To investigate the effect of the alternative training scheme introduced in Section 3.3, we apply our method to compress ResNet-56 using a joint training scheme and an alternative training scheme on CIFAR-100. Here, the joint training scheme denotes that we train the binary gates of weights and activations jointly. From the results of Table 5, the model trained with the alternative scheme achieves better performance than the joint one, which demonstrates the effectiveness of the alternative training scheme. + +Table 5: Effect of the alternative training scheme. We report the results of ResNet-56 on CIFAR-100. + +
NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56Joint71.3±0.291.6±0.31942.4
Alternative71.6±0.191.8±0.41918.8
+ +Resource-constrained compression. To demonstrate the effectiveness of our ABS on hardware devices, we further apply our methods to compress MobileNetV2 under the resource constraints on the BitFusion architecture (Sharma et al., 2018). Instead of using BOPs, we use the latency and energy on a simulator of the BitFusion to measure the computational costs. We report the results in Table 6. Compared with fixed-precision quantization, ABS achieves better performance with lower latency and energy. Specifically, ABS compressed MobileNetV2 with much lower latency and energy even outperforms 6-bit MobileNetV2 by $0 . 2 \%$ in the Top-1 accuracy. + +Table 6: Resource-constrained compression on BitFusion. We evaluate the proposed ABS under the latency- and energy-constrained and report the Top-1 and Top-5 accuracy on ImageNet. + +
NetworkMethodLatency-constrainedEnergy-constrained
Acc.-1 (%)Acc.-5 (%)Latency (ms)Acc.-1 (%)Acc.-5 (%)Energy (mJ)
MobileNetV26-bit precision71.890.324.971.890.332.8
ABS (Ours)72.090.417.272.090.326.3
+ +# 5 CONCLUSION AND FUTURE WORK + +In this paper, we have proposed a novel model compression method called Automatically Bit Sharing (ABS). Specifically, our ABS is based on the observation that quantized values of a high bitwidth share the ones of lower bitwidths under some constraints. We therefore have proposed the decomposition of quantization that encapsulates all candidate bitwidths. Starting from a low bitwidth in the search space, we sequentially increase the effective bitwidth by recursively adding re-assignment offsets. Based on this, we have further introduced learnable binary gates to encode the choice of different compression policies. By training the binary gates, the optimal compression ratio of each layer can be automatically determined. Experiments on CIFAR-100 and ImageNet have shown that our methods are able to achieve significant cost reduction while preserving the performance. In the future, we plan to work on a joint search for architecture, pruning, and quantization to find a compact model with better performance. + +# REFERENCES + +Yu Bai, Yu-Xiang Wang, and Edo Liberty. Proxquant: Quantized neural networks via proximal operators. In Proc. Int. Conf. Learn. Repren., 2019. + +Yoshua Bengio, Nicholas Leonard, and Aaron Courville. Estimating or propagating gradients ´ through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013. + +Yash Bhalgat, Jinwon Lee, Markus Nagel, Tijmen Blankevoort, and Nojun Kwak. Lsq $^ +$ : Improving low-bit quantization through learnable offsets and better initialization. In Proc. IEEE Conf. Comp. Vis. Patt. Recogn., 2020. + +Han Cai, Ligeng Zhu, and Song Han. ProxylessNAS: Direct neural architecture search on target task and hardware. In Proc. Int. Conf. Learn. Repren., 2019. + +Zhaowei Cai and Nuno Vasconcelos. Rethinking differentiable search for mixed-precision neural networks. In Proc. 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Structured binary neural networks for accurate image classification and semantic segmentation. In Proc. IEEE Conf. Comp. Vis. Patt. Recogn., 2019. + +Bohan Zhuang, Lingqiao Liu, Mingkui Tan, Chunhua Shen, and Ian Reid. Training quantized neural networks with a full-precision auxiliary module. In Proc. IEEE Conf. Comp. Vis. Patt. Recogn., 2020. + +Zhuangwei Zhuang, Mingkui Tan, Bohan Zhuang, Jing Liu, Yong Guo, Qingyao Wu, Junzhou Huang, and Jinhui Zhu. Discrimination-aware channel pruning for deep neural networks. In Proc. Adv. Neural Inf. Process. Syst., 2018b. + +# Appendix for ABS: Automatic Bit Sharing for Model Compression + +A HARDWARE-FRIENDLY DECOMPOSITION + +As mentioned in Sec. 3.2, $b _ { 1 }$ can be set to arbitrary appropriate integer values (e.g., 1, 2, 3, etc.). By default, we set $b _ { 1 } = 2$ for better hardware utilization. On general purpose computing devices (e.g., CPU, GPU), byte (8 bits) is the lowest data type for operations. Other data types and ALU registers are all composed with multiple bytes in width. By setting $b _ { 1 } = 2$ , 2-bit/ 4-bit/ 8-bit quantization values can be packed into byte (or short, int, long) data type without bit wasting. Otherwise, if $b _ { 1 } = 1$ or $b _ { 1 } = 3$ , it is inevitable to have wasted bits when packing mixed-precision quantized tensors on general purpose devices. For example, one 32-bit int data type can be used to store ten 3-bit quantized values with 2 bits wasted. One might argue that these 2 bits can be leveraged with the next group of 3-bit data, but it will result in irregular memory access patterns, which will degrade the hardware utilization more seriously. Moreover, 8-bit quantization has been demonstrated to own similar performance with the full precision counterparts for many networks. Therefore, there is no need to consider a bitwidth larger than 8. + +# B FORMULATION OF DIFFERENTIABLE COMPUTATIONAL LOSS + +In this section, we introduce the differentiable computational loss mentioned in Section 3.3. Unlike the cross-entropy loss in Eq. (17), the computational costs $R ( \mathbf { W } , \alpha ^ { p } , \alpha ^ { q } )$ is non-differentiable. To solve this issue, we model the computational costs as a function of binary gates as: + +$$ +R ( \mathbf { W } , \alpha ^ { p } , \alpha ^ { q } ) = \sum _ { c = 1 } ^ { G } g _ { c } ^ { p } \left( R _ { x _ { c , b _ { 1 } } } + g _ { b _ { 2 } } ^ { q } \left( R _ { x _ { c , b _ { 2 } } } - R _ { x _ { c , b _ { 1 } } } + \cdot \cdot + \cdot + g _ { b _ { K } } ^ { q } \left( R _ { x _ { c , b _ { K } } } - R _ { x _ { c , b _ { K - 1 } } } \right) \right) \right) , +$$ + +where Rxc,b is the computational cost for the $c$ -th group of filters with $b _ { j }$ -bit quantization and $G$ is the number of groups in total. + +# C QUANTIZATION CONFIGURATIONS + +All the methods in Tables 1 and 2 use layer-wise and symmetric quantization schemes and the compared methods strictly follow the quantization configurations in their original papers. Specifically, for DQ (Uhlich et al., 2020), we parameterize the fixed-point quantizer using case U3 with $\theta = [ d , q _ { \mathrm { m a x } } ]$ . We initialize the weights using a pre-trained model. The initial step size is set to $d = \bar { 2 } ^ { \lfloor \log _ { 2 } ( \operatorname* { m a x } ( | \mathbf { W } | ) / ( 2 ^ { b - 1 } - 1 ) \rfloor }$ for weights and $2 ^ { - 3 }$ for activations. The remaining quantization parameters are set such that the initial bitwidth is 4-bit. For HAQ (Wang et al., 2019), we first truncate the weights and activations into the range of $[ - v _ { w } , v _ { w } ]$ and $[ 0 , v _ { x } ]$ , respectively. We then perform linear quantization for both weights and activations. To find more proper $v _ { w }$ and $v _ { x }$ , we minimize the KL-divergence between the original weight distribution W and the quantized weight distribution $Q ( \mathbf { W } )$ . For DNAS (Wu et al., 2018), we follow DoReFa-Net (Zhou et al., 2016) to quantize weights and follow PACT (Choi et al., 2018) to quantize activations. We initialize the learnable upper bound to 1. For DJPQ (Ying et al., 2020) and Bayesian Bits (van Baalen et al., 2020), we directly get the results from original papers. For other methods in Tables 1 and 2, we use the quantization function introduced in Section 3.1. The trainable quantization intervals $v _ { x }$ and $v _ { w }$ are initialized to 1. + +# D SEARCH COST COMPARISONS + +To evaluate the efficiency of the proposed ABS, we compare the search cost of different methods and report the results in Table 7. From the results, the search costs of the proposed ABS is much smaller than the state-of-the-art methods. Moreover, compared with ABS-Q, ABS only introduces a small amount of computational overhead, which demonstrates the efficiency of the proposed methods. + +Table 7: Comparisons of the search costs on CIFAR-100. The search costs are measured on a GPU device (NVIDIA TITAN Xp). + +
NetworkMethodSearch Cost (GPU hours)
ResNet-20HAQ DQ5.8 3.0
DNAS2.8
ABS-Q (Ours) ABS-P (Ours) ABS (Ours)0.8 0.2
+ +Table 8: Comparisons of different methods w.r.t. memory footprints. We compress ResNet-56 using different methods and report the results on CIFAR-100. + +
MethodMemory footprints(KB)M.f. comp. ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision5653.41.071.792.2
4-bit precision711.77.970.9±0.391.2±0.4
DNAS708.98.071.5±0.291.3±0.1
HAQ700.08.171.3±0.191.1±0.1
ABS-Q (Ours)674.58.471.5±0.291.6±0.2
ABS (Ours)657.38.671.6±0.191.8±0.4
+ +Table 9: Comparisons of different methods with MobileNetV3 on CIFAR-100. + +
MethodBOPs (M)BOP comp.ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision68170.11.076.193.9
6-bit precision2412.628.376.1±0.093.7±0.0
DQ2136.331.975.9±0.193.7±0.1
HAQ2191.731.176.1±0.193.5±0.0
DNAS2051.933.276.1±0.193.7±0.1
ABS-P(Ours)59465.81.176.0±0.093.5±0.0
ABS-Q(Ours)2021.933.776.1±0.193.7±0.1
ABS (Ours)2006.634.076.1±0.193.7±0.1
+ +# E MORE RESULTS ON MEMORY FOOTPRINTS + +To further demonstrate the effectiveness of the proposed ABS, we replace BOPs with total weights and activations memory footprints (Uhlich et al., 2020). We apply different methods to compress ResNet-56 and report the results in Table 8. From the results, ABS compressed ResNet-56 outperforms other methods with fewer memory footprints. These results show the effectiveness of our proposed ABS in terms of memory footprints reduction. + +# F DETAILED STRUCTURE OF THE COMPRESSED NETWORK + +We illustrate the pruning rate and bitwidth of each layer’s weights and activations of the compressed ResNet-18 and MobileNetV2 in Figure 3 and Figure 4, respectively. From the results, we observe that our ABS assigns more bitwidths to the weights in the downsampling convolutional layer in ResNet-18 and depthwise convolutional layer in MobileNetV2. Intuitively, this is because the number of parameters of these layers is much smaller than other layers. Moreover, our ABS inclines to prune more filters in the shallower layers, which can significantly reduce the number of parameters and computational overhead. Finally, we also observe that the correlation between the bitwidth and pruning rate is as follows. If a layer is set to a high pruning rate, our ABS tends to select a higher bitwidth to compensate for the performance drop. In contrast, if a layer is with a low pruning rate, our ABS tends to select a lower bitwidth to reduce the model size and computational costs. + +# G MORE RESULTS ON MOBILENETV3 + +To evaluate the proposed ABS on the lightweight model, we apply our methods to MobileNetV3 on CIFAR-100. Following $\mathrm { L S Q + }$ (Bhalgat et al., 2020), we introduce a learnable offset to handle the negative activations in hard-swish. We show the results in Table 9. From the results of MobileNetV3, our proposed ABS still outperforms the compared methods, which demonstrates its effectiveness. + +![](images/284f562ab05ccb482bd329fad017446ba815605269415a114484ae378c8d00c3.jpg) +(a) Bitwidth configuration of the compressed ResNet-18. + +![](images/d02d26b45dc7eafeecbf604a33ffcc12f1d0ec1a1c0956af5f33cccaa8b129c8.jpg) +Figure 3: Detailed configurations of the compressed ResNet-18. + +(b) Pruning rate configuration of the compressed ResNet-18. The pruning rate is defined as the ratio between #pruned weights of the compressed models and #weights of the uncompressed models. + +![](images/529215833fa76e311a61fb639d37fa31e0fc915f7db262f5a57182c66e854ec9.jpg) +Figure 4: Detailed configurations of the compressed MobileNetV2. + +(b) Pruning rate configuration of the compressed MobileNetV2. The pruning rate is defined as the ratio between #pruned weights of the compressed models and #weights of the uncompressed models. \ No newline at end of file diff --git a/parse/train/QjINdYOfq0b/QjINdYOfq0b_content_list.json b/parse/train/QjINdYOfq0b/QjINdYOfq0b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..092b5a87b64bb6dc9557e4554221771bf40daa16 --- /dev/null +++ b/parse/train/QjINdYOfq0b/QjINdYOfq0b_content_list.json @@ -0,0 +1,1967 @@ +[ + { + "type": "text", + "text": "ABS: AUTOMATIC BIT SHARING FOR MODEL COMPRESSION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present Automatic Bit Sharing (ABS) to automatically search for optimal model compression configurations (e.g., pruning ratio and bitwidth). Unlike previous works that consider model pruning and quantization separately, we seek to optimize them jointly. To deal with the resultant large designing space, we propose a novel super-bit model, a single-path method, to encode all candidate compression configurations, rather than maintaining separate paths for each configuration. Specifically, we first propose a novel decomposition of quantization that encapsulates all the candidate bitwidths in the search space. Starting from a low bitwidth, we sequentially consider higher bitwidths by recursively adding reassignment offsets. We then introduce learnable binary gates to encode the choice of bitwidth, including filter-wise 0-bit for pruning. By jointly training the binary gates in conjunction with network parameters, the compression configurations of each layer can be automatically determined. Our ABS brings two benefits for model compression: 1) It avoids the combinatorially large design space, with a reduced number of trainable parameters and search costs. 2) It also averts directly fitting an extremely low bit quantizer to the data, hence greatly reducing the optimization difficulty due to the non-differentiable quantization. Experiments on CIFAR-100 and ImageNet show that our methods achieve significant computational cost reduction while preserving promising performance. ", + "bbox": [ + 233, + 266, + 764, + 530 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 558, + 336, + 574 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks (DNNs) have achieved great success in many challenging computer vision tasks, including image classification (Krizhevsky et al., 2012; He et al., 2016) and object detection (Lin et al., 2017a;b). However, a deep model usually has a large number of parameters and consumes huge amounts of computational resources, which remains great obstacles for many applications, especially on resource-limited devices with limited memory and computational resources, such as smartphones. To reduce the number of parameters and computational overhead, many methods (He et al., 2019; Zhou et al., 2016) have been proposed to conduct model compression by removing the redundancy while maintaining the performance. ", + "bbox": [ + 174, + 589, + 823, + 702 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In the last decades, we have witnessed a lot of model compression methods, such as network pruning (He et al., 2017; 2019) and quantization (Zhou et al., 2016; Hubara et al., 2016). Specifically, network pruning reduces the model size and computational costs by removing redundant modules while network quantization maps the full-precision values to low-precision ones. It has been shown that sequentially perform network pruning and quantization is able to get a compressed network with small model size and lower computational overhead (Han et al., 2016). However, performing pruning and quantization in a separate step may lead to sub-optimal results. For example, the best quantization strategy for the uncompressed network is not necessarily the optimal one after network pruning. Therefore, we need to consider performing pruning and quantization simultaneously. ", + "bbox": [ + 174, + 708, + 823, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, many attempts have been made to automatically determine the compression configurations of each layer (i.e., pruning ratios, and/or bitwidths), either based on reinforcement learning (RL) (Wang et al., 2019), evolutionary search (ES) (Wang et al., 2020), Bayesian optimization (BO) (Tung & Mori, 2018) or differentiable methods (Wu et al., 2018; Dong & Yang, 2019). In particular, previous differentiable methods formulate model compression as a differentiable searching problem to explore the search space using gradient-based optimization. As shown in Figure 1(a), each candidate operation is maintained as a separate path, which leads to a huge number of trainable parameters and high computational overhead when the search space becomes combinatorially large. Moreover, due to the non-differentiable quantizer and pruning process, the optimization of heavily compressed candidate networks can be more challenging than that in the conventional search problem. ", + "bbox": [ + 174, + 840, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/d661d86f6402cd19d79021a39a0281ea92dacd0285f0d8f2e32e077e497f6529.jpg", + "image_caption": [ + "Figure 1: Multi-path v.s. single-path compression scheme. (a) Multi-path search scheme (Wu et al., 2018): represents each candidate configuration as a separate path and formulates the compression problem as a path selection problem, which gives rise to huge numbers of trainable parameters and high computational overhead when the search space becomes combinatorially large. Here, $z _ { k }$ is the $k$ -bit quantized version of $z$ and $\\alpha _ { k } ^ { q }$ is the architecture parameters corresponding to the path of $k$ -bit quantization. (b) Single-path search scheme (Ours): represents each candidate configuration as a subset of a “super-bit” and formulates the compression problem as a subset selection problem, which greatly reduces the computational costs and optimization difficulty from the discontinuity of quantization. Here, the super-bit denotes the highest bitwidth in the search space, $g _ { k } ^ { q }$ is a binary gate that controls the decision of bitwidth, and $\\epsilon _ { k }$ is the re-assignment offset (quantized residual error). " + ], + "image_footnote": [], + "bbox": [ + 181, + 99, + 816, + 287 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 460, + 825, + 516 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we propose a simple yet effective model compression method named Automatic Bit Sharing (ABS) to reduce the search cost and ease the optimization for the compressed candidates. Inspired by recent single-path neural architecture search (NAS) methods (Stamoulis et al., 2019; Guo et al., 2020), the proposed ABS introduces a novel single-path super-bit to encode all effective bitwidths in the search space instead of formulating each candidate operation as a separate path, as shown in Figure 1(b). Specifically, we build upon the observation that the quantized values of a high bitwidth can share the ones of low bitwidths under some conditions. Therefore, we are able to decompose the quantized representation into the sum of the lowest bit quantization and a series of re-assignment offsets. We then introduce learnable binary gates to encode the choice of bitwidth, including filter-wise 0-bit for pruning. By jointly training the binary gates and network parameters, the compression ratio of each layer can be automatically determined. The proposed scheme has several advantages. First, we only need to solve the search problem as finding which subset of the super-bit to use for each layer’s weights and activations rather than selecting from different paths. Second, we enforce the candidate bitwidths to share the common quantized values. Hence, we are able to optimize them jointly instead of separately, which greatly reduces the optimization difficulty from the discontinuity of discretization. ", + "bbox": [ + 173, + 523, + 825, + 744 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our main contributions are summarized as follows: ", + "bbox": [ + 174, + 752, + 509, + 766 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We devise a novel super-bit scheme that encapsulates multiple compression configurations in a unified single-path framework. Relying on the super-bit scheme, we further introduce learnable binary gates to determine the optimal bitwidths (including filter-wise 0-bit for \npruning). The proposed ABS casts the search problem as subset selection problem, hence significantly reducing the search cost. \n• We formulate the quantized representation as a gated combination of the lowest bitwidth quantization and a series of re-assignment offsets, in which we explicitly share the quantized values between different bitwidths. In this way, we enable the candidate operations to learn jointly rather than separately, hence greatly easing the optimization, especially in the non-differentiable quantization scenario. ", + "bbox": [ + 217, + 779, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We evaluate our ABS on CIFAR-100 and ImageNet over various network architectures. Extensive experiments show that the proposed method achieves the state-of-the-art performance. For example, on ImageNet, our ABS compressed MobileNetV2 achieves $2 8 . 5 \\times$ Bit-Operation (BOP) reduction with only $0 . 2 \\%$ performance drop on the Top-1 accuracy. ", + "bbox": [ + 218, + 103, + 823, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 186, + 339, + 203 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Network quantization. Network quantization represents the weights, activations and even gradients in low-precision to yield compact DNNs. With low-precision integers or power-of-two representations, the heavy matrix multiplications can be replaced by efficient bitwise operations, leading to much faster test-time inference and lower power consumption. To improve the quantization performance, current methods either focus on designing accurate quantizers by fitting the quantizer to the data (Jung et al., 2019; Zhang et al., 2018; Choi et al., 2018; Cai et al., 2017), or seek to approximate the gradients due to the non-differentiable discretization (Ding et al., 2019; Louizos et al., 2019; Zhuang et al., 2020). Moreover, most previous works assign the same bitwidth for all layers (Zhou et al., 2016; Zhuang et al., 2018a; 2019; Jung et al., 2019; Jin et al., 2019; Li et al., 2020; Esser et al., 2020). Though attractive for simplicity, setting a uniform precision places no guarantee on optimizing network performance since different layers have different redundancy and arithmetic intensity. Therefore, several studies proposed mixed-precision quantization (Wang et al., 2019; Dong et al., 2019; Wu et al., 2018; Uhlich et al., 2020) to set different bitwidths according to the redundancy of each layer. In this paper, based on the proposed quantization decomposition, we devise an approach that can effectively learn appropriate bitwidths for each layer through gradient-based optimization. ", + "bbox": [ + 173, + 222, + 825, + 431 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "NAS and pruning. Neural architecture search (NAS) aims to automatically design efficient architectures with low model size and computational costs, either based on reinforcement learning (Pham et al., 2018; Guo et al., 2019), evolutionary search (Real et al., 2019) or gradient-based methods (Liu et al., 2019a). In particular, gradient-based NAS has gained increased popularity, where the search space can be divided into the multi-path design (Liu et al., 2019a; Cai et al., 2019) and single-path formulation (Stamoulis et al., 2019; Guo et al., 2020), depending on whether adding each operation as a separate path or not. While prevailing NAS methods optimize the network topology, the focus of this paper is to search optimal compression ratios for a given architecture. Moreover, network pruning can be treated as fine-grained NAS, which aims at removing redundant modules to accelerate the run-time inference speed, giving rise to methods based on unstructured weight pruning (Han et al., 2016; Guo et al., 2016) or structured channel pruning (He et al., 2017; Zhuang et al., 2018b; Luo et al., 2017). Based on channel pruning, our paper further takes quantization into consideration to generate more compact networks. ", + "bbox": [ + 173, + 438, + 825, + 617 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "AutoML for model compression. Recently, much effort has been put into automatically determining either the optimal pruning rate (Tung & Mori, 2018; Dong & Yang, 2019; He et al., 2018), or the bitwidth (Lou et al., 2019; Cai & Vasconcelos, 2020) of each layer via hyper-parameter search, without relying on heuristics. In particular, HAQ (Wang et al., 2019) employs reinforcement learning to search bitwidth strategies with the hardware accelerator’s feedback. Meta-pruning (Liu et al., 2019b) uses meta-learning to generate the weight parameters of the pruned networks and then adopts an evolutionary search algorithm to find the layer-wise sparsity for channel pruning. More recently, several studies (Wu et al., 2018; Cai & Vasconcelos, 2020) have focused on using differentiable schemes via gradient-based optimization. ", + "bbox": [ + 174, + 625, + 825, + 750 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Closely related methods. To further improve the compression ratio, several methods propose to jointly optimize pruning and quantization strategies. In particular, some works only support weight quantization (Tung & Mori, 2018; Ye et al., 2019) or use fine-grained pruning (Yang et al., 2020). However, the resultant networks cannot be implemented efficiently on edge devices. Recently, several methods (Wu et al., 2018; Wang et al., 2020; Ying et al., 2020) have been proposed to consider filter pruning, weight quantization, and activation quantization jointly. In contrast to these methods, we carefully design the compression search space by sharing the quantized values between different candidate configurations, which significantly reduces the search cost and eases the optimization. Compared with those methods that share the similarities of using quantized residual errors (Chen et al., 2010; Gong et al., 2014; Li et al., 2017b; van Baalen et al., 2020), our proposed method recursively uses quantized residual errors to decompose a quantized representation as a set of candidate bitwidths and parameterize the selection of optimal bitwidth via binary gates. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our proposed ABS and Bayesian Bits (van Baalen et al., 2020) are developed concurrently that share a similar idea of quantization decomposition. Critically, our ABS differs from Bayesian Bits in several aspects: 1) The quantization decomposition in our methods can be extended to non-powerof-two bit widths (i.e., $b _ { 1 }$ can be set to arbitrary appropriate integer values), which is a general case of the one in Bayesian Bits. 2) The optimization problems are different. Specifically, we formulate model compression as a single-path subset selection problem while Bayesian Bits casts the optimization of the binary gates to a variational inference problem that requires more relaxations and hyperparameters. 3) Our compressed models with less or comparable BOPs outperform those of Bayesian Bits by a large margin on ImageNet (See Table 2). ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 PROPOSED METHOD ", + "text_level": 1, + "bbox": [ + 176, + 248, + 377, + 265 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 PRELIMINARY: NORMALIZATION AND QUANTIZATION FUNCTION ", + "text_level": 1, + "bbox": [ + 176, + 279, + 668, + 295 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Without loss of generality, given a convolutional layer, let $x$ and $w$ be the activations of the last layer and its weight parameters, respectively. First, for convenience, following (Choi et al., 2018; Bai et al., 2019), we can normalize $x$ and $w$ into scale [0, 1] by $T _ { x }$ and $T _ { w }$ , respectively: ", + "bbox": [ + 174, + 305, + 825, + 348 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2573da2b5b5d827baac8354428914e1b25eb057d4ddb055e96e4f18735945829.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { z _ { x } = T _ { x } ( x ) = \\mathrm { c l i p } \\left( \\frac { x } { v _ { x } } , 0 , 1 \\right) , } } \\\\ { \\displaystyle { z _ { w } = T _ { w } ( w ) = \\frac { 1 } { 2 } \\left( \\mathrm { c l i p } \\left( \\frac { w } { v _ { w } } , - 1 , 1 \\right) + 1 \\right) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 352, + 650, + 421 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the function $\\mathrm { c l i p } \\left( v , v _ { \\mathrm { l o w } } , v _ { \\mathrm { u p } } \\right) = \\mathrm { m i n } ( \\mathrm { m a x } ( v , v _ { \\mathrm { l o w } } ) , v _ { \\mathrm { u p } } )$ clips any number $v$ into the range $[ v _ { \\mathrm { l o w } } , v _ { \\mathrm { u p } } ]$ , and $v _ { x }$ and $v _ { w }$ are trainable quantization intervals which indicate the range of weights and activations to be quantized. Then, we can apply the following function to quantize the normalized activations and parameters, namely $z _ { x } \\in [ 0 , 1 ]$ and $z _ { w } \\in [ 0 , 1 ]$ , to discretized ones: ", + "bbox": [ + 174, + 424, + 825, + 481 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c7b94efe6892e1eaffb3328fea85965e5da9ef48d151d2d456531ac41eec5a7d.jpg", + "text": "$$\nD ( z , s ) = s \\cdot { \\mathrm { r o u n d } } \\left( { \\frac { z } { s } } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 410, + 484, + 588, + 512 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where round $( \\cdot )$ returns the nearest integer of a given value and $s$ denotes the normalized step size. Typically, for $k$ -bit quantization, the normalized step size $s$ can be computed by ", + "bbox": [ + 169, + 515, + 821, + 544 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/080af8693f3bf1c61c28f4ece25b83a78b3de46f3fdf692d93ed31b3a17a6e46.jpg", + "text": "$$\ns = { \\frac { 1 } { 2 ^ { k } - 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 455, + 546, + 542, + 577 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "After doing the $k$ -bit quantization, we shall have $2 ^ { k } - 1$ quantized values. Specifically, we obtain the quantization $Q ( w )$ and $Q ( x )$ by ", + "bbox": [ + 176, + 580, + 823, + 611 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/de34fed150bebc5a2f2a8fec9383cbd2e94cc6bcbefe5909e35923da5d5d3c9b.jpg", + "text": "$$\n\\begin{array} { r l } & { Q ( w ) = { \\cal T } _ { w } ^ { - 1 } ( D ( z _ { w } , s ) ) = v _ { w } \\cdot ( 2 \\cdot D ( z _ { w } , s ) - 1 ) , } \\\\ & { ~ Q ( x ) = { \\cal T } _ { x } ^ { - 1 } ( D ( z _ { x } , s ) ) = v _ { x } \\cdot D ( z _ { x } , s ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 613, + 668, + 655 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $T _ { w } ^ { - 1 }$ and $T _ { x } ^ { - 1 }$ denote the inverse functions of $T _ { w }$ and $T _ { x }$ , respectively. ", + "bbox": [ + 176, + 657, + 676, + 672 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 BIT SHARING DECOMPOSITION ", + "text_level": 1, + "bbox": [ + 174, + 688, + 433, + 703 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Previous methods consider different compression configurations as different paths and reformulate model compression as a path selection problem, which gives rise to a huge number of trainable parameters and high computational costs. In this paper, we seek to conduct filter pruning and quantization simultaneously by solving the following problem: ", + "bbox": [ + 173, + 714, + 823, + 770 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/acdc9ee7f8532885cb70959a5f2d7dcdee3d75c6506976b74d9dc8d67715230d.jpg", + "text": "$$\n\\begin{array} { r } { \\underset { \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } } { \\operatorname* { m i n } } \\mathcal { L } \\left( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 416, + 773, + 581, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathcal { L } ( \\cdot )$ denotes some losses, and $\\mathbf { W }$ is the parameters of the network. $\\alpha ^ { p }$ and $\\alpha ^ { q }$ are the pruning and quantization configurations, respectively. As shown in Eq. (7), we propose to encode all compression configurations in a single-path super-bit model (See Figure 1(b)). In the following, we first introduce the bit sharing decomposition and then describe how to learn for compression. ", + "bbox": [ + 174, + 803, + 825, + 859 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To illustrate the bit sharing decomposition, we begin with an example of 2-bit quantization for $z \\in \\{ z _ { x } , z _ { w } \\}$ . Specifically, we consider using the following equation to quantize $z$ to 2-bit: ", + "bbox": [ + 171, + 864, + 821, + 895 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/feff15567b30f65afec1b3655c3c5609d798002c2b467c341bbd4fb7e0cf0d06.jpg", + "text": "$$\nz _ { 2 } = D ( z , s _ { 2 } ) , \\quad s _ { 2 } = \\frac { 1 } { 2 ^ { 2 } - 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 392, + 897, + 604, + 928 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $z _ { 2 }$ and $s _ { 2 }$ are the quantized value and the step size of 2-bit quantization, respectively. Due to the large step size, the residual error $z - z _ { 2 } \\in [ - s _ { 2 } / 2 , s _ { 2 } / 2 ]$ may be big and result in a significant performance drop. To reduce the residual error, an intuitive way is to use a smaller step size, which indicates that we quantize $z$ to a higher bitwidth. Since the step size $s _ { 4 } = 1 / ( 2 ^ { 4 } - 1 )$ in 4-bit quantization is a divisor of the step size $s _ { 2 }$ in 2-bit quantization, the quantized values of 2-bit quantization are among the ones of 4-bit quantization. In fact, based on 2-bit quantization, the 4-bit counterpart introduces additional unshared quantized values. In particular, if $z _ { 2 }$ has zero residual error, then 4-bit quantization maps $z$ to the shared quantized values (i.e., $z _ { 2 }$ ). In contrast, if $z _ { 2 }$ is with non-zero residual error, 4-bit quantization is likely to map $z$ to the unshared quantized values. In this case, 4-bit quantization can be regarded as performing quantized value re-assignment based on $z _ { 2 }$ . Such a re-assignment process can be formulated as follows: ", + "bbox": [ + 173, + 103, + 825, + 256 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/914ddc0fcb66dfa7a994e7f4cbeb637f164debdfb8f0ff4345e1fafe1a1e463e.jpg", + "text": "$$\nz _ { 4 } = z _ { 2 } + \\epsilon _ { 4 } ,\n$$", + "text_format": "latex", + "bbox": [ + 452, + 260, + 545, + 275 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $z _ { 4 }$ is the 4-bit quantized value and $\\epsilon _ { 4 }$ is the re-assignment offset based on $z _ { 2 }$ . To ensure that the results of re-assignment fall into the unshared quantized values, the re-assignment offset $\\epsilon _ { 4 }$ must be an integer multiplying of the 4-bit step size $s _ { 4 }$ . Formally, $\\epsilon _ { 4 }$ can be computed by performing 4-bit quantization on the residual error of $z _ { 2 }$ : ", + "bbox": [ + 173, + 276, + 825, + 333 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/40a85ea4827df5a7f1402b5df91135764a8572f329a4aae1266af84f9f000bbb.jpg", + "text": "$$\n\\epsilon _ { 4 } = D ( z - z _ { 2 } , s _ { 4 } ) , ~ s _ { 4 } = \\frac { s _ { 2 } } { 2 ^ { 2 } + 1 } = \\frac { 1 } { 2 ^ { 4 } - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 333, + 658, + 364 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Therefore, according to Eq. (9), a 4-bit quantized value can be decomposed into the 2-bit representation and its re-assignment offset. Similarly, an 8-bit quantized value can also be decomposed into the 4-bit representation and its corresponding re-assignment offset. In this way, we can generalize the idea of decomposition to arbitrary effective bitwidths as follows. ", + "bbox": [ + 174, + 366, + 823, + 421 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 1 (Quantization decomposition) Let $z \\in [ 0 , 1 ]$ be a normalized full-precision input, $\\big \\{ b _ { 1 } , . . . , b _ { K } \\big \\}$ be a sequence of candidate bitwidths, and $b _ { 1 } ~ < ~ b _ { 2 } , . . . , < ~ b _ { K - 1 } < b _ { K }$ . We use the following quantized $\\widehat { z }$ to approximate $z$ : ", + "bbox": [ + 173, + 430, + 823, + 473 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/633098f514838f33081ed1e04b334ceb82f4b71b545ba123f44581a75f7b07e8.jpg", + "text": "$$\n{ \\widehat { z } } = z _ { b _ { 1 } } + \\sum _ { j = 2 } ^ { K } \\epsilon _ { b _ { j } } , \\quad { \\mathrm { w h e r e ~ } } \\epsilon _ { b _ { j } } = D ( z - z _ { b _ { j - 1 } } , s _ { b _ { j } } ) , \\quad s _ { b _ { j } } = { \\frac { s _ { b _ { j - 1 } } } { 2 ^ { b _ { j - 1 } } + 1 } } = { \\frac { 1 } { 2 ^ { b _ { j } } - 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 205, + 474, + 764, + 520 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In other words, the quantized approximation $\\widehat { z }$ can be decomposed into the sum of the lowest bit bquantization and a series of recursive re-assignment offsets. In Definition (1), to enable quantized value re-assignment, we need to constrain that $s _ { b _ { j - 1 } }$ is divisible by $s _ { b _ { j } }$ , which requires the bitwidths $b _ { j } ( j > 1 )$ to satisfy the following relation: ", + "bbox": [ + 173, + 526, + 823, + 584 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/bb664006b4710defac80587b2c5d3afe38f85df0df482cbc05e2781b2b779a62.jpg", + "text": "$$\nb _ { j } = 2 ^ { j - 1 } \\cdot b _ { 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 447, + 587, + 550, + 604 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In fact, the bitwidth $b _ { 1 }$ can be set to arbitrary appropriate integer values (e.g., 1, 2, 3, etc.). To get a hardware-friendly compressed network1, we set $b _ { 1 }$ to 2, which ensures that all the decomposition bitwidths are power-of-two. Moreover, since 8-bit quantization achieves lossless performance compared with the full-precision counterpart (Zhou et al., 2016), we only consider those candidate bitwidths that are not greater than 8-bit. In other words, we constrain the value of $j$ to [1, 3]. ", + "bbox": [ + 173, + 606, + 825, + 676 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Remark 1 The proposed bit sharing decomposition has several advantages. First, the proposed method only needs to maintain a small number of trainable parameters, which greatly reduces the computational costs during search. Second, we are able to directly extract a low-precision representation from its higher precision, which allows optimizing different bitwidths jointly and ease the discontinuous optimization due to quantization. ", + "bbox": [ + 173, + 683, + 825, + 753 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 LEARNING FOR COMPRESSION ", + "text_level": 1, + "bbox": [ + 176, + 768, + 426, + 782 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that different layers have different levels of redundancy, which indicates that different layers may choose different subsets of the quantized values. To learn the quantized approximation for each layer, we introduce a layer-wise binary quantization gate $g _ { b _ { j } } ^ { q } \\in \\{ 0 , 1 \\}$ on each of the re-assignment offsets in Eq. (11) to encode the choice of the quantization bitwidth, which can be formulated as ", + "bbox": [ + 174, + 794, + 825, + 853 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3a5ead49ac99e46e4fe8ada6614ccf92852f3f37ef9879db303981db00ecea64.jpg", + "text": "$$\n\\begin{array} { c } { g _ { b _ { j } } ^ { q } = \\mathbb { 1 } \\left( | | z - z _ { b _ { j - 1 } } | | - \\alpha _ { b _ { j } } ^ { q } > 0 \\right) , } \\\\ { \\widehat { z } = z _ { b _ { 1 } } + g _ { b _ { 2 } } ^ { q } \\big ( \\epsilon _ { b _ { 2 } } + \\cdot \\cdot \\cdot + g _ { b _ { K - 1 } } ^ { q } \\big ( \\epsilon _ { b _ { K - 1 } } + g _ { b _ { K } } ^ { q } \\epsilon _ { b _ { K } } \\big ) \\big ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 854, + 676, + 905 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathbb { 1 } ( \\cdot )$ is the indicator function and $\\alpha _ { b _ { j } } ^ { q }$ is a layer-wise threshold that controls the choice of bitwidth. Specifically, if the quantization error $\\left| \\left| z \\textrm { -- } z _ { b _ { j - 1 } } \\right| \\right|$ is greater than the threshold $\\alpha _ { b _ { j } } ^ { q }$ , we activate the corresponding quantization gate to increase the bitwidth so that the residual error can be reduced, and vice versa. ", + "bbox": [ + 173, + 102, + 825, + 165 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that from Eq. (13), we can consider the filter pruning as 0-bit filter-wise quantization. To avoid the prohibitively large filter-wise search space, we propose to divide the filters into groups based on indexes and consider the group-wise sparsity instead. To be specific, we introduce a binary gate $g _ { c } ^ { p }$ for each group to encode the choice of pruning, which can be formulated as follows: ", + "bbox": [ + 173, + 171, + 825, + 229 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/a6585f6d9ffddc4613cbc5663b3c97eacad3a67b7f424952bd88029e09f6d1e9.jpg", + "text": "$$\n\\begin{array} { c } { g _ { c } ^ { p } = \\mathbb { 1 } ( | | w _ { c } | | - \\alpha ^ { p } > 0 ) , } \\\\ { \\widehat { z } _ { c } = g _ { c } ^ { p } \\cdot \\big ( z _ { c , b _ { 1 } } + g _ { b _ { 2 } } ^ { q } \\big ( \\epsilon _ { c , b _ { 2 } } + \\cdot \\cdot \\cdot + g _ { b _ { K - 1 } } ^ { q } ( \\epsilon _ { c , b _ { K - 1 } } + g _ { b _ { K } } ^ { q } \\epsilon _ { c , b _ { K } } ) \\big ) \\big ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 237, + 717, + 280 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\widehat { z } _ { c }$ is the $c$ -th group of quantized filters and $\\epsilon _ { c , b _ { j } }$ is the corresponding re-assignment offset by bquantizing the residual error $z _ { c } - z _ { c , b _ { j - 1 } }$ . Here, $\\alpha ^ { p }$ is a layer-wise threshold for filter pruning. Following PFEC (Li et al., 2017a), we use $\\ell _ { 1 }$ -norm to evaluate the importance of the filter. Specifically, if a group of filters is important, the corresponding pruning gate will be activated and vice versa. ", + "bbox": [ + 174, + 287, + 825, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that both quantization and pruning have their corresponding thresholds. Instead of manually setting the thresholds, we propose to learn them via gradient descent. However, the indicator function in Eq. (13) is non-differentiable. To address this, we use straight-through estimator (STE) (Bengio et al., 2013; Zhou et al., 2016) to approximate the gradient of the indicator function $\\mathbb { 1 } ( \\cdot )$ using the gradient of the sigmoid function $\\sigma ( \\cdot )$ , which can be formulated as: ", + "bbox": [ + 173, + 349, + 825, + 421 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/348f53bd9dc487c866d7aa0cc55a7f93a8d8ad7609ca0c65a75ca5303db279f1.jpg", + "text": "$$\n\\frac { \\partial g } { \\partial \\alpha } = \\frac { \\partial \\mathbb { 1 } \\left( A - \\alpha \\right) } { \\partial \\alpha } \\approx \\frac { \\partial \\sigma \\left( A - \\alpha \\right) } { \\partial \\alpha } = - \\sigma \\left( A - \\alpha \\right) \\left( 1 - \\sigma ( A - \\alpha ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 272, + 430, + 722, + 463 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $g$ is the output of a binary gate, $\\alpha \\in \\{ \\alpha ^ { p } , \\alpha ^ { q } \\}$ is the corresponding threshold and $A$ denotes some specific metrics (i.e., $\\ell _ { 1 }$ -norm of the filter or the quantization error). By jointly training the binary gates and the network parameters, the pruning ratio and bitwidth of each layer can be automatically determined. However, the gradient approximation of the binary gate inevitably introduces noisy signals, which can be even more severe when we quantize both weights and activations. Thus, we propose to train the binary gates of weights and activations in an alternative manner. Specifically, when training the binary gates of weights, we fix the binary gates of activations, and vice versa. ", + "bbox": [ + 173, + 470, + 825, + 570 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Search Space for Model Compression. Given an uncompressed network with $L$ layers, we use $C _ { l }$ to denote the number of filters at the $l$ -th layer. To obtain the compressed model, we first divide the filters of each layer into groups and then search for the optimal bitwidths for the considered layer. Let $B$ be the number of filters in a group. For any layer $l$ , there would be $\\displaystyle \\left\\lfloor \\frac { C _ { l } } { B } \\right\\rfloor$ groups in total. Since we quantize both weights and activations, given $K$ candidate bitwidths, there are $K ^ { 2 }$ different quantization configurations for each layer. Thus, for the whole network with $L$ layers, the size of the search space $\\Omega$ can be computed by ", + "bbox": [ + 173, + 575, + 825, + 678 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2561053e1746674833c80011cd7c51e168cc9c093d44f69baa48713a1505b224.jpg", + "text": "$$\n\\left| \\Omega \\right| = \\prod _ { l = 1 } ^ { L } \\left( K ^ { 2 } \\times \\left\\lfloor \\frac { C _ { l } } { B } \\right\\rfloor \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 686, + 591, + 731 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Eq. (16) indicates that the search space is large enough to cover the potentially good configurations. ", + "bbox": [ + 171, + 739, + 823, + 755 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training Objective. To design a hardware-efficient network, the objective function in Eq. (7) should reflect both the accuracy of the compressed network and its computational costs. Following (Cai et al., 2019), we train the network and architecture by minimizing following loss function: ", + "bbox": [ + 173, + 761, + 825, + 803 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/bacf4d1b5c584d7debfc105f296c85a3c9f0eccd4e93446014b877a693a6d7cd.jpg", + "text": "$$\n{ \\mathcal { L } } ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) = { \\mathcal { L } } _ { c e } ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) + \\lambda \\log R ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 308, + 813, + 689, + 830 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\mathcal { L } _ { c e } ( \\cdot )$ is the cross-entropy loss, $R ( \\cdot )$ is the computational costs of the network and $\\lambda$ is a balancing hyper-parameter. Following single-path NAS (Stamoulis et al., 2019), we use a similar formulation of computational costs to preserve the differentiability of the objective function. The details of the differentiable computational loss can be found in Appendix B. Once the training is finished, we can obtain the compressed network by selecting those filters and bitwidths with activated binary gates. Then, we fine-tune the compressed network to compensate the accuracy loss. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/b520055e2de6343a058a0e4cbe6b3a1717d07972b3935b429d10b685e84c68cc.jpg", + "table_caption": [ + "Table 1: Comparisons of different methods on CIFAR-100. “W” and “A” represent the average quantization bitwidth of the weights and activations, respectively. " + ], + "table_footnote": [], + "table_body": "
NetworkMethodBOPs (M)BOP comp.ratioWIATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-20Full-precision41798.61.032.0/32.067.590.8
4-bit precision674.662.04.0/4.067.8±0.390.4±0.2
DQ1180.035.45.3/6.167.7±0.690.4±0.5
HAQ653.464.03.7/4.267.7±0.190.4±0.3
DNAS660.062.94.6/3.867.8±0.390.4±0.2
ABS-P(Ours)28586.51.532.0/32.067.9±0.190.7±0.2
ABS-Q(Ours) ABS (Ours)649.564.44.4/4.268.1±0.190.5±0.0
630.666.34.4/4.268.1±0.390.6±0.2
ResNet-56Full-precision128771.71.032.0/32.071.792.2
4-bit precision2033.663.34.0/4.070.9±0.391.2±0.4
DQ2222.957.93.8 /4.670.7±0.291.4±0.4
HAQ2014.963.93.3/4.971.2±0.191.1±0.2
DNAS2035.765.35.3/3.271.2±0.291.3±0.3
ABS-P(Ours)87021.61.532.0/32.071.5±0.191.8±0.2
ABS-Q(Ours)1970.765.34.1/4.071.5±0.291.5±0.2
ABS (Ours)1918.867.14.2/4.171.6±0.191.8±0.4
", + "bbox": [ + 232, + 146, + 764, + 314 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/6c07349b87f9e6d672e29a547e2de1141f2b382554b25e27f434c7c0ff27512b.jpg", + "image_caption": [ + "Figure 2: Results of different compressed networks with different BOPs and memory footprints. We use different methods to compress ResNet-56 and report the results on CIFAR-100. " + ], + "image_footnote": [], + "bbox": [ + 253, + 329, + 736, + 497 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 553, + 326, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Compared methods. To investigate the effectiveness of the proposed method, we consider the following methods for comparisons: ABS: our proposed method with joint pruning and quantization; ABS-Q: ABS with quantization only; ABS-P: ABS with pruning only; and several state-ofthe-art model compression methods including HAQ (Wang et al., 2019), DQ (Uhlich et al., 2020), DJPQ (Ying et al., 2020), Bayesian Bits (van Baalen et al., 2020) and DNAS (Wu et al., 2018). We measure the performance of different methods in terms of the Top-1 and Top-5 accuracy. Following (Guo et al., 2020; Ying et al., 2020), we measure the computational costs by the Bit-Operation (BOP) count. The BOP compression ratio is defined as the ratio between the total BOPs of the uncompressed and compressed models. We can also measure the computational costs with the total weights and activations memory footprints following DQ (Uhlich et al., 2020). Moreover, following (Stamoulis et al., 2019; Liu et al., 2019a), we use the search cost to measure the time of finding an optimal compressed model. ", + "bbox": [ + 174, + 583, + 825, + 750 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Implementation details. Following HAQ (Wang et al., 2019), we quantize all the layers, in which the first and the last layers are quantized to 8-bit. Following ThiNet (Luo et al., 2017), we only conduct filter pruning for the first layer in the residual block. For ResNet-20 and ResNet-56 on CIFAR-100 (Krizhevsky et al., 2009), we set $B$ to 4. For ResNet-18 and MobileNetV2 on ImageNet (Russakovsky et al., 2015), $B$ is set to 16 and 8, respectively. We first train the full-precision models and then use the pretrained weights to initialize the compressed models. Following (Li et al., 2020; Esser et al., 2020), we introduce weight normalization during training. We use SGD with nesterov (Nesterov, 1983) for optimization, with a momentum of 0.9. For CIFAR-100, we use the same data augmentation as in (He et al., 2016), including translation and horizontal flipping. For ImageNet, images are resized to $2 5 6 \\times 2 5 6$ , and then a $2 2 4 \\times 2 2 4$ patch is randomly cropped from an image or its horizontal flip for training. For testing, a $2 2 4 \\times 2 2 4$ center cropped is chosen. We first train the uncompressed network for 30 epochs on CIFAR-100 and 10 epochs on ImageNet. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/b48ed4cdd06f980611dd8f000a83d2078455f970f9577aafb1e0cbad855ca289.jpg", + "table_caption": [ + "Table 2: Comparisons on ImageNet. “\\*” denotes that we get the results from the figures in (van Baalen et al., 2020) and “–” denotes that the results are not reported. Moreover, “W” and “A” represent the average quantization bitwidth of the weights and activations, respectively. " + ], + "table_footnote": [], + "table_body": "
NetworkMethodBOPs (G)BOP comp. ratioW/ATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-18Full-precision1857.61.032.0/32.070.789.8
4-bit precision34.753.54.0/4.071.089.8
DQ40.740.61/168.51
DJPQ35.552.3-/169.1
HAQ34.753.54.0/4.070.289.5
Bayesian Bits*35.951.71/169.5
ABS-Q(Ours)33.156.14.5/3.870.989.7
ABS (Ours)32.357.54.6/4.270.889.6
MobileNetV2Full-precision308.01.032.0/32.071.990.3
6-bit precision11.227.56.0/6.071.890.3
DQ19.61.96.8/8.070.489.7
HAQ10.828.55.61/6.2771.290.0
Bayesian Bits *10.828.5-/170.9一
ABS-Q(Ours)10.928.36.8/6.871.890.4
ABS (Ours)10.828.56.1/7.171.790.3
", + "bbox": [ + 217, + 159, + 781, + 319 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The learning rate is set to 0.001. We then fine-tune the searched compressed network to recover the performance drop. On CIFAR-100, we train the searched network for 200 epochs with a mini-batch size of 128. The learning rate is initialized to 0.1 and is divided by 10 at 80-th and 120-th epochs. Experiments on CIFAR-100 are repeated for 5 times and we report the mean and standard deviation. For ResNet-18 on ImageNet, we finetune the searched network for 90 epochs with a mini-batch size of 256. For MobileNetV2 on ImageNet, we fine-tune for 150 epochs. For all models on ImageNet, the learning rate starts at 0.01 and decays with cosine annealing (Loshchilov & Hutter, 2017). ", + "bbox": [ + 173, + 329, + 825, + 426 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.1 MAIN RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 444, + 320, + 458 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We apply the proposed methods to compress ResNet-20, ResNet-56 on CIFAR-100 and ResNet18, MobileNetV2 on ImageNet. We compare the performance of different methods in Table 1 and Table 2. We also show the results of the compressed ResNet-56 with different BOPs and memory footprints in Figure 2. From the results, we can see that 4-bit quantized networks achieve lossless performance. Also, 6-bit MobileNetV2 only leads to a $0 . 1 \\%$ performance drop on the Top-1 Accuracy. Compared with fixed-precision quantization, mixed-precision methods are able to reduce the BOPs while preserving the performance. Critically, our proposed ABS-Q outperforms the stateof-the-arts baselines with less computational costs. Specifically, ABS-Q compressed ResNet-18 outperforms the one compressed by HAQ with more BOPs reduction. More critically, our proposed ABS achieves significant improvement in terms of BOPs and memory footprints. For example, in Figure 2(a), our ABS compressed ResNet-56 model yields much fewer BOPs (395.25 vs. 536.24) but achieves comparable performance compared with the fixed-precision counterpart. Moreover, by combing pruning and quantization, ABS achieves nearly lossless performance while further reducing the computational costs of ABS-Q. For example, ABS compressed ResNet-18 reduces the BOPs by $5 7 . 5 \\times$ while still outperforming the full-precision network by $0 . 1 \\%$ in terms of the Top-1 accuracy on ImageNet. ", + "bbox": [ + 174, + 469, + 825, + 693 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 FURTHER STUDIES ", + "text_level": 1, + "bbox": [ + 176, + 709, + 343, + 723 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Effect of the bit-sharing scheme. To investigate the effect of the bit-sharing scheme, we apply our methods to quantize ResNet-20 and ResNet-56 with and without the bit sharing scheme on CIFAR100. We report the testing accuracy and BOPs in Table 3. We also present the search costs and consumed GPU memory measured on a GPU device (NVIDIA TITAN $\\mathrm { X p }$ ). It can be seen from the results that the method with the bit sharing scheme consistently outperforms the ones without the bit sharing scheme while significantly reducing the search costs and GPU memory. ", + "bbox": [ + 174, + 734, + 825, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Effect of the one-stage compression. To investigate the effect of the one-stage compression scheme (perform pruning and quantization jointly), we extend ABS to two-stage optimization, where we sequentially do filter pruning and quantization, denoted as ABS- $\\mathbf { \\partial } \\cdot \\mathbf { P } { } \\mathbf { A } \\mathbf { B } \\mathbf { S } { - } \\mathbf { Q }$ . The results are shown in Table 4. Compared with the two-stage counterpart, ABS achieves better performance with less computational costs, which shows the superiority of the one-stage optimization. For example, ABS compressed ResNet-56 outperforms the counterpart by $0 . 4 \\%$ on the Top-1 accuracy with less computational overhead. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/fba71f434fd93322196b5659241f168f9bef427d8d408c34cf59414a5be498aa.jpg", + "table_caption": [ + "Table 3: Effect of the bit-sharing scheme. We report the testing accuracy, BOPs, and search costs on CIFAR-100. The search costs are measured on a GPU device (NVIDIA TITAN Xp). " + ], + "table_footnote": [], + "table_body": "
NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)Search Cost (GPU hours)GPUMemory (GB)
ResNet-20 w/o bit sharing67.8±0.190.5±0.2664.22.84.4
w/bit sharing68.1±0.190.5±0.0649.50.81.5
ResNet-56w/o bit sharing71.3±0.391.4±0.42001.18.710.9
w/ bit sharing71.5±0.291.5±0.21970.71.93.0
", + "bbox": [ + 176, + 156, + 830, + 217 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/4ee4e05e37def2d9dafbcf83c7a9b3d59b162ecc0fff01bb6add475c78516cc8.jpg", + "table_caption": [ + "Table 4: Effect of the one-stage compression. We report the results of ResNet-56 on CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56ABS-P→ABS-Q70.4±0.190.8±0.21077.7
ABS70.8±0.491.2±0.11042.5
", + "bbox": [ + 297, + 257, + 699, + 295 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Effect of the alternative training scheme. To investigate the effect of the alternative training scheme introduced in Section 3.3, we apply our method to compress ResNet-56 using a joint training scheme and an alternative training scheme on CIFAR-100. Here, the joint training scheme denotes that we train the binary gates of weights and activations jointly. From the results of Table 5, the model trained with the alternative scheme achieves better performance than the joint one, which demonstrates the effectiveness of the alternative training scheme. ", + "bbox": [ + 173, + 306, + 825, + 390 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/f4d725e2b661e75c85d8956fee28fbdec2e76a5511411532d1a7fc88dd926513.jpg", + "table_caption": [ + "Table 5: Effect of the alternative training scheme. We report the results of ResNet-56 on CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56Joint71.3±0.291.6±0.31942.4
Alternative71.6±0.191.8±0.41918.8
", + "bbox": [ + 312, + 439, + 684, + 476 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Resource-constrained compression. To demonstrate the effectiveness of our ABS on hardware devices, we further apply our methods to compress MobileNetV2 under the resource constraints on the BitFusion architecture (Sharma et al., 2018). Instead of using BOPs, we use the latency and energy on a simulator of the BitFusion to measure the computational costs. We report the results in Table 6. Compared with fixed-precision quantization, ABS achieves better performance with lower latency and energy. Specifically, ABS compressed MobileNetV2 with much lower latency and energy even outperforms 6-bit MobileNetV2 by $0 . 2 \\%$ in the Top-1 accuracy. ", + "bbox": [ + 173, + 500, + 825, + 598 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/07222d8161e00c3b009ed5364b8f27623fac1635c8c3ae64048239311f66d522.jpg", + "table_caption": [ + "Table 6: Resource-constrained compression on BitFusion. We evaluate the proposed ABS under the latency- and energy-constrained and report the Top-1 and Top-5 accuracy on ImageNet. " + ], + "table_footnote": [], + "table_body": "
NetworkMethodLatency-constrainedEnergy-constrained
Acc.-1 (%)Acc.-5 (%)Latency (ms)Acc.-1 (%)Acc.-5 (%)Energy (mJ)
MobileNetV26-bit precision71.890.324.971.890.332.8
ABS (Ours)72.090.417.272.090.326.3
", + "bbox": [ + 176, + 659, + 821, + 708 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 737, + 495, + 753 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we have proposed a novel model compression method called Automatically Bit Sharing (ABS). Specifically, our ABS is based on the observation that quantized values of a high bitwidth share the ones of lower bitwidths under some constraints. We therefore have proposed the decomposition of quantization that encapsulates all candidate bitwidths. Starting from a low bitwidth in the search space, we sequentially increase the effective bitwidth by recursively adding re-assignment offsets. Based on this, we have further introduced learnable binary gates to encode the choice of different compression policies. By training the binary gates, the optimal compression ratio of each layer can be automatically determined. Experiments on CIFAR-100 and ImageNet have shown that our methods are able to achieve significant cost reduction while preserving the performance. In the future, we plan to work on a joint search for architecture, pruning, and quantization to find a compact model with better performance. ", + "bbox": [ + 173, + 770, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Yu Bai, Yu-Xiang Wang, and Edo Liberty. Proxquant: Quantized neural networks via proximal operators. In Proc. Int. Conf. Learn. Repren., 2019. ", + "bbox": [ + 176, + 126, + 823, + 155 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Yoshua Bengio, Nicholas Leonard, and Aaron Courville. Estimating or propagating gradients ´ through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013. 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", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Zhuangwei Zhuang, Mingkui Tan, Bohan Zhuang, Jing Liu, Yong Guo, Qingyao Wu, Junzhou Huang, and Jinhui Zhu. Discrimination-aware channel pruning for deep neural networks. In Proc. Adv. Neural Inf. Process. Syst., 2018b. ", + "bbox": [ + 174, + 155, + 825, + 196 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Appendix for ABS: Automatic Bit Sharing for Model Compression ", + "text_level": 1, + "bbox": [ + 235, + 98, + 764, + 137 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A HARDWARE-FRIENDLY DECOMPOSITION ", + "bbox": [ + 174, + 160, + 545, + 175 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "As mentioned in Sec. 3.2, $b _ { 1 }$ can be set to arbitrary appropriate integer values (e.g., 1, 2, 3, etc.). By default, we set $b _ { 1 } = 2$ for better hardware utilization. On general purpose computing devices (e.g., CPU, GPU), byte (8 bits) is the lowest data type for operations. Other data types and ALU registers are all composed with multiple bytes in width. By setting $b _ { 1 } = 2$ , 2-bit/ 4-bit/ 8-bit quantization values can be packed into byte (or short, int, long) data type without bit wasting. Otherwise, if $b _ { 1 } = 1$ or $b _ { 1 } = 3$ , it is inevitable to have wasted bits when packing mixed-precision quantized tensors on general purpose devices. For example, one 32-bit int data type can be used to store ten 3-bit quantized values with 2 bits wasted. One might argue that these 2 bits can be leveraged with the next group of 3-bit data, but it will result in irregular memory access patterns, which will degrade the hardware utilization more seriously. Moreover, 8-bit quantization has been demonstrated to own similar performance with the full precision counterparts for many networks. Therefore, there is no need to consider a bitwidth larger than 8. ", + "bbox": [ + 173, + 193, + 825, + 359 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B FORMULATION OF DIFFERENTIABLE COMPUTATIONAL LOSS ", + "text_level": 1, + "bbox": [ + 174, + 383, + 710, + 400 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we introduce the differentiable computational loss mentioned in Section 3.3. Unlike the cross-entropy loss in Eq. (17), the computational costs $R ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } )$ is non-differentiable. To solve this issue, we model the computational costs as a function of binary gates as: ", + "bbox": [ + 176, + 417, + 825, + 460 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7590e53268bad8e7d62ef6993005292a6c60de605c3423b5c0575b214ea137bb.jpg", + "text": "$$\nR ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) = \\sum _ { c = 1 } ^ { G } g _ { c } ^ { p } \\left( R _ { x _ { c , b _ { 1 } } } + g _ { b _ { 2 } } ^ { q } \\left( R _ { x _ { c , b _ { 2 } } } - R _ { x _ { c , b _ { 1 } } } + \\cdot \\cdot + \\cdot + g _ { b _ { K } } ^ { q } \\left( R _ { x _ { c , b _ { K } } } - R _ { x _ { c , b _ { K - 1 } } } \\right) \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 469, + 812, + 513 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where Rxc,b is the computational cost for the $c$ -th group of filters with $b _ { j }$ -bit quantization and $G$ is the number of groups in total. ", + "bbox": [ + 176, + 523, + 825, + 555 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C QUANTIZATION CONFIGURATIONS ", + "text_level": 1, + "bbox": [ + 176, + 579, + 491, + 594 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "All the methods in Tables 1 and 2 use layer-wise and symmetric quantization schemes and the compared methods strictly follow the quantization configurations in their original papers. Specifically, for DQ (Uhlich et al., 2020), we parameterize the fixed-point quantizer using case U3 with $\\theta = [ d , q _ { \\mathrm { m a x } } ]$ . We initialize the weights using a pre-trained model. The initial step size is set to $d = \\bar { 2 } ^ { \\lfloor \\log _ { 2 } ( \\operatorname* { m a x } ( | \\mathbf { W } | ) / ( 2 ^ { b - 1 } - 1 ) \\rfloor }$ for weights and $2 ^ { - 3 }$ for activations. The remaining quantization parameters are set such that the initial bitwidth is 4-bit. For HAQ (Wang et al., 2019), we first truncate the weights and activations into the range of $[ - v _ { w } , v _ { w } ]$ and $[ 0 , v _ { x } ]$ , respectively. We then perform linear quantization for both weights and activations. To find more proper $v _ { w }$ and $v _ { x }$ , we minimize the KL-divergence between the original weight distribution W and the quantized weight distribution $Q ( \\mathbf { W } )$ . For DNAS (Wu et al., 2018), we follow DoReFa-Net (Zhou et al., 2016) to quantize weights and follow PACT (Choi et al., 2018) to quantize activations. We initialize the learnable upper bound to 1. For DJPQ (Ying et al., 2020) and Bayesian Bits (van Baalen et al., 2020), we directly get the results from original papers. For other methods in Tables 1 and 2, we use the quantization function introduced in Section 3.1. The trainable quantization intervals $v _ { x }$ and $v _ { w }$ are initialized to 1. ", + "bbox": [ + 173, + 612, + 825, + 810 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D SEARCH COST COMPARISONS ", + "text_level": 1, + "bbox": [ + 176, + 834, + 452, + 851 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "To evaluate the efficiency of the proposed ABS, we compare the search cost of different methods and report the results in Table 7. From the results, the search costs of the proposed ABS is much smaller than the state-of-the-art methods. Moreover, compared with ABS-Q, ABS only introduces a small amount of computational overhead, which demonstrates the efficiency of the proposed methods. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/63bc429059acd23a603b64a5a9b060d08bb4e83ce42f4e52582bcc5cca119845.jpg", + "table_caption": [ + "Table 7: Comparisons of the search costs on CIFAR-100. The search costs are measured on a GPU device (NVIDIA TITAN Xp). " + ], + "table_footnote": [], + "table_body": "
NetworkMethodSearch Cost (GPU hours)
ResNet-20HAQ DQ5.8 3.0
DNAS2.8
ABS-Q (Ours) ABS-P (Ours) ABS (Ours)0.8 0.2
", + "bbox": [ + 366, + 145, + 630, + 244 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/8ec06803eb6b7d86b9b0d0db7863d5655fb504a7503b868ff04b0319df621273.jpg", + "table_caption": [ + "Table 8: Comparisons of different methods w.r.t. memory footprints. We compress ResNet-56 using different methods and report the results on CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
MethodMemory footprints(KB)M.f. comp. ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision5653.41.071.792.2
4-bit precision711.77.970.9±0.391.2±0.4
DNAS708.98.071.5±0.291.3±0.1
HAQ700.08.171.3±0.191.1±0.1
ABS-Q (Ours)674.58.471.5±0.291.6±0.2
ABS (Ours)657.38.671.6±0.191.8±0.4
", + "bbox": [ + 235, + 296, + 759, + 376 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/19f745059083282078a1849ace381285de78914473022219977cffadff023e57.jpg", + "table_caption": [ + "Table 9: Comparisons of different methods with MobileNetV3 on CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
MethodBOPs (M)BOP comp.ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision68170.11.076.193.9
6-bit precision2412.628.376.1±0.093.7±0.0
DQ2136.331.975.9±0.193.7±0.1
HAQ2191.731.176.1±0.193.5±0.0
DNAS2051.933.276.1±0.193.7±0.1
ABS-P(Ours)59465.81.176.0±0.093.5±0.0
ABS-Q(Ours)2021.933.776.1±0.193.7±0.1
ABS (Ours)2006.634.076.1±0.193.7±0.1
", + "bbox": [ + 281, + 415, + 717, + 512 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E MORE RESULTS ON MEMORY FOOTPRINTS ", + "text_level": 1, + "bbox": [ + 174, + 522, + 557, + 536 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To further demonstrate the effectiveness of the proposed ABS, we replace BOPs with total weights and activations memory footprints (Uhlich et al., 2020). We apply different methods to compress ResNet-56 and report the results in Table 8. From the results, ABS compressed ResNet-56 outperforms other methods with fewer memory footprints. These results show the effectiveness of our proposed ABS in terms of memory footprints reduction. ", + "bbox": [ + 173, + 551, + 825, + 622 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "F DETAILED STRUCTURE OF THE COMPRESSED NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 643, + 668, + 659 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We illustrate the pruning rate and bitwidth of each layer’s weights and activations of the compressed ResNet-18 and MobileNetV2 in Figure 3 and Figure 4, respectively. From the results, we observe that our ABS assigns more bitwidths to the weights in the downsampling convolutional layer in ResNet-18 and depthwise convolutional layer in MobileNetV2. Intuitively, this is because the number of parameters of these layers is much smaller than other layers. Moreover, our ABS inclines to prune more filters in the shallower layers, which can significantly reduce the number of parameters and computational overhead. Finally, we also observe that the correlation between the bitwidth and pruning rate is as follows. If a layer is set to a high pruning rate, our ABS tends to select a higher bitwidth to compensate for the performance drop. In contrast, if a layer is with a low pruning rate, our ABS tends to select a lower bitwidth to reduce the model size and computational costs. ", + "bbox": [ + 174, + 674, + 825, + 813 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "G MORE RESULTS ON MOBILENETV3 ", + "text_level": 1, + "bbox": [ + 174, + 833, + 506, + 849 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To evaluate the proposed ABS on the lightweight model, we apply our methods to MobileNetV3 on CIFAR-100. Following $\\mathrm { L S Q + }$ (Bhalgat et al., 2020), we introduce a learnable offset to handle the negative activations in hard-swish. We show the results in Table 9. From the results of MobileNetV3, our proposed ABS still outperforms the compared methods, which demonstrates its effectiveness. ", + "bbox": [ + 174, + 864, + 825, + 920 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/284f562ab05ccb482bd329fad017446ba815605269415a114484ae378c8d00c3.jpg", + "image_caption": [ + "(a) Bitwidth configuration of the compressed ResNet-18. " + ], + "image_footnote": [], + "bbox": [ + 204, + 195, + 794, + 474 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/d02d26b45dc7eafeecbf604a33ffcc12f1d0ec1a1c0956af5f33cccaa8b129c8.jpg", + "image_caption": [ + "Figure 3: Detailed configurations of the compressed ResNet-18. " + ], + "image_footnote": [], + "bbox": [ + 223, + 515, + 772, + 752 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(b) Pruning rate configuration of the compressed ResNet-18. The pruning rate is defined as the ratio between #pruned weights of the compressed models and #weights of the uncompressed models. ", + "bbox": [ + 199, + 768, + 797, + 796 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/529215833fa76e311a61fb639d37fa31e0fc915f7db262f5a57182c66e854ec9.jpg", + "image_caption": [ + "Figure 4: Detailed configurations of the compressed MobileNetV2. " + ], + "image_footnote": [], + "bbox": [ + 173, + 247, + 831, + 714 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(b) Pruning rate configuration of the compressed MobileNetV2. The pruning rate is defined as the ratio between #pruned weights of the compressed models and #weights of the uncompressed models. ", + "bbox": [ + 173, + 712, + 834, + 738 + ], + "page_idx": 16 + } +] \ No newline at end of file diff --git a/parse/train/QjINdYOfq0b/QjINdYOfq0b_middle.json b/parse/train/QjINdYOfq0b/QjINdYOfq0b_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..8113a0e5bd18975a07e95a81444f507f06d0edd1 --- /dev/null +++ b/parse/train/QjINdYOfq0b/QjINdYOfq0b_middle.json @@ -0,0 +1,40828 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "score": 1.0, + "content": "ABS: AUTOMATIC BIT SHARING FOR MODEL COM-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 101, + 181, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 181, + 117 + ], + "score": 1.0, + "content": "PRESSION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 187, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 211, + 468, + 420 + ], + "lines": [ + { + "bbox": [ + 142, + 212, + 469, + 225 + ], + "spans": [ + { + "bbox": [ + 142, + 212, + 469, + 225 + ], + "score": 1.0, + "content": "We present Automatic Bit Sharing (ABS) to automatically search for optimal", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 222, + 469, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 469, + 236 + ], + "score": 1.0, + "content": "model compression configurations (e.g., pruning ratio and bitwidth). 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Experiments on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "score": 1.0, + "content": "CIFAR-100 and ImageNet show that our methods achieve significant computa-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 408, + 394, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 394, + 422 + ], + "score": 1.0, + "content": "tional cost reduction while preserving promising performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 442, + 206, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 208, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 208, + 458 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 504, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "Deep neural networks (DNNs) have achieved great success in many challenging computer vision", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "tasks, including image classification (Krizhevsky et al., 2012; He et al., 2016) and object detec-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "tion (Lin et al., 2017a;b). 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To reduce the number of parameters and computational overhead, many meth-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "ods (He et al., 2019; Zhou et al., 2016) have been proposed to conduct model compression by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 355, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 355, + 557 + ], + "score": 1.0, + "content": "removing the redundancy while maintaining the performance.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "In the last decades, we have witnessed a lot of model compression methods, such as network prun-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "ing (He et al., 2017; 2019) and quantization (Zhou et al., 2016; Hubara et al., 2016). 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It has been shown", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "that sequentially perform network pruning and quantization is able to get a compressed network", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 613, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 506, + 631 + ], + "score": 1.0, + "content": "with small model size and lower computational overhead (Han et al., 2016). However, performing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "pruning and quantization in a separate step may lead to sub-optimal results. 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Unlike pre-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 233, + 470, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 233, + 470, + 247 + ], + "score": 1.0, + "content": "vious works that consider model pruning and quantization separately, we seek", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 470, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 470, + 258 + ], + "score": 1.0, + "content": "to optimize them jointly. To deal with the resultant large designing space, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 470, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 470, + 268 + ], + "score": 1.0, + "content": "propose a novel super-bit model, a single-path method, to encode all candidate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "compression configurations, rather than maintaining separate paths for each con-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "score": 1.0, + "content": "figuration. Specifically, we first propose a novel decomposition of quantization", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 288, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 288, + 470, + 301 + ], + "score": 1.0, + "content": "that encapsulates all the candidate bitwidths in the search space. Starting from a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 469, + 312 + ], + "score": 1.0, + "content": "low bitwidth, we sequentially consider higher bitwidths by recursively adding re-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 470, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 324 + ], + "score": 1.0, + "content": "assignment offsets. We then introduce learnable binary gates to encode the choice", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 469, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 469, + 335 + ], + "score": 1.0, + "content": "of bitwidth, including filter-wise 0-bit for pruning. By jointly training the binary", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "score": 1.0, + "content": "gates in conjunction with network parameters, the compression configurations of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 356 + ], + "score": 1.0, + "content": "each layer can be automatically determined. Our ABS brings two benefits for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 470, + 367 + ], + "score": 1.0, + "content": "model compression: 1) It avoids the combinatorially large design space, with a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "score": 1.0, + "content": "reduced number of trainable parameters and search costs. 2) It also averts directly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 469, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 469, + 390 + ], + "score": 1.0, + "content": "fitting an extremely low bit quantizer to the data, hence greatly reducing the op-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 387, + 469, + 400 + ], + "spans": [ + { + "bbox": [ + 141, + 387, + 469, + 400 + ], + "score": 1.0, + "content": "timization difficulty due to the non-differentiable quantization. Experiments on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "score": 1.0, + "content": "CIFAR-100 and ImageNet show that our methods achieve significant computa-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 408, + 394, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 394, + 422 + ], + "score": 1.0, + "content": "tional cost reduction while preserving promising performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 212, + 470, + 422 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 442, + 206, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 208, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 208, + 458 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 504, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "Deep neural networks (DNNs) have achieved great success in many challenging computer vision", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "tasks, including image classification (Krizhevsky et al., 2012; He et al., 2016) and object detec-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "tion (Lin et al., 2017a;b). However, a deep model usually has a large number of parameters and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "consumes huge amounts of computational resources, which remains great obstacles for many appli-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "cations, especially on resource-limited devices with limited memory and computational resources,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "such as smartphones. To reduce the number of parameters and computational overhead, many meth-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "ods (He et al., 2019; Zhou et al., 2016) have been proposed to conduct model compression by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 355, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 355, + 557 + ], + "score": 1.0, + "content": "removing the redundancy while maintaining the performance.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 467, + 506, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "In the last decades, we have witnessed a lot of model compression methods, such as network prun-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 586 + ], + "score": 1.0, + "content": "ing (He et al., 2017; 2019) and quantization (Zhou et al., 2016; Hubara et al., 2016). Specifically,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "network pruning reduces the model size and computational costs by removing redundant modules", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "while network quantization maps the full-precision values to low-precision ones. It has been shown", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "score": 1.0, + "content": "that sequentially perform network pruning and quantization is able to get a compressed network", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 613, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 613, + 506, + 631 + ], + "score": 1.0, + "content": "with small model size and lower computational overhead (Han et al., 2016). However, performing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "pruning and quantization in a separate step may lead to sub-optimal results. For example, the best", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 650 + ], + "score": 1.0, + "content": "quantization strategy for the uncompressed network is not necessarily the optimal one after network", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 483, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 483, + 662 + ], + "score": 1.0, + "content": "pruning. Therefore, we need to consider performing pruning and quantization simultaneously.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 561, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Recently, many attempts have been made to automatically determine the compression configura-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "tions of each layer (i.e., pruning ratios, and/or bitwidths), either based on reinforcement learning", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "(RL) (Wang et al., 2019), evolutionary search (ES) (Wang et al., 2020), Bayesian optimization (BO)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "(Tung & Mori, 2018) or differentiable methods (Wu et al., 2018; Dong & Yang, 2019). In particular,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "previous differentiable methods formulate model compression as a differentiable searching problem", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "to explore the search space using gradient-based optimization. As shown in Figure 1(a), each candi-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "date operation is maintained as a separate path, which leads to a huge number of trainable parameters", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "and high computational overhead when the search space becomes combinatorially large. Moreover,", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "due to the non-differentiable quantizer and pruning process, the optimization of heavily compressed", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 398, + 466, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 466, + 410 + ], + "score": 1.0, + "content": "candidate networks can be more challenging than that in the conventional search problem.", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 666, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 79, + 500, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 79, + 500, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 79, + 500, + 228 + ], + "spans": [ + { + "bbox": [ + 111, + 79, + 500, + 228 + ], + "score": 0.967, + "type": "image", + "image_path": "d661d86f6402cd19d79021a39a0281ea92dacd0285f0d8f2e32e077e497f6529.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 79, + 500, + 128.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 128.66666666666666, + 500, + 178.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 178.33333333333331, + 500, + 227.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 241, + 505, + 351 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Figure 1: Multi-path v.s. single-path compression scheme. (a) Multi-path search scheme (Wu et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 251, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 265 + ], + "score": 1.0, + "content": "2018): represents each candidate configuration as a separate path and formulates the compression", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 275 + ], + "score": 1.0, + "content": "problem as a path selection problem, which gives rise to huge numbers of trainable parameters and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 482, + 287 + ], + "score": 1.0, + "content": "high computational overhead when the search space becomes combinatorially large. Here,", + "type": "text" + }, + { + "bbox": [ + 483, + 275, + 493, + 285 + ], + "score": 0.84, + "content": "z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 121, + 298 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 285, + 128, + 295 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 284, + 228, + 298 + ], + "score": 1.0, + "content": "-bit quantized version of", + "type": "text" + }, + { + "bbox": [ + 228, + 287, + 235, + 295 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 284, + 253, + 298 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 285, + 266, + 297 + ], + "score": 0.9, + "content": "\\alpha _ { k } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "is the architecture parameters corresponding to the path of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 295, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 113, + 306 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 295, + 506, + 309 + ], + "score": 1.0, + "content": "-bit quantization. (b) Single-path search scheme (Ours): represents each candidate configuration", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "as a subset of a “super-bit” and formulates the compression problem as a subset selection problem,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "which greatly reduces the computational costs and optimization difficulty from the discontinuity of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 430, + 342 + ], + "score": 1.0, + "content": "quantization. Here, the super-bit denotes the highest bitwidth in the search space,", + "type": "text" + }, + { + "bbox": [ + 430, + 329, + 442, + 341 + ], + "score": 0.89, + "content": "g _ { k } ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "is a binary gate", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 339, + 500, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 274, + 353 + ], + "score": 1.0, + "content": "that controls the decision of bitwidth, and", + "type": "text" + }, + { + "bbox": [ + 275, + 341, + 285, + 351 + ], + "score": 0.86, + "content": "\\epsilon _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 339, + 500, + 353 + ], + "score": 1.0, + "content": "is the re-assignment offset (quantized residual error).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + } + ], + "index": 4.25 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "date operation is maintained as a separate path, which leads to a huge number of trainable parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "and high computational overhead when the search space becomes combinatorially large. 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Specifically, we build upon the observation that the quantized values of a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "high bitwidth can share the ones of low bitwidths under some conditions. Therefore, we are able to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 492, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 504 + ], + "score": 1.0, + "content": "decompose the quantized representation into the sum of the lowest bit quantization and a series of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "re-assignment offsets. 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In this way, we enable the candidate operations to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 141, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 141, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "learn jointly rather than separately, hence greatly easing the optimization, especially in the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 721, + 306, + 732 + ], + "spans": [ + { + "bbox": [ + 141, + 721, + 306, + 732 + ], + "score": 1.0, + "content": "non-differentiable quantization scenario.", + "type": "text" + } + ], + "index": 43, + "is_list_end_line": true + } + ], + "index": 38.5, + "bbox_fs": [ + 133, + 616, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 134, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 133, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 133, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "• We evaluate our ABS on CIFAR-100 and ImageNet over various network architectures.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "Extensive experiments show that the proposed method achieves the state-of-the-art perfor-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 141, + 105, + 477, + 117 + ], + "score": 1.0, + "content": "mance. For example, on ImageNet, our ABS compressed MobileNetV2 achieves", + "type": "text" + }, + { + "bbox": [ + 477, + 105, + 504, + 115 + ], + "score": 0.85, + "content": "2 8 . 5 \\times", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 114, + 500, + 129 + ], + "spans": [ + { + "bbox": [ + 141, + 114, + 309, + 129 + ], + "score": 1.0, + "content": "Bit-Operation (BOP) reduction with only", + "type": "text" + }, + { + "bbox": [ + 309, + 115, + 331, + 126 + ], + "score": 0.86, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 114, + 500, + 129 + ], + "score": 1.0, + "content": "performance drop on the Top-1 accuracy.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 148, + 208, + 161 + ], + "lines": [ + { + "bbox": [ + 104, + 147, + 211, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 211, + 163 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "Network quantization. Network quantization represents the weights, activations and even gradients", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "in low-precision to yield compact DNNs. With low-precision integers or power-of-two representa-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "tions, the heavy matrix multiplications can be replaced by efficient bitwise operations, leading to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "much faster test-time inference and lower power consumption. To improve the quantization perfor-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "mance, current methods either focus on designing accurate quantizers by fitting the quantizer to the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "data (Jung et al., 2019; Zhang et al., 2018; Choi et al., 2018; Cai et al., 2017), or seek to approximate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "the gradients due to the non-differentiable discretization (Ding et al., 2019; Louizos et al., 2019;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "Zhuang et al., 2020). Moreover, most previous works assign the same bitwidth for all layers (Zhou", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "et al., 2016; Zhuang et al., 2018a; 2019; Jung et al., 2019; Jin et al., 2019; Li et al., 2020; Esser et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "2020). Though attractive for simplicity, setting a uniform precision places no guarantee on optimiz-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "ing network performance since different layers have different redundancy and arithmetic intensity.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "Therefore, several studies proposed mixed-precision quantization (Wang et al., 2019; Dong et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "2019; Wu et al., 2018; Uhlich et al., 2020) to set different bitwidths according to the redundancy of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "each layer. In this paper, based on the proposed quantization decomposition, we devise an approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 502, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 502, + 343 + ], + "score": 1.0, + "content": "that can effectively learn appropriate bitwidths for each layer through gradient-based optimization.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 504, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 504, + 359 + ], + "score": 1.0, + "content": "NAS and pruning. Neural architecture search (NAS) aims to automatically design efficient archi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "tectures with low model size and computational costs, either based on reinforcement learning (Pham", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "et al., 2018; Guo et al., 2019), evolutionary search (Real et al., 2019) or gradient-based methods (Liu", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "et al., 2019a). In particular, gradient-based NAS has gained increased popularity, where the search", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "space can be divided into the multi-path design (Liu et al., 2019a; Cai et al., 2019) and single-path", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "formulation (Stamoulis et al., 2019; Guo et al., 2020), depending on whether adding each operation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "as a separate path or not. While prevailing NAS methods optimize the network topology, the focus", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "of this paper is to search optimal compression ratios for a given architecture. Moreover, network", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "pruning can be treated as fine-grained NAS, which aims at removing redundant modules to acceler-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 504, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 504, + 457 + ], + "score": 1.0, + "content": "ate the run-time inference speed, giving rise to methods based on unstructured weight pruning (Han", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 457, + 504, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 504, + 469 + ], + "score": 1.0, + "content": "et al., 2016; Guo et al., 2016) or structured channel pruning (He et al., 2017; Zhuang et al., 2018b;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 466, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 481 + ], + "score": 1.0, + "content": "Luo et al., 2017). Based on channel pruning, our paper further takes quantization into consideration", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 479, + 252, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 252, + 491 + ], + "score": 1.0, + "content": "to generate more compact networks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "AutoML for model compression. Recently, much effort has been put into automatically determin-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "ing either the optimal pruning rate (Tung & Mori, 2018; Dong & Yang, 2019; He et al., 2018), or", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "score": 1.0, + "content": "the bitwidth (Lou et al., 2019; Cai & Vasconcelos, 2020) of each layer via hyper-parameter search,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "without relying on heuristics. In particular, HAQ (Wang et al., 2019) employs reinforcement learn-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "ing to search bitwidth strategies with the hardware accelerator’s feedback. Meta-pruning (Liu et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 504, + 563 + ], + "score": 1.0, + "content": "2019b) uses meta-learning to generate the weight parameters of the pruned networks and then adopts", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 504, + 574 + ], + "score": 1.0, + "content": "an evolutionary search algorithm to find the layer-wise sparsity for channel pruning. More recently,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "several studies (Wu et al., 2018; Cai & Vasconcelos, 2020) have focused on using differentiable", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 273, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 273, + 596 + ], + "score": 1.0, + "content": "schemes via gradient-based optimization.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Closely related methods. To further improve the compression ratio, several methods propose to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "jointly optimize pruning and quantization strategies. In particular, some works only support weight", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "quantization (Tung & Mori, 2018; Ye et al., 2019) or use fine-grained pruning (Yang et al., 2020).", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "However, the resultant networks cannot be implemented efficiently on edge devices. Recently, sev-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "eral methods (Wu et al., 2018; Wang et al., 2020; Ying et al., 2020) have been proposed to consider", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "filter pruning, weight quantization, and activation quantization jointly. In contrast to these methods,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "we carefully design the compression search space by sharing the quantized values between differ-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "ent candidate configurations, which significantly reduces the search cost and eases the optimization.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Compared with those methods that share the similarities of using quantized residual errors (Chen", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "et al., 2010; Gong et al., 2014; Li et al., 2017b; van Baalen et al., 2020), our proposed method recur-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "sively uses quantized residual errors to decompose a quantized representation as a set of candidate", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 417, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 417, + 733 + ], + "score": 1.0, + "content": "bitwidths and parameterize the selection of optimal bitwidth via binary gates.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 47.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 134, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 133, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 133, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "• We evaluate our ABS on CIFAR-100 and ImageNet over various network architectures.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "Extensive experiments show that the proposed method achieves the state-of-the-art perfor-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 105, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 141, + 105, + 477, + 117 + ], + "score": 1.0, + "content": "mance. For example, on ImageNet, our ABS compressed MobileNetV2 achieves", + "type": "text" + }, + { + "bbox": [ + 477, + 105, + 504, + 115 + ], + "score": 0.85, + "content": "2 8 . 5 \\times", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 114, + 500, + 129 + ], + "spans": [ + { + "bbox": [ + 141, + 114, + 309, + 129 + ], + "score": 1.0, + "content": "Bit-Operation (BOP) reduction with only", + "type": "text" + }, + { + "bbox": [ + 309, + 115, + 331, + 126 + ], + "score": 0.86, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 114, + 500, + 129 + ], + "score": 1.0, + "content": "performance drop on the Top-1 accuracy.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 133, + 82, + 506, + 129 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 148, + 208, + 161 + ], + "lines": [ + { + "bbox": [ + 104, + 147, + 211, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 211, + 163 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "Network quantization. Network quantization represents the weights, activations and even gradients", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 201 + ], + "score": 1.0, + "content": "in low-precision to yield compact DNNs. With low-precision integers or power-of-two representa-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "tions, the heavy matrix multiplications can be replaced by efficient bitwise operations, leading to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "much faster test-time inference and lower power consumption. To improve the quantization perfor-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "mance, current methods either focus on designing accurate quantizers by fitting the quantizer to the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "data (Jung et al., 2019; Zhang et al., 2018; Choi et al., 2018; Cai et al., 2017), or seek to approximate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "the gradients due to the non-differentiable discretization (Ding et al., 2019; Louizos et al., 2019;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "Zhuang et al., 2020). Moreover, most previous works assign the same bitwidth for all layers (Zhou", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "et al., 2016; Zhuang et al., 2018a; 2019; Jung et al., 2019; Jin et al., 2019; Li et al., 2020; Esser et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "2020). Though attractive for simplicity, setting a uniform precision places no guarantee on optimiz-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "ing network performance since different layers have different redundancy and arithmetic intensity.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "Therefore, several studies proposed mixed-precision quantization (Wang et al., 2019; Dong et al.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "2019; Wu et al., 2018; Uhlich et al., 2020) to set different bitwidths according to the redundancy of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "each layer. In this paper, based on the proposed quantization decomposition, we devise an approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 502, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 502, + 343 + ], + "score": 1.0, + "content": "that can effectively learn appropriate bitwidths for each layer through gradient-based optimization.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 176, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 504, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 504, + 359 + ], + "score": 1.0, + "content": "NAS and pruning. Neural architecture search (NAS) aims to automatically design efficient archi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "tectures with low model size and computational costs, either based on reinforcement learning (Pham", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "et al., 2018; Guo et al., 2019), evolutionary search (Real et al., 2019) or gradient-based methods (Liu", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "et al., 2019a). In particular, gradient-based NAS has gained increased popularity, where the search", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "space can be divided into the multi-path design (Liu et al., 2019a; Cai et al., 2019) and single-path", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "formulation (Stamoulis et al., 2019; Guo et al., 2020), depending on whether adding each operation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "as a separate path or not. While prevailing NAS methods optimize the network topology, the focus", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "of this paper is to search optimal compression ratios for a given architecture. Moreover, network", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "pruning can be treated as fine-grained NAS, which aims at removing redundant modules to acceler-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 504, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 504, + 457 + ], + "score": 1.0, + "content": "ate the run-time inference speed, giving rise to methods based on unstructured weight pruning (Han", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 457, + 504, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 504, + 469 + ], + "score": 1.0, + "content": "et al., 2016; Guo et al., 2016) or structured channel pruning (He et al., 2017; Zhuang et al., 2018b;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 466, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 481 + ], + "score": 1.0, + "content": "Luo et al., 2017). Based on channel pruning, our paper further takes quantization into consideration", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 479, + 252, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 252, + 491 + ], + "score": 1.0, + "content": "to generate more compact networks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 348, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "AutoML for model compression. Recently, much effort has been put into automatically determin-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "ing either the optimal pruning rate (Tung & Mori, 2018; Dong & Yang, 2019; He et al., 2018), or", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "score": 1.0, + "content": "the bitwidth (Lou et al., 2019; Cai & Vasconcelos, 2020) of each layer via hyper-parameter search,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "without relying on heuristics. In particular, HAQ (Wang et al., 2019) employs reinforcement learn-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "ing to search bitwidth strategies with the hardware accelerator’s feedback. Meta-pruning (Liu et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 504, + 563 + ], + "score": 1.0, + "content": "2019b) uses meta-learning to generate the weight parameters of the pruned networks and then adopts", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 504, + 574 + ], + "score": 1.0, + "content": "an evolutionary search algorithm to find the layer-wise sparsity for channel pruning. More recently,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "several studies (Wu et al., 2018; Cai & Vasconcelos, 2020) have focused on using differentiable", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 273, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 273, + 596 + ], + "score": 1.0, + "content": "schemes via gradient-based optimization.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 495, + 506, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Closely related methods. To further improve the compression ratio, several methods propose to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "jointly optimize pruning and quantization strategies. In particular, some works only support weight", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "quantization (Tung & Mori, 2018; Ye et al., 2019) or use fine-grained pruning (Yang et al., 2020).", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "However, the resultant networks cannot be implemented efficiently on edge devices. Recently, sev-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "eral methods (Wu et al., 2018; Wang et al., 2020; Ying et al., 2020) have been proposed to consider", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "filter pruning, weight quantization, and activation quantization jointly. 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In this paper, we seek to conduct filter pruning and quan-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 600, + 340, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 340, + 611 + ], + "score": 1.0, + "content": "tization simultaneously by solving the following problem:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 566, + 506, + 611 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 613, + 356, + 632 + ], + "lines": [ + { + "bbox": [ + 255, + 613, + 356, + 632 + ], + "spans": [ + { + "bbox": [ + 255, + 613, + 356, + 632 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\underset { \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } } { \\operatorname* { m i n } } \\mathcal { L } \\left( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "acdc9ee7f8532885cb70959a5f2d7dcdee3d75c6506976b74d9dc8d67715230d.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 255, + 613, + 356, + 632 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 134, + 648 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 636, + 154, + 648 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 636, + 264, + 648 + ], + "score": 1.0, + "content": "denotes some losses, and", + "type": "text" + }, + { + "bbox": [ + 264, + 636, + 278, + 646 + ], + "score": 0.37, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 636, + 425, + 648 + ], + "score": 1.0, + "content": "is the parameters of the network.", + "type": "text" + }, + { + "bbox": [ + 426, + 637, + 438, + 646 + ], + "score": 0.86, + "content": "\\alpha ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 636, + 459, + 648 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 460, + 637, + 471, + 646 + ], + "score": 0.87, + "content": "\\alpha ^ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "are the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "pruning and quantization configurations, respectively. As shown in Eq. (7), we propose to encode", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "all compression configurations in a single-path super-bit model (See Figure 1(b)). In the following,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 669, + 493, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 493, + 681 + ], + "score": 1.0, + "content": "we first introduce the bit sharing decomposition and then describe how to learn for compression.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 636, + 506, + 681 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 685, + 503, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "To illustrate the bit sharing decomposition, we begin with an example of 2-bit quantization for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 697, + 471, + 710 + ], + "spans": [ + { + "bbox": [ + 107, + 697, + 160, + 709 + ], + "score": 0.93, + "content": "z \\in \\{ z _ { x } , z _ { w } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 697, + 429, + 710 + ], + "score": 1.0, + "content": ". Specifically, we consider using the following equation to quantize", + "type": "text" + }, + { + "bbox": [ + 429, + 699, + 436, + 707 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 697, + 471, + 710 + ], + "score": 1.0, + "content": "to 2-bit:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 686, + 505, + 710 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 711, + 370, + 735 + ], + "lines": [ + { + "bbox": [ + 240, + 711, + 370, + 735 + ], + "spans": [ + { + "bbox": [ + 240, + 711, + 370, + 735 + ], + "score": 0.93, + "content": "z _ { 2 } = D ( z , s _ { 2 } ) , \\quad s _ { 2 } = \\frac { 1 } { 2 ^ { 2 } - 1 } ,", + "type": "interline_equation", + "image_path": "feff15567b30f65afec1b3655c3c5609d798002c2b467c341bbd4fb7e0cf0d06.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 240, + 711, + 370, + 735 + ], + "spans": [], + "index": 43 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 133, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 84, + 144, + 93 + ], + "score": 0.84, + "content": "z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 82, + 162, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 84, + 172, + 93 + ], + "score": 0.85, + "content": "s _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "are the quantized value and the step size of 2-bit quantization, respectively. Due to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 255, + 106 + ], + "score": 1.0, + "content": "the large step size, the residual error", + "type": "text" + }, + { + "bbox": [ + 255, + 93, + 352, + 106 + ], + "score": 0.93, + "content": "z - z _ { 2 } \\in [ - s _ { 2 } / 2 , s _ { 2 } / 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "may be big and result in a significant", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "performance drop. To reduce the residual error, an intuitive way is to use a smaller step size, which", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 211, + 128 + ], + "score": 1.0, + "content": "indicates that we quantize", + "type": "text" + }, + { + "bbox": [ + 211, + 117, + 218, + 125 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 115, + 376, + 128 + ], + "score": 1.0, + "content": "to a higher bitwidth. Since the step size", + "type": "text" + }, + { + "bbox": [ + 376, + 115, + 442, + 127 + ], + "score": 0.93, + "content": "s _ { 4 } = 1 / ( 2 ^ { 4 } - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "in 4-bit quanti-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 240, + 139 + ], + "score": 1.0, + "content": "zation is a divisor of the step size", + "type": "text" + }, + { + "bbox": [ + 241, + 128, + 251, + 137 + ], + "score": 0.85, + "content": "s _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "in 2-bit quantization, the quantized values of 2-bit quantization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "are among the ones of 4-bit quantization. In fact, based on 2-bit quantization, the 4-bit counterpart", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 375, + 161 + ], + "score": 1.0, + "content": "introduces additional unshared quantized values. 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In contrast, if", + "type": "text" + }, + { + "bbox": [ + 426, + 160, + 437, + 170 + ], + "score": 0.86, + "content": "z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "is with non-zero", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 306, + 183 + ], + "score": 1.0, + "content": "residual error, 4-bit quantization is likely to map", + "type": "text" + }, + { + "bbox": [ + 306, + 173, + 313, + 180 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "to the unshared quantized values. In this case,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 467, + 194 + ], + "score": 1.0, + "content": "4-bit quantization can be regarded as performing quantized value re-assignment based on", + "type": "text" + }, + { + "bbox": [ + 467, + 183, + 477, + 192 + ], + "score": 0.83, + "content": "z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 181, + 505, + 194 + ], + "score": 1.0, + "content": ". Such", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 193, + 324, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 324, + 203 + ], + "score": 1.0, + "content": "a re-assignment process can be formulated as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 277, + 206, + 334, + 218 + ], + "lines": [ + { + "bbox": [ + 277, + 206, + 334, + 218 + ], + "spans": [ + { + "bbox": [ + 277, + 206, + 334, + 218 + ], + "score": 0.91, + "content": "z _ { 4 } = z _ { 2 } + \\epsilon _ { 4 } ,", + "type": "interline_equation", + "image_path": "914ddc0fcb66dfa7a994e7f4cbeb637f164debdfb8f0ff4345e1fafe1a1e463e.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 277, + 206, + 334, + 218 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 133, + 231 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 221, + 144, + 231 + ], + "score": 0.84, + "content": "z _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 219, + 273, + 231 + ], + "score": 1.0, + "content": "is the 4-bit quantized value and", + "type": "text" + }, + { + "bbox": [ + 273, + 221, + 283, + 231 + ], + "score": 0.87, + "content": "\\epsilon _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 219, + 430, + 231 + ], + "score": 1.0, + "content": "is the re-assignment offset based on", + "type": "text" + }, + { + "bbox": [ + 430, + 221, + 441, + 231 + ], + "score": 0.84, + "content": "z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 219, + 505, + 231 + ], + "score": 1.0, + "content": ". To ensure that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 472, + 243 + ], + "score": 1.0, + "content": "the results of re-assignment fall into the unshared quantized values, the re-assignment offset", + "type": "text" + }, + { + "bbox": [ + 472, + 232, + 482, + 241 + ], + "score": 0.85, + "content": "\\epsilon _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "must", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 289, + 254 + ], + "score": 1.0, + "content": "be an integer multiplying of the 4-bit step size", + "type": "text" + }, + { + "bbox": [ + 290, + 243, + 300, + 252 + ], + "score": 0.85, + "content": "s _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 241, + 344, + 254 + ], + "score": 1.0, + "content": ". Formally,", + "type": "text" + }, + { + "bbox": [ + 345, + 243, + 354, + 252 + ], + "score": 0.85, + "content": "\\epsilon _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "can be computed by performing 4-bit", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 267, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 252, + 265 + ], + "score": 1.0, + "content": "quantization on the residual error of", + "type": "text" + }, + { + "bbox": [ + 252, + 254, + 262, + 263 + ], + "score": 0.85, + "content": "z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 252, + 267, + 265 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 264, + 403, + 289 + ], + "lines": [ + { + "bbox": [ + 208, + 264, + 403, + 289 + ], + "spans": [ + { + "bbox": [ + 208, + 264, + 403, + 289 + ], + "score": 0.92, + "content": "\\epsilon _ { 4 } = D ( z - z _ { 2 } , s _ { 4 } ) , ~ s _ { 4 } = \\frac { s _ { 2 } } { 2 ^ { 2 } + 1 } = \\frac { 1 } { 2 ^ { 4 } - 1 } .", + "type": "interline_equation", + "image_path": "40a85ea4827df5a7f1402b5df91135764a8572f329a4aae1266af84f9f000bbb.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 208, + 264, + 403, + 289 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 290, + 504, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "Therefore, according to Eq. (9), a 4-bit quantized value can be decomposed into the 2-bit represen-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "tation and its re-assignment offset. Similarly, an 8-bit quantized value can also be decomposed into", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "the 4-bit representation and its corresponding re-assignment offset. In this way, we can generalize", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 381, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 381, + 334 + ], + "score": 1.0, + "content": "the idea of decomposition to arbitrary effective bitwidths as follows.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 504, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 311, + 355 + ], + "score": 1.0, + "content": "Definition 1 (Quantization decomposition) Let", + "type": "text" + }, + { + "bbox": [ + 311, + 342, + 352, + 353 + ], + "score": 0.9, + "content": "z \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 340, + 505, + 355 + ], + "score": 1.0, + "content": "be a normalized full-precision input,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 353, + 155, + 365 + ], + "score": 0.91, + "content": "\\big \\{ b _ { 1 } , . . . , b _ { K } \\big \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 352, + 334, + 366 + ], + "score": 1.0, + "content": "be a sequence of candidate bitwidths, and", + "type": "text" + }, + { + "bbox": [ + 335, + 353, + 450, + 364 + ], + "score": 0.91, + "content": "b _ { 1 } ~ < ~ b _ { 2 } , . . . , < ~ b _ { K - 1 } < b _ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 352, + 506, + 366 + ], + "score": 1.0, + "content": ". We use the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 269, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 188, + 376 + ], + "score": 1.0, + "content": "following quantized", + "type": "text" + }, + { + "bbox": [ + 188, + 364, + 194, + 373 + ], + "score": 0.82, + "content": "\\widehat { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 363, + 258, + 376 + ], + "score": 1.0, + "content": "to approximate", + "type": "text" + }, + { + "bbox": [ + 258, + 366, + 264, + 373 + ], + "score": 0.69, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 363, + 269, + 376 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 376, + 468, + 412 + ], + "lines": [ + { + "bbox": [ + 126, + 376, + 468, + 412 + ], + "spans": [ + { + "bbox": [ + 126, + 376, + 468, + 412 + ], + "score": 0.91, + "content": "{ \\widehat { z } } = z _ { b _ { 1 } } + \\sum _ { j = 2 } ^ { K } \\epsilon _ { b _ { j } } , \\quad { \\mathrm { w h e r e ~ } } \\epsilon _ { b _ { j } } = D ( z - z _ { b _ { j - 1 } } , s _ { b _ { j } } ) , \\quad s _ { b _ { j } } = { \\frac { s _ { b _ { j - 1 } } } { 2 ^ { b _ { j - 1 } } + 1 } } = { \\frac { 1 } { 2 ^ { b _ { j } } - 1 } } .", + "type": "interline_equation", + "image_path": "633098f514838f33081ed1e04b334ceb82f4b71b545ba123f44581a75f7b07e8.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 126, + 376, + 468, + 388.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 126, + 388.0, + 468, + 400.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 126, + 400.0, + 468, + 412.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 504, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 290, + 430 + ], + "score": 1.0, + "content": "In other words, the quantized approximation", + "type": "text" + }, + { + "bbox": [ + 290, + 419, + 298, + 428 + ], + "score": 0.81, + "content": "\\widehat { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "can be decomposed into the sum of the lowest bit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "bquantization and a series of recursive re-assignment offsets. 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To get", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 312, + 504 + ], + "score": 1.0, + "content": "a hardware-friendly compressed network1, we set", + "type": "text" + }, + { + "bbox": [ + 312, + 492, + 322, + 502 + ], + "score": 0.88, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "to 2, which ensures that all the decomposi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "tion bitwidths are power-of-two. 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Such", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 193, + 324, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 324, + 203 + ], + "score": 1.0, + "content": "a re-assignment process can be formulated as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 82, + 506, + 203 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 277, + 206, + 334, + 218 + ], + "lines": [ + { + "bbox": [ + 277, + 206, + 334, + 218 + ], + "spans": [ + { + "bbox": [ + 277, + 206, + 334, + 218 + ], + "score": 0.91, + "content": "z _ { 4 } = z _ { 2 } + \\epsilon _ { 4 } ,", + "type": "interline_equation", + "image_path": "914ddc0fcb66dfa7a994e7f4cbeb637f164debdfb8f0ff4345e1fafe1a1e463e.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 277, + 206, + 334, + 218 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 133, + 231 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 221, + 144, + 231 + ], + "score": 0.84, + "content": "z _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 219, + 273, + 231 + ], + "score": 1.0, + "content": "is the 4-bit quantized value and", + "type": "text" + }, + { + "bbox": [ + 273, + 221, + 283, + 231 + ], + "score": 0.87, + "content": "\\epsilon _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 219, + 430, + 231 + ], + "score": 1.0, + "content": "is the re-assignment offset based on", + "type": "text" + }, + { + "bbox": [ + 430, + 221, + 441, + 231 + ], + "score": 0.84, + "content": "z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 219, + 505, + 231 + ], + "score": 1.0, + "content": ". To ensure that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 472, + 243 + ], + "score": 1.0, + "content": "the results of re-assignment fall into the unshared quantized values, the re-assignment offset", + "type": "text" + }, + { + "bbox": [ + 472, + 232, + 482, + 241 + ], + "score": 0.85, + "content": "\\epsilon _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "must", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 289, + 254 + ], + "score": 1.0, + "content": "be an integer multiplying of the 4-bit step size", + "type": "text" + }, + { + "bbox": [ + 290, + 243, + 300, + 252 + ], + "score": 0.85, + "content": "s _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 241, + 344, + 254 + ], + "score": 1.0, + "content": ". 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(9), a 4-bit quantized value can be decomposed into the 2-bit represen-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "tation and its re-assignment offset. Similarly, an 8-bit quantized value can also be decomposed into", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "the 4-bit representation and its corresponding re-assignment offset. In this way, we can generalize", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 381, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 381, + 334 + ], + "score": 1.0, + "content": "the idea of decomposition to arbitrary effective bitwidths as follows.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 289, + 506, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 504, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 311, + 355 + ], + "score": 1.0, + "content": "Definition 1 (Quantization decomposition) Let", + "type": "text" + }, + { + "bbox": [ + 311, + 342, + 352, + 353 + ], + "score": 0.9, + "content": "z \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 340, + 505, + 355 + ], + "score": 1.0, + "content": "be a normalized full-precision input,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 353, + 155, + 365 + ], + "score": 0.91, + "content": "\\big \\{ b _ { 1 } , . . . , b _ { K } \\big \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 352, + 334, + 366 + ], + "score": 1.0, + "content": "be a sequence of candidate bitwidths, and", + "type": "text" + }, + { + "bbox": [ + 335, + 353, + 450, + 364 + ], + "score": 0.91, + "content": "b _ { 1 } ~ < ~ b _ { 2 } , . . . , < ~ b _ { K - 1 } < b _ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 352, + 506, + 366 + ], + "score": 1.0, + "content": ". We use the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 269, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 188, + 376 + ], + "score": 1.0, + "content": "following quantized", + "type": "text" + }, + { + "bbox": [ + 188, + 364, + 194, + 373 + ], + "score": 0.82, + "content": "\\widehat { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 363, + 258, + 376 + ], + "score": 1.0, + "content": "to approximate", + "type": "text" + }, + { + "bbox": [ + 258, + 366, + 264, + 373 + ], + "score": 0.69, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 363, + 269, + 376 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 340, + 506, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 376, + 468, + 412 + ], + "lines": [ + { + "bbox": [ + 126, + 376, + 468, + 412 + ], + "spans": [ + { + "bbox": [ + 126, + 376, + 468, + 412 + ], + "score": 0.91, + "content": "{ \\widehat { z } } = z _ { b _ { 1 } } + \\sum _ { j = 2 } ^ { K } \\epsilon _ { b _ { j } } , \\quad { \\mathrm { w h e r e ~ } } \\epsilon _ { b _ { j } } = D ( z - z _ { b _ { j - 1 } } , s _ { b _ { j } } ) , \\quad s _ { b _ { j } } = { \\frac { s _ { b _ { j - 1 } } } { 2 ^ { b _ { j - 1 } } + 1 } } = { \\frac { 1 } { 2 ^ { b _ { j } } - 1 } } .", + "type": "interline_equation", + "image_path": "633098f514838f33081ed1e04b334ceb82f4b71b545ba123f44581a75f7b07e8.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 126, + 376, + 468, + 388.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 126, + 388.0, + 468, + 400.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 126, + 400.0, + 468, + 412.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 504, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 290, + 430 + ], + "score": 1.0, + "content": "In other words, the quantized approximation", + "type": "text" + }, + { + "bbox": [ + 290, + 419, + 298, + 428 + ], + "score": 0.81, + "content": "\\widehat { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "can be decomposed into the sum of the lowest bit", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "bquantization and a series of recursive re-assignment offsets. 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To get", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 312, + 504 + ], + "score": 1.0, + "content": "a hardware-friendly compressed network1, we set", + "type": "text" + }, + { + "bbox": [ + 312, + 492, + 322, + 502 + ], + "score": 0.88, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "to 2, which ensures that all the decomposi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "tion bitwidths are power-of-two. Moreover, since 8-bit quantization achieves lossless performance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "compared with the full-precision counterpart (Zhou et al., 2016), we only consider those candidate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 524, + 475, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 433, + 537 + ], + "score": 1.0, + "content": "bitwidths that are not greater than 8-bit. In other words, we constrain the value of", + "type": "text" + }, + { + "bbox": [ + 433, + 525, + 439, + 536 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 524, + 475, + 537 + ], + "score": 1.0, + "content": "to [1, 3].", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 479, + 506, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "Remark 1 The proposed bit sharing decomposition has several advantages. First, the proposed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "method only needs to maintain a small number of trainable parameters, which greatly reduces the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "computational costs during search. 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By jointly training the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "binary gates and the network parameters, the pruning ratio and bitwidth of each layer can be auto-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "matically determined. However, the gradient approximation of the binary gate inevitably introduces", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "noisy signals, which can be even more severe when we quantize both weights and activations. 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Following (Cai", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 624, + 469, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 469, + 638 + ], + "score": 1.0, + "content": "et al., 2019), we train the network and architecture by minimizing following loss function:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 644, + 422, + 658 + ], + "lines": [ + { + "bbox": [ + 189, + 644, + 422, + 658 + ], + "spans": [ + { + "bbox": [ + 189, + 644, + 422, + 658 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) = { \\mathcal { L } } _ { c e } ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) + \\lambda \\log R ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) ,", + "type": "interline_equation", + "image_path": "bacf4d1b5c584d7debfc105f296c85a3c9f0eccd4e93446014b877a693a6d7cd.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 189, + 644, + 422, + 658 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 134, + 678 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 666, + 160, + 678 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { c e } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 665, + 267, + 678 + ], + "score": 1.0, + "content": "is the cross-entropy loss,", + "type": "text" + }, + { + "bbox": [ + 268, + 666, + 286, + 678 + ], + "score": 0.92, + "content": "R ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 665, + 479, + 678 + ], + "score": 1.0, + "content": "is the computational costs of the network and", + "type": "text" + }, + { + "bbox": [ + 479, + 666, + 486, + 676 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "balancing hyper-parameter. 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Here,", + "type": "text" + }, + { + "bbox": [ + 300, + 240, + 312, + 249 + ], + "score": 0.86, + "content": "\\alpha ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 238, + 506, + 253 + ], + "score": 1.0, + "content": "is a layer-wise threshold for filter pruning. Fol-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 262, + 263 + ], + "score": 1.0, + "content": "lowing PFEC (Li et al., 2017a), we use", + "type": "text" + }, + { + "bbox": [ + 263, + 252, + 272, + 261 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "-norm to evaluate the importance of the filter. 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However, the indicator func-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "tion in Eq. (13) is non-differentiable. To address this, we use straight-through estimator (STE) (Ben-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 309, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 462, + 326 + ], + "score": 1.0, + "content": "gio et al., 2013; Zhou et al., 2016) to approximate the gradient of the indicator function", + "type": "text" + }, + { + "bbox": [ + 462, + 311, + 479, + 323 + ], + "score": 0.76, + "content": "\\mathbb { 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 309, + 506, + 326 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 388, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 252, + 334 + ], + "score": 1.0, + "content": "the gradient of the sigmoid function", + "type": "text" + }, + { + "bbox": [ + 252, + 322, + 270, + 334 + ], + "score": 0.9, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 322, + 388, + 334 + ], + "score": 1.0, + "content": ", which can be formulated as:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 277, + 506, + 334 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 341, + 442, + 367 + ], + "lines": [ + { + "bbox": [ + 167, + 341, + 442, + 367 + ], + "spans": [ + { + "bbox": [ + 167, + 341, + 442, + 367 + ], + "score": 0.93, + "content": "\\frac { \\partial g } { \\partial \\alpha } = \\frac { \\partial \\mathbb { 1 } \\left( A - \\alpha \\right) } { \\partial \\alpha } \\approx \\frac { \\partial \\sigma \\left( A - \\alpha \\right) } { \\partial \\alpha } = - \\sigma \\left( A - \\alpha \\right) \\left( 1 - \\sigma ( A - \\alpha ) \\right) ,", + "type": "interline_equation", + "image_path": "348f53bd9dc487c866d7aa0cc55a7f93a8d8ad7609ca0c65a75ca5303db279f1.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 167, + 341, + 442, + 367 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 373, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 133, + 387 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 376, + 140, + 385 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 373, + 262, + 387 + ], + "score": 1.0, + "content": "is the output of a binary gate,", + "type": "text" + }, + { + "bbox": [ + 263, + 374, + 319, + 386 + ], + "score": 0.93, + "content": "\\alpha \\in \\{ \\alpha ^ { p } , \\alpha ^ { q } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 373, + 462, + 387 + ], + "score": 1.0, + "content": "is the corresponding threshold and", + "type": "text" + }, + { + "bbox": [ + 462, + 374, + 471, + 384 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 217, + 397 + ], + "score": 1.0, + "content": "some specific metrics (i.e.,", + "type": "text" + }, + { + "bbox": [ + 218, + 385, + 228, + 396 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "-norm of the filter or the quantization error). By jointly training the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "binary gates and the network parameters, the pruning ratio and bitwidth of each layer can be auto-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "matically determined. However, the gradient approximation of the binary gate inevitably introduces", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "noisy signals, which can be even more severe when we quantize both weights and activations. Thus,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "we propose to train the binary gates of weights and activations in an alternative manner. Specifically,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 440, + 490, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 490, + 452 + ], + "score": 1.0, + "content": "when training the binary gates of weights, we fix the binary gates of activations, and vice versa.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 373, + 506, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 504, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 424, + 469 + ], + "score": 1.0, + "content": "Search Space for Model Compression. Given an uncompressed network with", + "type": "text" + }, + { + "bbox": [ + 424, + 457, + 433, + 466 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 456, + 492, + 469 + ], + "score": 1.0, + "content": "layers, we use", + "type": "text" + }, + { + "bbox": [ + 492, + 457, + 504, + 468 + ], + "score": 0.85, + "content": "C _ { l }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 254, + 480 + ], + "score": 1.0, + "content": "to denote the number of filters at the", + "type": "text" + }, + { + "bbox": [ + 254, + 468, + 259, + 477 + ], + "score": 0.7, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "-th layer. To obtain the compressed model, we first divide the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "filters of each layer into groups and then search for the optimal bitwidths for the considered layer.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 488, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 488, + 123, + 505 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 491, + 132, + 500 + ], + "score": 0.77, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 488, + 343, + 505 + ], + "score": 1.0, + "content": "be the number of filters in a group. For any layer", + "type": "text" + }, + { + "bbox": [ + 343, + 491, + 348, + 501 + ], + "score": 0.47, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 488, + 417, + 505 + ], + "score": 1.0, + "content": ", there would be", + "type": "text" + }, + { + "bbox": [ + 418, + 489, + 438, + 504 + ], + "score": 0.91, + "content": "\\displaystyle \\left\\lfloor \\frac { C _ { l } } { B } \\right\\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 488, + 506, + 505 + ], + "score": 1.0, + "content": "groups in total.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 322, + 515 + ], + "score": 1.0, + "content": "Since we quantize both weights and activations, given", + "type": "text" + }, + { + "bbox": [ + 322, + 503, + 333, + 513 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 502, + 452, + 515 + ], + "score": 1.0, + "content": "candidate bitwidths, there are", + "type": "text" + }, + { + "bbox": [ + 453, + 502, + 467, + 513 + ], + "score": 0.88, + "content": "K ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "different", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 421, + 527 + ], + "score": 1.0, + "content": "quantization configurations for each layer. Thus, for the whole network with", + "type": "text" + }, + { + "bbox": [ + 421, + 515, + 429, + 524 + ], + "score": 0.83, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "layers, the size of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 525, + 265, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 174, + 538 + ], + "score": 1.0, + "content": "the search space", + "type": "text" + }, + { + "bbox": [ + 174, + 525, + 182, + 535 + ], + "score": 0.81, + "content": "\\Omega", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 525, + 265, + 538 + ], + "score": 1.0, + "content": "can be computed by", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 456, + 506, + 538 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 544, + 362, + 579 + ], + "lines": [ + { + "bbox": [ + 249, + 544, + 362, + 579 + ], + "spans": [ + { + "bbox": [ + 249, + 544, + 362, + 579 + ], + "score": 0.95, + "content": "\\left| \\Omega \\right| = \\prod _ { l = 1 } ^ { L } \\left( K ^ { 2 } \\times \\left\\lfloor \\frac { C _ { l } } { B } \\right\\rfloor \\right) .", + "type": "interline_equation", + "image_path": "2561053e1746674833c80011cd7c51e168cc9c093d44f69baa48713a1505b224.jpg" + } + ] + } + ], + "index": 35.5, + "virtual_lines": [ + { + "bbox": [ + 249, + 544, + 362, + 561.5 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 249, + 561.5, + 362, + 579.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 586, + 504, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 600 + ], + "score": 1.0, + "content": "Eq. (16) indicates that the search space is large enough to cover the potentially good configurations.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 584, + 505, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 504, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 504, + 615 + ], + "score": 1.0, + "content": "Training Objective. To design a hardware-efficient network, the objective function in Eq. (7) should", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "reflect both the accuracy of the compressed network and its computational costs. Following (Cai", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 624, + 469, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 469, + 638 + ], + "score": 1.0, + "content": "et al., 2019), we train the network and architecture by minimizing following loss function:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 603, + 505, + 638 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 644, + 422, + 658 + ], + "lines": [ + { + "bbox": [ + 189, + 644, + 422, + 658 + ], + "spans": [ + { + "bbox": [ + 189, + 644, + 422, + 658 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) = { \\mathcal { L } } _ { c e } ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) + \\lambda \\log R ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) ,", + "type": "interline_equation", + "image_path": "bacf4d1b5c584d7debfc105f296c85a3c9f0eccd4e93446014b877a693a6d7cd.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 189, + 644, + 422, + 658 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 134, + 678 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 666, + 160, + 678 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { c e } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 665, + 267, + 678 + ], + "score": 1.0, + "content": "is the cross-entropy loss,", + "type": "text" + }, + { + "bbox": [ + 268, + 666, + 286, + 678 + ], + "score": 0.92, + "content": "R ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 665, + 479, + 678 + ], + "score": 1.0, + "content": "is the computational costs of the network and", + "type": "text" + }, + { + "bbox": [ + 479, + 666, + 486, + 676 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "balancing hyper-parameter. Following single-path NAS (Stamoulis et al., 2019), we use a similar", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "formulation of computational costs to preserve the differentiability of the objective function. The", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "details of the differentiable computational loss can be found in Appendix B. Once the training is fin-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "ished, we can obtain the compressed network by selecting those filters and bitwidths with activated", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 470, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 470, + 733 + ], + "score": 1.0, + "content": "binary gates. 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NetworkMethodBOPs (M)BOP comp.ratioWIATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-20Full-precision41798.61.032.0/32.067.590.8
4-bit precision674.662.04.0/4.067.8±0.390.4±0.2
DQ1180.035.45.3/6.167.7±0.690.4±0.5
HAQ653.464.03.7/4.267.7±0.190.4±0.3
DNAS660.062.94.6/3.867.8±0.390.4±0.2
ABS-P(Ours)28586.51.532.0/32.067.9±0.190.7±0.2
ABS-Q(Ours) ABS (Ours)649.564.44.4/4.268.1±0.190.5±0.0
630.666.34.4/4.268.1±0.390.6±0.2
ResNet-56Full-precision128771.71.032.0/32.071.792.2
4-bit precision2033.663.34.0/4.070.9±0.391.2±0.4
DQ2222.957.93.8 /4.670.7±0.291.4±0.4
HAQ2014.963.93.3/4.971.2±0.191.1±0.2
DNAS2035.765.35.3/3.271.2±0.291.3±0.3
ABS-P(Ours)87021.61.532.0/32.071.5±0.191.8±0.2
ABS-Q(Ours)1970.765.34.1/4.071.5±0.291.5±0.2
ABS (Ours)1918.867.14.2/4.171.6±0.191.8±0.4
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We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 418, + 441, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 441, + 430 + ], + "score": 1.0, + "content": "use different methods to compress ResNet-56 and report the results on CIFAR-100.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "title", + "bbox": [ + 108, + 438, + 200, + 450 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 201, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 201, + 452 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "Compared methods. 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NetworkMethodBOPs (M)BOP comp.ratioWIATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-20Full-precision41798.61.032.0/32.067.590.8
4-bit precision674.662.04.0/4.067.8±0.390.4±0.2
DQ1180.035.45.3/6.167.7±0.690.4±0.5
HAQ653.464.03.7/4.267.7±0.190.4±0.3
DNAS660.062.94.6/3.867.8±0.390.4±0.2
ABS-P(Ours)28586.51.532.0/32.067.9±0.190.7±0.2
ABS-Q(Ours) ABS (Ours)649.564.44.4/4.268.1±0.190.5±0.0
630.666.34.4/4.268.1±0.390.6±0.2
ResNet-56Full-precision128771.71.032.0/32.071.792.2
4-bit precision2033.663.34.0/4.070.9±0.391.2±0.4
DQ2222.957.93.8 /4.670.7±0.291.4±0.4
HAQ2014.963.93.3/4.971.2±0.191.1±0.2
DNAS2035.765.35.3/3.271.2±0.291.3±0.3
ABS-P(Ours)87021.61.532.0/32.071.5±0.191.8±0.2
ABS-Q(Ours)1970.765.34.1/4.071.5±0.291.5±0.2
ABS (Ours)1918.867.14.2/4.171.6±0.191.8±0.4
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Following HAQ (Wang et al., 2019), we quantize all the layers, in which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "the first and the last layers are quantized to 8-bit. Following ThiNet (Luo et al., 2017), we only", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "conduct filter pruning for the first layer in the residual block. For ResNet-20 and ResNet-56 on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 294, + 645 + ], + "score": 1.0, + "content": "CIFAR-100 (Krizhevsky et al., 2009), we set", + "type": "text" + }, + { + "bbox": [ + 294, + 633, + 304, + 643 + ], + "score": 0.77, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 633, + 504, + 645 + ], + "score": 1.0, + "content": "to 4. For ResNet-18 and MobileNetV2 on Ima-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 243, + 657 + ], + "score": 1.0, + "content": "geNet (Russakovsky et al., 2015),", + "type": "text" + }, + { + "bbox": [ + 244, + 644, + 253, + 654 + ], + "score": 0.78, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "is set to 16 and 8, respectively. We first train the full-precision", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "models and then use the pretrained weights to initialize the compressed models. Following (Li", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "et al., 2020; Esser et al., 2020), we introduce weight normalization during training. We use SGD", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "with nesterov (Nesterov, 1983) for optimization, with a momentum of 0.9. For CIFAR-100, we use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "the same data augmentation as in (He et al., 2016), including translation and horizontal flipping.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 256, + 712 + ], + "score": 1.0, + "content": "For ImageNet, images are resized to", + "type": "text" + }, + { + "bbox": [ + 256, + 699, + 300, + 709 + ], + "score": 0.89, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 697, + 349, + 712 + ], + "score": 1.0, + "content": ", and then a", + "type": "text" + }, + { + "bbox": [ + 349, + 699, + 394, + 709 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "patch is randomly cropped", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 356, + 722 + ], + "score": 1.0, + "content": "from an image or its horizontal flip for training. For testing, a", + "type": "text" + }, + { + "bbox": [ + 356, + 710, + 400, + 720 + ], + "score": 0.89, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "center cropped is chosen.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "We first train the uncompressed network for 30 epochs on CIFAR-100 and 10 epochs on ImageNet.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 599, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 133, + 126, + 478, + 253 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 505, + 123 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 506, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 506, + 102 + ], + "score": 1.0, + "content": "Table 2: Comparisons on ImageNet. “*” denotes that we get the results from the figures in (van", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "score": 1.0, + "content": "Baalen et al., 2020) and “–” denotes that the results are not reported. Moreover, “W” and “A”", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 457, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 457, + 124 + ], + "score": 1.0, + "content": "represent the average quantization bitwidth of the weights and activations, respectively.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 133, + 126, + 478, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 126, + 478, + 253 + ], + "spans": [ + { + "bbox": [ + 133, + 126, + 478, + 253 + ], + "score": 0.985, + "html": "
NetworkMethodBOPs (G)BOP comp. ratioW/ATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-18Full-precision1857.61.032.0/32.070.789.8
4-bit precision34.753.54.0/4.071.089.8
DQ40.740.61/168.51
DJPQ35.552.3-/169.1
HAQ34.753.54.0/4.070.289.5
Bayesian Bits*35.951.71/169.5
ABS-Q(Ours)33.156.14.5/3.870.989.7
ABS (Ours)32.357.54.6/4.270.889.6
MobileNetV2Full-precision308.01.032.0/32.071.990.3
6-bit precision11.227.56.0/6.071.890.3
DQ19.61.96.8/8.070.489.7
HAQ10.828.55.61/6.2771.290.0
Bayesian Bits *10.828.5-/170.9一
ABS-Q(Ours)10.928.36.8/6.871.890.4
ABS (Ours)10.828.56.1/7.171.790.3
", + "type": "table", + "image_path": "b48ed4cdd06f980611dd8f000a83d2078455f970f9577aafb1e0cbad855ca289.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 133, + 126, + 478, + 168.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 133, + 168.33333333333334, + 478, + 210.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 133, + 210.66666666666669, + 478, + 253.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 261, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "score": 1.0, + "content": "The learning rate is set to 0.001. We then fine-tune the searched compressed network to recover the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "performance drop. On CIFAR-100, we train the searched network for 200 epochs with a mini-batch", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "size of 128. The learning rate is initialized to 0.1 and is divided by 10 at 80-th and 120-th epochs.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "Experiments on CIFAR-100 are repeated for 5 times and we report the mean and standard deviation.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "For ResNet-18 on ImageNet, we finetune the searched network for 90 epochs with a mini-batch size", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "of 256. For MobileNetV2 on ImageNet, we fine-tune for 150 epochs. For all models on ImageNet,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 327, + 483, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 483, + 340 + ], + "score": 1.0, + "content": "the learning rate starts at 0.01 and decays with cosine annealing (Loshchilov & Hutter, 2017).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 352, + 196, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 198, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 198, + 365 + ], + "score": 1.0, + "content": "4.1 MAIN RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "We apply the proposed methods to compress ResNet-20, ResNet-56 on CIFAR-100 and ResNet-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "18, MobileNetV2 on ImageNet. We compare the performance of different methods in Table 1 and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "score": 1.0, + "content": "Table 2. We also show the results of the compressed ResNet-56 with different BOPs and memory", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "footprints in Figure 2. From the results, we can see that 4-bit quantized networks achieve lossless", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 333, + 429 + ], + "score": 1.0, + "content": "performance. Also, 6-bit MobileNetV2 only leads to a", + "type": "text" + }, + { + "bbox": [ + 333, + 417, + 355, + 428 + ], + "score": 0.86, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "performance drop on the Top-1 Ac-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "curacy. Compared with fixed-precision quantization, mixed-precision methods are able to reduce", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "the BOPs while preserving the performance. Critically, our proposed ABS-Q outperforms the state-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "of-the-arts baselines with less computational costs. Specifically, ABS-Q compressed ResNet-18", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "outperforms the one compressed by HAQ with more BOPs reduction. More critically, our proposed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "ABS achieves significant improvement in terms of BOPs and memory footprints. For example, in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "Figure 2(a), our ABS compressed ResNet-56 model yields much fewer BOPs (395.25 vs. 536.24)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "score": 1.0, + "content": "but achieves comparable performance compared with the fixed-precision counterpart. Moreover,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "by combing pruning and quantization, ABS achieves nearly lossless performance while further re-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "ducing the computational costs of ABS-Q. For example, ABS compressed ResNet-18 reduces the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 145, + 540 + ], + "score": 1.0, + "content": "BOPs by", + "type": "text" + }, + { + "bbox": [ + 145, + 527, + 172, + 537 + ], + "score": 0.87, + "content": "5 7 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 526, + 394, + 540 + ], + "score": 1.0, + "content": "while still outperforming the full-precision network by", + "type": "text" + }, + { + "bbox": [ + 394, + 527, + 416, + 537 + ], + "score": 0.85, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "in terms of the Top-1", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 537, + 201, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 201, + 550 + ], + "score": 1.0, + "content": "accuracy on ImageNet.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 108, + 562, + 210, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 212, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 212, + 574 + ], + "score": 1.0, + "content": "4.2 FURTHER STUDIES", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Effect of the bit-sharing scheme. To investigate the effect of the bit-sharing scheme, we apply our", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "methods to quantize ResNet-20 and ResNet-56 with and without the bit sharing scheme on CIFAR-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "100. We report the testing accuracy and BOPs in Table 3. We also present the search costs and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 390, + 629 + ], + "score": 1.0, + "content": "consumed GPU memory measured on a GPU device (NVIDIA TITAN", + "type": "text" + }, + { + "bbox": [ + 390, + 617, + 405, + 628 + ], + "score": 0.27, + "content": "\\mathrm { X p }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "). It can be seen from the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "results that the method with the bit sharing scheme consistently outperforms the ones without the bit", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 426, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 426, + 651 + ], + "score": 1.0, + "content": "sharing scheme while significantly reducing the search costs and GPU memory.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "Effect of the one-stage compression. To investigate the effect of the one-stage compression scheme", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "(perform pruning and quantization jointly), we extend ABS to two-stage optimization, where we se-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 361, + 690 + ], + "score": 1.0, + "content": "quentially do filter pruning and quantization, denoted as ABS-", + "type": "text" + }, + { + "bbox": [ + 362, + 677, + 408, + 688 + ], + "score": 0.26, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { P } { } \\mathbf { A } \\mathbf { B } \\mathbf { S } { - } \\mathbf { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ". The results are shown", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "in Table 4. Compared with the two-stage counterpart, ABS achieves better performance with less", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "computational costs, which shows the superiority of the one-stage optimization. For example, ABS", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 328, + 723 + ], + "score": 1.0, + "content": "compressed ResNet-56 outperforms the counterpart by", + "type": "text" + }, + { + "bbox": [ + 329, + 710, + 351, + 720 + ], + "score": 0.87, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "on the Top-1 accuracy with less com-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 190, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 190, + 732 + ], + "score": 1.0, + "content": "putational overhead.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 133, + 126, + 478, + 253 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 505, + 123 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 506, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 506, + 102 + ], + "score": 1.0, + "content": "Table 2: Comparisons on ImageNet. “*” denotes that we get the results from the figures in (van", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "score": 1.0, + "content": "Baalen et al., 2020) and “–” denotes that the results are not reported. Moreover, “W” and “A”", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 457, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 457, + 124 + ], + "score": 1.0, + "content": "represent the average quantization bitwidth of the weights and activations, respectively.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 133, + 126, + 478, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 126, + 478, + 253 + ], + "spans": [ + { + "bbox": [ + 133, + 126, + 478, + 253 + ], + "score": 0.985, + "html": "
NetworkMethodBOPs (G)BOP comp. ratioW/ATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-18Full-precision1857.61.032.0/32.070.789.8
4-bit precision34.753.54.0/4.071.089.8
DQ40.740.61/168.51
DJPQ35.552.3-/169.1
HAQ34.753.54.0/4.070.289.5
Bayesian Bits*35.951.71/169.5
ABS-Q(Ours)33.156.14.5/3.870.989.7
ABS (Ours)32.357.54.6/4.270.889.6
MobileNetV2Full-precision308.01.032.0/32.071.990.3
6-bit precision11.227.56.0/6.071.890.3
DQ19.61.96.8/8.070.489.7
HAQ10.828.55.61/6.2771.290.0
Bayesian Bits *10.828.5-/170.9一
ABS-Q(Ours)10.928.36.8/6.871.890.4
ABS (Ours)10.828.56.1/7.171.790.3
", + "type": "table", + "image_path": "b48ed4cdd06f980611dd8f000a83d2078455f970f9577aafb1e0cbad855ca289.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 133, + 126, + 478, + 168.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 133, + 168.33333333333334, + 478, + 210.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 133, + 210.66666666666669, + 478, + 253.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 261, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 504, + 273 + ], + "score": 1.0, + "content": "The learning rate is set to 0.001. We then fine-tune the searched compressed network to recover the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "performance drop. On CIFAR-100, we train the searched network for 200 epochs with a mini-batch", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "size of 128. The learning rate is initialized to 0.1 and is divided by 10 at 80-th and 120-th epochs.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "Experiments on CIFAR-100 are repeated for 5 times and we report the mean and standard deviation.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "For ResNet-18 on ImageNet, we finetune the searched network for 90 epochs with a mini-batch size", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "of 256. For MobileNetV2 on ImageNet, we fine-tune for 150 epochs. For all models on ImageNet,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 327, + 483, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 483, + 340 + ], + "score": 1.0, + "content": "the learning rate starts at 0.01 and decays with cosine annealing (Loshchilov & Hutter, 2017).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 262, + 505, + 340 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 352, + 196, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 198, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 198, + 365 + ], + "score": 1.0, + "content": "4.1 MAIN RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 372, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "We apply the proposed methods to compress ResNet-20, ResNet-56 on CIFAR-100 and ResNet-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "18, MobileNetV2 on ImageNet. We compare the performance of different methods in Table 1 and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 408 + ], + "score": 1.0, + "content": "Table 2. We also show the results of the compressed ResNet-56 with different BOPs and memory", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "footprints in Figure 2. From the results, we can see that 4-bit quantized networks achieve lossless", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 333, + 429 + ], + "score": 1.0, + "content": "performance. Also, 6-bit MobileNetV2 only leads to a", + "type": "text" + }, + { + "bbox": [ + 333, + 417, + 355, + 428 + ], + "score": 0.86, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "performance drop on the Top-1 Ac-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "curacy. Compared with fixed-precision quantization, mixed-precision methods are able to reduce", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "the BOPs while preserving the performance. Critically, our proposed ABS-Q outperforms the state-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "of-the-arts baselines with less computational costs. Specifically, ABS-Q compressed ResNet-18", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "outperforms the one compressed by HAQ with more BOPs reduction. More critically, our proposed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "ABS achieves significant improvement in terms of BOPs and memory footprints. For example, in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "Figure 2(a), our ABS compressed ResNet-56 model yields much fewer BOPs (395.25 vs. 536.24)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "score": 1.0, + "content": "but achieves comparable performance compared with the fixed-precision counterpart. Moreover,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "by combing pruning and quantization, ABS achieves nearly lossless performance while further re-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "ducing the computational costs of ABS-Q. For example, ABS compressed ResNet-18 reduces the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 145, + 540 + ], + "score": 1.0, + "content": "BOPs by", + "type": "text" + }, + { + "bbox": [ + 145, + 527, + 172, + 537 + ], + "score": 0.87, + "content": "5 7 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 526, + 394, + 540 + ], + "score": 1.0, + "content": "while still outperforming the full-precision network by", + "type": "text" + }, + { + "bbox": [ + 394, + 527, + 416, + 537 + ], + "score": 0.85, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "in terms of the Top-1", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 537, + 201, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 201, + 550 + ], + "score": 1.0, + "content": "accuracy on ImageNet.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 372, + 506, + 550 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 562, + 210, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 212, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 212, + 574 + ], + "score": 1.0, + "content": "4.2 FURTHER STUDIES", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Effect of the bit-sharing scheme. To investigate the effect of the bit-sharing scheme, we apply our", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "methods to quantize ResNet-20 and ResNet-56 with and without the bit sharing scheme on CIFAR-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "100. We report the testing accuracy and BOPs in Table 3. We also present the search costs and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 390, + 629 + ], + "score": 1.0, + "content": "consumed GPU memory measured on a GPU device (NVIDIA TITAN", + "type": "text" + }, + { + "bbox": [ + 390, + 617, + 405, + 628 + ], + "score": 0.27, + "content": "\\mathrm { X p }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "). It can be seen from the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "results that the method with the bit sharing scheme consistently outperforms the ones without the bit", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 426, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 426, + 651 + ], + "score": 1.0, + "content": "sharing scheme while significantly reducing the search costs and GPU memory.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 582, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "Effect of the one-stage compression. To investigate the effect of the one-stage compression scheme", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "(perform pruning and quantization jointly), we extend ABS to two-stage optimization, where we se-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 361, + 690 + ], + "score": 1.0, + "content": "quentially do filter pruning and quantization, denoted as ABS-", + "type": "text" + }, + { + "bbox": [ + 362, + 677, + 408, + 688 + ], + "score": 0.26, + "content": "\\mathbf { \\partial } \\cdot \\mathbf { P } { } \\mathbf { A } \\mathbf { B } \\mathbf { S } { - } \\mathbf { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ". The results are shown", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "in Table 4. Compared with the two-stage counterpart, ABS achieves better performance with less", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "computational costs, which shows the superiority of the one-stage optimization. For example, ABS", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 328, + 723 + ], + "score": 1.0, + "content": "compressed ResNet-56 outperforms the counterpart by", + "type": "text" + }, + { + "bbox": [ + 329, + 710, + 351, + 720 + ], + "score": 0.87, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "on the Top-1 accuracy with less com-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 190, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 190, + 732 + ], + "score": 1.0, + "content": "putational overhead.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 654, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 124, + 508, + 172 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 504, + 112 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "score": 1.0, + "content": "Table 3: Effect of the bit-sharing scheme. We report the testing accuracy, BOPs, and search costs on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 445, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 445, + 113 + ], + "score": 1.0, + "content": "CIFAR-100. The search costs are measured on a GPU device (NVIDIA TITAN Xp).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 124, + 508, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 124, + 508, + 172 + ], + "spans": [ + { + "bbox": [ + 108, + 124, + 508, + 172 + ], + "score": 0.978, + "html": "
NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)Search Cost (GPU hours)GPUMemory (GB)
ResNet-20 w/o bit sharing67.8±0.190.5±0.2664.22.84.4
w/bit sharing68.1±0.190.5±0.0649.50.81.5
ResNet-56w/o bit sharing71.3±0.391.4±0.42001.18.710.9
w/ bit sharing71.5±0.291.5±0.21970.71.93.0
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56ABS-P→ABS-Q70.4±0.190.8±0.21077.7
ABS70.8±0.491.2±0.11042.5
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56Joint71.3±0.291.6±0.31942.4
Alternative71.6±0.191.8±0.41918.8
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NetworkMethodLatency-constrainedEnergy-constrained
Acc.-1 (%)Acc.-5 (%)Latency (ms)Acc.-1 (%)Acc.-5 (%)Energy (mJ)
MobileNetV26-bit precision71.890.324.971.890.332.8
ABS (Ours)72.090.417.272.090.326.3
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)Search Cost (GPU hours)GPUMemory (GB)
ResNet-20 w/o bit sharing67.8±0.190.5±0.2664.22.84.4
w/bit sharing68.1±0.190.5±0.0649.50.81.5
ResNet-56w/o bit sharing71.3±0.391.4±0.42001.18.710.9
w/ bit sharing71.5±0.291.5±0.21970.71.93.0
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56ABS-P→ABS-Q70.4±0.190.8±0.21077.7
ABS70.8±0.491.2±0.11042.5
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56Joint71.3±0.291.6±0.31942.4
Alternative71.6±0.191.8±0.41918.8
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NetworkMethodLatency-constrainedEnergy-constrained
Acc.-1 (%)Acc.-5 (%)Latency (ms)Acc.-1 (%)Acc.-5 (%)Energy (mJ)
MobileNetV26-bit precision71.890.324.971.890.332.8
ABS (Ours)72.090.417.272.090.326.3
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In the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "future, we plan to work on a joint search for architecture, pruning, and quantization to find a compact", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 235, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 235, + 733 + ], + "score": 1.0, + "content": "model with better performance.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 611, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 176, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 100, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "Yu Bai, Yu-Xiang Wang, and Edo Liberty. 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Specif-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "ically, for DQ (Uhlich et al., 2020), we parameterize the fixed-point quantizer using case U3 with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 107, + 518, + 164, + 530 + ], + "score": 0.91, + "content": "\\theta = [ d , q _ { \\mathrm { m a x } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 518, + 506, + 532 + ], + "score": 1.0, + "content": ". We initialize the weights using a pre-trained model. 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For HAQ (Wang et al., 2019), we first truncate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 552, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 289, + 568 + ], + "score": 1.0, + "content": "the weights and activations into the range of", + "type": "text" + }, + { + "bbox": [ + 289, + 554, + 330, + 566 + ], + "score": 0.93, + "content": "[ - v _ { w } , v _ { w } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 552, + 349, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 554, + 375, + 566 + ], + "score": 0.93, + "content": "[ 0 , v _ { x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 552, + 505, + 568 + ], + "score": 1.0, + "content": ", respectively. We then perform", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 403, + 579 + ], + "score": 1.0, + "content": "linear quantization for both weights and activations. To find more proper", + "type": "text" + }, + { + "bbox": [ + 403, + 567, + 416, + 576 + ], + "score": 0.86, + "content": "v _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 564, + 434, + 579 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 435, + 567, + 446, + 576 + ], + "score": 0.85, + "content": "v _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 564, + 506, + 579 + ], + "score": 1.0, + "content": ", we minimize", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "the KL-divergence between the original weight distribution W and the quantized weight distribution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 135, + 599 + ], + "score": 0.9, + "content": "Q ( \\mathbf { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 586, + 505, + 600 + ], + "score": 1.0, + "content": ". For DNAS (Wu et al., 2018), we follow DoReFa-Net (Zhou et al., 2016) to quantize weights", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "and follow PACT (Choi et al., 2018) to quantize activations. We initialize the learnable upper bound", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 608, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 504, + 621 + ], + "score": 1.0, + "content": "to 1. For DJPQ (Ying et al., 2020) and Bayesian Bits (van Baalen et al., 2020), we directly get the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "results from original papers. For other methods in Tables 1 and 2, we use the quantization function", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 477, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 356, + 643 + ], + "score": 1.0, + "content": "introduced in Section 3.1. The trainable quantization intervals", + "type": "text" + }, + { + "bbox": [ + 356, + 632, + 367, + 641 + ], + "score": 0.88, + "content": "v _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 630, + 385, + 643 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 632, + 397, + 641 + ], + "score": 0.87, + "content": "v _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 630, + 477, + 643 + ], + "score": 1.0, + "content": "are initialized to 1.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 661, + 277, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 280, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 280, + 676 + ], + "score": 1.0, + "content": "D SEARCH COST COMPARISONS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "To evaluate the efficiency of the proposed ABS, we compare the search cost of different methods and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "report the results in Table 7. From the results, the search costs of the proposed ABS is much smaller", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "than the state-of-the-art methods. Moreover, compared with ABS-Q, ABS only introduces a small", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 491, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 491, + 733 + ], + "score": 1.0, + "content": "amount of computational overhead, which demonstrates the efficiency of the proposed methods.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 144, + 78, + 468, + 109 + ], + "lines": [ + { + "bbox": [ + 141, + 77, + 470, + 95 + ], + "spans": [ + { + "bbox": [ + 141, + 77, + 470, + 95 + ], + "score": 1.0, + "content": "Appendix for ABS: Automatic Bit Sharing for Model", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 263, + 92, + 348, + 111 + ], + "spans": [ + { + "bbox": [ + 263, + 92, + 348, + 111 + ], + "score": 1.0, + "content": "Compression", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 127, + 334, + 139 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 335, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 335, + 142 + ], + "score": 1.0, + "content": "A HARDWARE-FRIENDLY DECOMPOSITION", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 127, + 335, + 142 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 153, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 211, + 167 + ], + "score": 1.0, + "content": "As mentioned in Sec. 3.2,", + "type": "text" + }, + { + "bbox": [ + 212, + 154, + 222, + 164 + ], + "score": 0.83, + "content": "b _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "can be set to arbitrary appropriate integer values (e.g., 1, 2, 3, etc.). By", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 163, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 167, + 178 + ], + "score": 1.0, + "content": "default, we set", + "type": "text" + }, + { + "bbox": [ + 167, + 165, + 196, + 176 + ], + "score": 0.91, + "content": "b _ { 1 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 163, + 505, + 178 + ], + "score": 1.0, + "content": "for better hardware utilization. On general purpose computing devices (e.g.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 175, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 188 + ], + "score": 1.0, + "content": "CPU, GPU), byte (8 bits) is the lowest data type for operations. Other data types and ALU registers", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 346, + 199 + ], + "score": 1.0, + "content": "are all composed with multiple bytes in width. By setting", + "type": "text" + }, + { + "bbox": [ + 346, + 186, + 377, + 198 + ], + "score": 0.9, + "content": "b _ { 1 } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 186, + 505, + 199 + ], + "score": 1.0, + "content": ", 2-bit/ 4-bit/ 8-bit quantization", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "values can be packed into byte (or short, int, long) data type without bit wasting. Otherwise, if", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 208, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 139, + 219 + ], + "score": 0.91, + "content": "b _ { 1 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 208, + 153, + 220 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 153, + 208, + 185, + 219 + ], + "score": 0.91, + "content": "b _ { 1 } = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 208, + 506, + 220 + ], + "score": 1.0, + "content": ", it is inevitable to have wasted bits when packing mixed-precision quantized", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "tensors on general purpose devices. For example, one 32-bit int data type can be used to store ten", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "3-bit quantized values with 2 bits wasted. One might argue that these 2 bits can be leveraged with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "the next group of 3-bit data, but it will result in irregular memory access patterns, which will degrade", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "the hardware utilization more seriously. Moreover, 8-bit quantization has been demonstrated to own", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "similar performance with the full precision counterparts for many networks. Therefore, there is no", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 274, + 273, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 273, + 286 + ], + "score": 1.0, + "content": "need to consider a bitwidth larger than 8.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 153, + 506, + 286 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 304, + 435, + 317 + ], + "lines": [ + { + "bbox": [ + 104, + 303, + 437, + 319 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 437, + 319 + ], + "score": 1.0, + "content": "B FORMULATION OF DIFFERENTIABLE COMPUTATIONAL LOSS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 108, + 331, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "In this section, we introduce the differentiable computational loss mentioned in Section 3.3. Unlike", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 344, + 354 + ], + "score": 1.0, + "content": "the cross-entropy loss in Eq. (17), the computational costs", + "type": "text" + }, + { + "bbox": [ + 344, + 342, + 403, + 354 + ], + "score": 0.93, + "content": "R ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "is non-differentiable. To", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 352, + 439, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 439, + 366 + ], + "score": 1.0, + "content": "solve this issue, we model the computational costs as a function of binary gates as:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 331, + 505, + 366 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 372, + 497, + 407 + ], + "lines": [ + { + "bbox": [ + 111, + 372, + 497, + 407 + ], + "spans": [ + { + "bbox": [ + 111, + 372, + 497, + 407 + ], + "score": 0.93, + "content": "R ( \\mathbf { W } , \\alpha ^ { p } , \\alpha ^ { q } ) = \\sum _ { c = 1 } ^ { G } g _ { c } ^ { p } \\left( R _ { x _ { c , b _ { 1 } } } + g _ { b _ { 2 } } ^ { q } \\left( R _ { x _ { c , b _ { 2 } } } - R _ { x _ { c , b _ { 1 } } } + \\cdot \\cdot + \\cdot + g _ { b _ { K } } ^ { q } \\left( R _ { x _ { c , b _ { K } } } - R _ { x _ { c , b _ { K - 1 } } } \\right) \\right) \\right) ,", + "type": "interline_equation", + "image_path": "7590e53268bad8e7d62ef6993005292a6c60de605c3423b5c0575b214ea137bb.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 111, + 372, + 497, + 383.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 111, + 383.6666666666667, + 497, + 395.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 111, + 395.33333333333337, + 497, + 407.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 415, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 160, + 429 + ], + "score": 1.0, + "content": "where Rxc,b", + "type": "text" + }, + { + "bbox": [ + 160, + 415, + 292, + 428 + ], + "score": 1.0, + "content": "is the computational cost for the", + "type": "text" + }, + { + "bbox": [ + 292, + 417, + 298, + 425 + ], + "score": 0.77, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 415, + 392, + 428 + ], + "score": 1.0, + "content": "-th group of filters with", + "type": "text" + }, + { + "bbox": [ + 392, + 416, + 401, + 428 + ], + "score": 0.87, + "content": "b _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 415, + 486, + 428 + ], + "score": 1.0, + "content": "-bit quantization and", + "type": "text" + }, + { + "bbox": [ + 486, + 417, + 495, + 425 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 428, + 228, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 228, + 441 + ], + "score": 1.0, + "content": "the number of groups in total.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 414, + 505, + 441 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 459, + 301, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 304, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 304, + 475 + ], + "score": 1.0, + "content": "C QUANTIZATION CONFIGURATIONS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "All the methods in Tables 1 and 2 use layer-wise and symmetric quantization schemes and the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "compared methods strictly follow the quantization configurations in their original papers. Specif-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "ically, for DQ (Uhlich et al., 2020), we parameterize the fixed-point quantizer using case U3 with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 107, + 518, + 164, + 530 + ], + "score": 0.91, + "content": "\\theta = [ d , q _ { \\mathrm { m a x } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 518, + 506, + 532 + ], + "score": 1.0, + "content": ". We initialize the weights using a pre-trained model. The initial step size is set to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 526, + 507, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 228, + 542 + ], + "score": 0.88, + "content": "d = \\bar { 2 } ^ { \\lfloor \\log _ { 2 } ( \\operatorname* { m a x } ( | \\mathbf { W } | ) / ( 2 ^ { b - 1 } - 1 ) \\rfloor }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 526, + 295, + 547 + ], + "score": 1.0, + "content": "for weights and", + "type": "text" + }, + { + "bbox": [ + 295, + 531, + 312, + 542 + ], + "score": 0.89, + "content": "2 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 526, + 507, + 547 + ], + "score": 1.0, + "content": "for activations. The remaining quantization pa-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "rameters are set such that the initial bitwidth is 4-bit. For HAQ (Wang et al., 2019), we first truncate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 552, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 289, + 568 + ], + "score": 1.0, + "content": "the weights and activations into the range of", + "type": "text" + }, + { + "bbox": [ + 289, + 554, + 330, + 566 + ], + "score": 0.93, + "content": "[ - v _ { w } , v _ { w } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 552, + 349, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 554, + 375, + 566 + ], + "score": 0.93, + "content": "[ 0 , v _ { x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 552, + 505, + 568 + ], + "score": 1.0, + "content": ", respectively. We then perform", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 403, + 579 + ], + "score": 1.0, + "content": "linear quantization for both weights and activations. To find more proper", + "type": "text" + }, + { + "bbox": [ + 403, + 567, + 416, + 576 + ], + "score": 0.86, + "content": "v _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 564, + 434, + 579 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 435, + 567, + 446, + 576 + ], + "score": 0.85, + "content": "v _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 564, + 506, + 579 + ], + "score": 1.0, + "content": ", we minimize", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "the KL-divergence between the original weight distribution W and the quantized weight distribution", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 135, + 599 + ], + "score": 0.9, + "content": "Q ( \\mathbf { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 586, + 505, + 600 + ], + "score": 1.0, + "content": ". For DNAS (Wu et al., 2018), we follow DoReFa-Net (Zhou et al., 2016) to quantize weights", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "and follow PACT (Choi et al., 2018) to quantize activations. We initialize the learnable upper bound", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 608, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 504, + 621 + ], + "score": 1.0, + "content": "to 1. For DJPQ (Ying et al., 2020) and Bayesian Bits (van Baalen et al., 2020), we directly get the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "results from original papers. For other methods in Tables 1 and 2, we use the quantization function", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 477, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 356, + 643 + ], + "score": 1.0, + "content": "introduced in Section 3.1. The trainable quantization intervals", + "type": "text" + }, + { + "bbox": [ + 356, + 632, + 367, + 641 + ], + "score": 0.88, + "content": "v _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 630, + 385, + 643 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 632, + 397, + 641 + ], + "score": 0.87, + "content": "v _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 630, + 477, + 643 + ], + "score": 1.0, + "content": "are initialized to 1.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 484, + 507, + 643 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 661, + 277, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 280, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 280, + 676 + ], + "score": 1.0, + "content": "D SEARCH COST COMPARISONS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "To evaluate the efficiency of the proposed ABS, we compare the search cost of different methods and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "report the results in Table 7. From the results, the search costs of the proposed ABS is much smaller", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "than the state-of-the-art methods. Moreover, compared with ABS-Q, ABS only introduces a small", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 491, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 491, + 733 + ], + "score": 1.0, + "content": "amount of computational overhead, which demonstrates the efficiency of the proposed methods.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 688, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 224, + 115, + 386, + 194 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 503, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "score": 1.0, + "content": "Table 7: Comparisons of the search costs on CIFAR-100. The search costs are measured on a GPU", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 227, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 227, + 112 + ], + "score": 1.0, + "content": "device (NVIDIA TITAN Xp).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 224, + 115, + 386, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 224, + 115, + 386, + 194 + ], + "spans": [ + { + "bbox": [ + 224, + 115, + 386, + 194 + ], + "score": 0.973, + "html": "
NetworkMethodSearch Cost (GPU hours)
ResNet-20HAQ DQ5.8 3.0
DNAS2.8
ABS-Q (Ours) ABS-P (Ours) ABS (Ours)0.8 0.2
", + "type": "table", + "image_path": "63bc429059acd23a603b64a5a9b060d08bb4e83ce42f4e52582bcc5cca119845.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 115, + 386, + 128.16666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 224, + 128.16666666666666, + 386, + 141.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 224, + 141.33333333333331, + 386, + 154.49999999999997 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 224, + 154.49999999999997, + 386, + 167.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 224, + 167.66666666666663, + 386, + 180.8333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 224, + 180.8333333333333, + 386, + 193.99999999999994 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 144, + 235, + 465, + 298 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 209, + 504, + 231 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "Table 8: Comparisons of different methods w.r.t. memory footprints. We compress ResNet-56", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 220, + 354, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 354, + 231 + ], + "score": 1.0, + "content": "using different methods and report the results on CIFAR-100.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "table_body", + "bbox": [ + 144, + 235, + 465, + 298 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 144, + 235, + 465, + 298 + ], + "spans": [ + { + "bbox": [ + 144, + 235, + 465, + 298 + ], + "score": 0.977, + "html": "
MethodMemory footprints(KB)M.f. comp. ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision5653.41.071.792.2
4-bit precision711.77.970.9±0.391.2±0.4
DNAS708.98.071.5±0.291.3±0.1
HAQ700.08.171.3±0.191.1±0.1
ABS-Q (Ours)674.58.471.5±0.291.6±0.2
ABS (Ours)657.38.671.6±0.191.8±0.4
", + "type": "table", + "image_path": "8ec06803eb6b7d86b9b0d0db7863d5655fb504a7503b868ff04b0319df621273.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 144, + 235, + 465, + 256.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 144, + 256.0, + 465, + 277.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 144, + 277.0, + 465, + 298.0 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 9.75 + }, + { + "type": "table", + "bbox": [ + 172, + 329, + 439, + 406 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 145, + 313, + 462, + 324 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 144, + 312, + 464, + 326 + ], + "spans": [ + { + "bbox": [ + 144, + 312, + 464, + 326 + ], + "score": 1.0, + "content": "Table 9: Comparisons of different methods with MobileNetV3 on CIFAR-100.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "table_body", + "bbox": [ + 172, + 329, + 439, + 406 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 172, + 329, + 439, + 406 + ], + "spans": [ + { + "bbox": [ + 172, + 329, + 439, + 406 + ], + "score": 0.961, + "html": "
MethodBOPs (M)BOP comp.ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision68170.11.076.193.9
6-bit precision2412.628.376.1±0.093.7±0.0
DQ2136.331.975.9±0.193.7±0.1
HAQ2191.731.176.1±0.193.5±0.0
DNAS2051.933.276.1±0.193.7±0.1
ABS-P(Ours)59465.81.176.0±0.093.5±0.0
ABS-Q(Ours)2021.933.776.1±0.193.7±0.1
ABS (Ours)2006.634.076.1±0.193.7±0.1
", + "type": "table", + "image_path": "19f745059083282078a1849ace381285de78914473022219977cffadff023e57.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 172, + 329, + 439, + 354.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 172, + 354.6666666666667, + 439, + 380.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 172, + 380.33333333333337, + 439, + 406.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 14.0 + }, + { + "type": "title", + "bbox": [ + 107, + 414, + 341, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 342, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 342, + 428 + ], + "score": 1.0, + "content": "E MORE RESULTS ON MEMORY FOOTPRINTS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 437, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "To further demonstrate the effectiveness of the proposed ABS, we replace BOPs with total weights", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "and activations memory footprints (Uhlich et al., 2020). We apply different methods to compress", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 460, + 504, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 504, + 472 + ], + "score": 1.0, + "content": "ResNet-56 and report the results in Table 8. From the results, ABS compressed ResNet-56 outper-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "forms other methods with fewer memory footprints. These results show the effectiveness of our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 482, + 333, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 333, + 495 + ], + "score": 1.0, + "content": "proposed ABS in terms of memory footprints reduction.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 510, + 409, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 412, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 412, + 526 + ], + "score": 1.0, + "content": "F DETAILED STRUCTURE OF THE COMPRESSED NETWORK", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "We illustrate the pruning rate and bitwidth of each layer’s weights and activations of the compressed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "ResNet-18 and MobileNetV2 in Figure 3 and Figure 4, respectively. From the results, we observe", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "that our ABS assigns more bitwidths to the weights in the downsampling convolutional layer in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "ResNet-18 and depthwise convolutional layer in MobileNetV2. Intuitively, this is because the num-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "ber of parameters of these layers is much smaller than other layers. Moreover, our ABS inclines to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "prune more filters in the shallower layers, which can significantly reduce the number of parameters", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "and computational overhead. Finally, we also observe that the correlation between the bitwidth and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "pruning rate is as follows. If a layer is set to a high pruning rate, our ABS tends to select a higher", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "bitwidth to compensate for the performance drop. In contrast, if a layer is with a low pruning rate,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 633, + 470, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 470, + 645 + ], + "score": 1.0, + "content": "our ABS tends to select a lower bitwidth to reduce the model size and computational costs.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 660, + 310, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 311, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 311, + 675 + ], + "score": 1.0, + "content": "G MORE RESULTS ON MOBILENETV3", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 505, + 729 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "To evaluate the proposed ABS on the lightweight model, we apply our methods to MobileNetV3 on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 696, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 203, + 708 + ], + "score": 1.0, + "content": "CIFAR-100. Following", + "type": "text" + }, + { + "bbox": [ + 204, + 696, + 230, + 708 + ], + "score": 0.82, + "content": "\\mathrm { L S Q + }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 696, + 505, + 708 + ], + "score": 1.0, + "content": "(Bhalgat et al., 2020), we introduce a learnable offset to handle the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 708, + 506, + 719 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 719 + ], + "score": 1.0, + "content": "negative activations in hard-swish. We show the results in Table 9. From the results of MobileNetV3,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 718, + 498, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 498, + 731 + ], + "score": 1.0, + "content": "our proposed ABS still outperforms the compared methods, which demonstrates its effectiveness.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 224, + 115, + 386, + 194 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 503, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "score": 1.0, + "content": "Table 7: Comparisons of the search costs on CIFAR-100. The search costs are measured on a GPU", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 227, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 227, + 112 + ], + "score": 1.0, + "content": "device (NVIDIA TITAN Xp).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 224, + 115, + 386, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 224, + 115, + 386, + 194 + ], + "spans": [ + { + "bbox": [ + 224, + 115, + 386, + 194 + ], + "score": 0.973, + "html": "
NetworkMethodSearch Cost (GPU hours)
ResNet-20HAQ DQ5.8 3.0
DNAS2.8
ABS-Q (Ours) ABS-P (Ours) ABS (Ours)0.8 0.2
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MethodMemory footprints(KB)M.f. comp. ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision5653.41.071.792.2
4-bit precision711.77.970.9±0.391.2±0.4
DNAS708.98.071.5±0.291.3±0.1
HAQ700.08.171.3±0.191.1±0.1
ABS-Q (Ours)674.58.471.5±0.291.6±0.2
ABS (Ours)657.38.671.6±0.191.8±0.4
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MethodBOPs (M)BOP comp.ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision68170.11.076.193.9
6-bit precision2412.628.376.1±0.093.7±0.0
DQ2136.331.975.9±0.193.7±0.1
HAQ2191.731.176.1±0.193.5±0.0
DNAS2051.933.276.1±0.193.7±0.1
ABS-P(Ours)59465.81.176.0±0.093.5±0.0
ABS-Q(Ours)2021.933.776.1±0.193.7±0.1
ABS (Ours)2006.634.076.1±0.193.7±0.1
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We apply different methods to compress", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 460, + 504, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 504, + 472 + ], + "score": 1.0, + "content": "ResNet-56 and report the results in Table 8. From the results, ABS compressed ResNet-56 outper-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "forms other methods with fewer memory footprints. These results show the effectiveness of our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 482, + 333, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 333, + 495 + ], + "score": 1.0, + "content": "proposed ABS in terms of memory footprints reduction.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 437, + 505, + 495 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 510, + 409, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 508, + 412, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 508, + 412, + 526 + ], + "score": 1.0, + "content": "F DETAILED STRUCTURE OF THE COMPRESSED NETWORK", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "We illustrate the pruning rate and bitwidth of each layer’s weights and activations of the compressed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "ResNet-18 and MobileNetV2 in Figure 3 and Figure 4, respectively. From the results, we observe", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "that our ABS assigns more bitwidths to the weights in the downsampling convolutional layer in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "ResNet-18 and depthwise convolutional layer in MobileNetV2. Intuitively, this is because the num-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "ber of parameters of these layers is much smaller than other layers. Moreover, our ABS inclines to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "prune more filters in the shallower layers, which can significantly reduce the number of parameters", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "and computational overhead. Finally, we also observe that the correlation between the bitwidth and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "pruning rate is as follows. If a layer is set to a high pruning rate, our ABS tends to select a higher", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "bitwidth to compensate for the performance drop. In contrast, if a layer is with a low pruning rate,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 633, + 470, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 470, + 645 + ], + "score": 1.0, + "content": "our ABS tends to select a lower bitwidth to reduce the model size and computational costs.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 534, + 506, + 645 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 660, + 310, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 311, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 311, + 675 + ], + "score": 1.0, + "content": "G MORE RESULTS ON MOBILENETV3", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 505, + 729 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "To evaluate the proposed ABS on the lightweight model, we apply our methods to MobileNetV3 on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 696, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 203, + 708 + ], + "score": 1.0, + "content": "CIFAR-100. Following", + "type": "text" + }, + { + "bbox": [ + 204, + 696, + 230, + 708 + ], + "score": 0.82, + "content": "\\mathrm { L S Q + }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 696, + 505, + 708 + ], + "score": 1.0, + "content": "(Bhalgat et al., 2020), we introduce a learnable offset to handle the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 708, + 506, + 719 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 719 + ], + "score": 1.0, + "content": "negative activations in hard-swish. We show the results in Table 9. From the results of MobileNetV3,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 718, + 498, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 498, + 731 + ], + "score": 1.0, + "content": "our proposed ABS still outperforms the compared methods, which demonstrates its effectiveness.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 685, + 506, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 125, + 155, + 486, + 376 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 155, + 486, + 376 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 155, + 486, + 376 + ], + "spans": [ + { + "bbox": [ + 125, + 155, + 486, + 376 + ], + "score": 0.884, + "type": "image", + "image_path": "284f562ab05ccb482bd329fad017446ba815605269415a114484ae378c8d00c3.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 125, + 155, + 486, + 228.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 125, + 228.66666666666669, + 486, + 302.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 125, + 302.33333333333337, + 486, + 376.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 201, + 379, + 407, + 390 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 201, + 378, + 409, + 390 + ], + "spans": [ + { + "bbox": [ + 201, + 378, + 409, + 390 + ], + "score": 1.0, + "content": "(a) Bitwidth configuration of the compressed ResNet-18.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "image", + "bbox": [ + 137, + 408, + 473, + 596 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 137, + 408, + 473, + 596 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 137, + 408, + 473, + 596 + ], + "spans": [ + { + "bbox": [ + 137, + 408, + 473, + 596 + ], + "score": 0.839, + "type": "image", + "image_path": "d02d26b45dc7eafeecbf604a33ffcc12f1d0ec1a1c0956af5f33cccaa8b129c8.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 137, + 408, + 473, + 470.6666666666667 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 137, + 470.6666666666667, + 473, + 533.3333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 137, + 533.3333333333334, + 473, + 596.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 174, + 643, + 432, + 656 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 174, + 643, + 433, + 657 + ], + "spans": [ + { + "bbox": [ + 174, + 643, + 433, + 657 + ], + "score": 1.0, + "content": "Figure 3: Detailed configurations of the compressed ResNet-18.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + } + ], + "index": 7.0 + }, + { + "type": "text", + "bbox": [ + 122, + 609, + 488, + 631 + ], + "lines": [ + { + "bbox": [ + 123, + 609, + 488, + 621 + ], + "spans": [ + { + "bbox": [ + 123, + 609, + 488, + 621 + ], + "score": 1.0, + "content": "(b) Pruning rate configuration of the compressed ResNet-18. 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NetworkMethodBOPs (M)BOP comp.ratioWIATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-20Full-precision41798.61.032.0/32.067.590.8
4-bit precision674.662.04.0/4.067.8±0.390.4±0.2
DQ1180.035.45.3/6.167.7±0.690.4±0.5
HAQ653.464.03.7/4.267.7±0.190.4±0.3
DNAS660.062.94.6/3.867.8±0.390.4±0.2
ABS-P(Ours)28586.51.532.0/32.067.9±0.190.7±0.2
ABS-Q(Ours) ABS (Ours)649.564.44.4/4.268.1±0.190.5±0.0
630.666.34.4/4.268.1±0.390.6±0.2
ResNet-56Full-precision128771.71.032.0/32.071.792.2
4-bit precision2033.663.34.0/4.070.9±0.391.2±0.4
DQ2222.957.93.8 /4.670.7±0.291.4±0.4
HAQ2014.963.93.3/4.971.2±0.191.1±0.2
DNAS2035.765.35.3/3.271.2±0.291.3±0.3
ABS-P(Ours)87021.61.532.0/32.071.5±0.191.8±0.2
ABS-Q(Ours)1970.765.34.1/4.071.5±0.291.5±0.2
ABS (Ours)1918.867.14.2/4.171.6±0.191.8±0.4
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NetworkMethodBOPs (G)BOP comp. ratioW/ATop-1 Acc. (%)Top-5 Acc. (%)
ResNet-18Full-precision1857.61.032.0/32.070.789.8
4-bit precision34.753.54.0/4.071.089.8
DQ40.740.61/168.51
DJPQ35.552.3-/169.1
HAQ34.753.54.0/4.070.289.5
Bayesian Bits*35.951.71/169.5
ABS-Q(Ours)33.156.14.5/3.870.989.7
ABS (Ours)32.357.54.6/4.270.889.6
MobileNetV2Full-precision308.01.032.0/32.071.990.3
6-bit precision11.227.56.0/6.071.890.3
DQ19.61.96.8/8.070.489.7
HAQ10.828.55.61/6.2771.290.0
Bayesian Bits *10.828.5-/170.9一
ABS-Q(Ours)10.928.36.8/6.871.890.4
ABS (Ours)10.828.56.1/7.171.790.3
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)Search Cost (GPU hours)GPUMemory (GB)
ResNet-20 w/o bit sharing67.8±0.190.5±0.2664.22.84.4
w/bit sharing68.1±0.190.5±0.0649.50.81.5
ResNet-56w/o bit sharing71.3±0.391.4±0.42001.18.710.9
w/ bit sharing71.5±0.291.5±0.21970.71.93.0
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NetworkMethodLatency-constrainedEnergy-constrained
Acc.-1 (%)Acc.-5 (%)Latency (ms)Acc.-1 (%)Acc.-5 (%)Energy (mJ)
MobileNetV26-bit precision71.890.324.971.890.332.8
ABS (Ours)72.090.417.272.090.326.3
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56ABS-P→ABS-Q70.4±0.190.8±0.21077.7
ABS70.8±0.491.2±0.11042.5
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NetworkMethodTop-1 Acc.Top-5 Acc.BOPs (M)
ResNet-56Joint71.3±0.291.6±0.31942.4
Alternative71.6±0.191.8±0.41918.8
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MethodMemory footprints(KB)M.f. comp. ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision5653.41.071.792.2
4-bit precision711.77.970.9±0.391.2±0.4
DNAS708.98.071.5±0.291.3±0.1
HAQ700.08.171.3±0.191.1±0.1
ABS-Q (Ours)674.58.471.5±0.291.6±0.2
ABS (Ours)657.38.671.6±0.191.8±0.4
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NetworkMethodSearch Cost (GPU hours)
ResNet-20HAQ DQ5.8 3.0
DNAS2.8
ABS-Q (Ours) ABS-P (Ours) ABS (Ours)0.8 0.2
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MethodBOPs (M)BOP comp.ratioTop-1 Acc. (%)Top-5 Acc. (%)
Full-precision68170.11.076.193.9
6-bit precision2412.628.376.1±0.093.7±0.0
DQ2136.331.975.9±0.193.7±0.1
HAQ2191.731.176.1±0.193.5±0.0
DNAS2051.933.276.1±0.193.7±0.1
ABS-P(Ours)59465.81.176.0±0.093.5±0.0
ABS-Q(Ours)2021.933.776.1±0.193.7±0.1
ABS (Ours)2006.634.076.1±0.193.7±0.1
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b/parse/train/SJCscQcge/SJCscQcge.md @@ -0,0 +1,323 @@ +# SIMPLE BLACK-BOX ADVERSARIAL PERTURBATIONS FOR DEEP NETWORKS + +Nina Narodytska Shiva Kasiviswanathan Samsung Research America Mountain View, CA 94043, USA {n.narodytska,kasivisw}@gmail.com + +# ABSTRACT + +Deep neural networks are powerful and popular learning models that achieve stateof-the-art pattern recognition performance on many computer vision, speech, and language processing tasks. However, these networks have also been shown susceptible to carefully crafted adversarial perturbations which force misclassification of the inputs. Adversarial examples enable adversaries to subvert the expected system behavior leading to undesired consequences and could pose a security risk when these systems are deployed in the real world. + +In this work, we focus on deep convolutional neural networks and demonstrate that adversaries can easily craft adversarial examples even without any internal knowledge of the target network. Our attacks treat the network as an oracle (blackbox) and only assume that the output of the network can be observed on the probed inputs. Our first attack is based on a simple idea of adding perturbation to a randomly selected single pixel or a small set of them. We then improve the effectiveness of this attack by carefully constructing a small set of pixels to perturb by using the idea of greedy local-search. Our proposed attacks also naturally extend to a stronger notion of misclassification. Our extensive experimental results illustrate that even these elementary attacks can reveal a deep neural network’s vulnerabilities. The simplicity and effectiveness of our proposed schemes mean that they could serve as a litmus test for designing robust networks. + +# 1 INTRODUCTION + +Convolutional neural networks (CNNs) are among the most popular techniques employed for computer vision tasks, including but not limited to image recognition, localization, video tracking, and image and video segmentation (Goodfellow et al., 2016). Though these deep networks have exhibited good performances for these tasks, they have recently been shown to be particularly susceptible to adversarial perturbations to the input images (Szegedy et al., 2014; Goodfellow et al., 2015; MoosaviDezfooli et al., 2016; Papernot et al., 2016c;b; Kurakin et al., 2016; Grosse et al., 2016; Zagoruyko, 2016b). Vulnerability of these networks to adversarial attacks can lead to undesirable consequences in many practical applications using them. For example, adversarial attacks can be used to subvert fraud detection, malware detection, or mislead autonomous navigation systems (Papernot et al., 2016c; Grosse et al., 2016). Further strengthening these results is a recent observation by Kurakin et al. (2016) who showed that a significant fraction of adversarial images crafted using the original network are misclassified even when fed to the classifier through a physical world system (such as a camera). + +In this paper, we investigate the problem of robustness of state-of-the-art convolutional neural networks (CNNs) to simple black-box adversarial attacks. The rough goal of adversarial attacks is as follows: Given an image $I$ that is correctly classified by a machine learning system (say, a CNN), is it possible to construct a transformation of $I$ (say, by adding a small perturbation to some or all the pixels) that now leads to misclassification by the system. Since large perturbations can trivially lead to misclassification, the attacks seek to limit the amount of perturbation applied under some chosen metric. More often than not, in these attacks, the modification done to the image is so subtle that the changes are imperceptible to a human eye. Our proposed attacks also share this property, in addition to being practical and simplistic, thus highlighting a worrying aspect about lack of robustness prevalent in these modern vision techniques. + +There are two main research directions in the literature on adversarial attacks based on different assumptions about the adversarial knowledge of the target network. The first line of work assumes that the adversary has detailed knowledge of the network architecture and the parameters resulting from training (or access to the labeled training set) (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c). Using this information, an adversary constructs a perturbation for a given image. The most effective methods are gradient-based: a small perturbation is constructed based on the gradients of the loss function w.r.t. the input image and a target label. Often, adding this small perturbation to the original image leads to a misclassification. In the second line of work an adversary has restricted knowledge about the network from being able to only observe the network’s output on some probed inputs (Papernot et al., 2016b). Our work falls into this category. While this black-box model is a much more realistic and applicable threat model, it is also more challenging because it considers weak adversaries without knowledge of the network architecture, parameters, or training data. Interestingly, our results suggest that this level of access and a small number of queries provide sufficient information to construct an adversarial image. + +![](images/8f8bd387fd79bbd926de42b8e1bd85f7e2cc447025bdada2d104ff3603f7faa7.jpg) +Table 1: The top row shows the original images and the bottom row shows the perturbed images. The misclassification is as follows: (a) a stingray misclassified as a sea lion, (b) an ostrich misclassified as a goose, (c) a jay misclassified as a junco, and (d) a water ouzel misclassified as a redshank. + +As we operate in a black-box setting, we use a gradient-free approach to adversarial image generation. Papernot et al. (2016b) were the first to discuss a black-box attack against deep learning systems. Their attack crucially relies on the observation that there is a transferability (generalization) property in adversarial examples, i.e., adversarial examples form one model transfers to another. Our proposed attacks on the other hand is much more simple and direct, does not require this transferability property, and hence is more effective in constructing adversarial images, in addition to having some other computational advantages. We demonstrate that our method is capable of constructing adversarial images for several network architectures trained on different datasets. In particular in this paper, we consider the CIFAR10, MNIST, SVHN, STL10, and ImageNet1000 datasets, and two popular network architectures, Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman, 2014). In Table 1, we show four images from the ImageNet1000 dataset. The original images are in the upper row. The bottom row shows the corresponding perturbed images produced by our algorithm which are misclassified by a VGG CNN-S network (Chatfield et al., 2014a). + +Our Contributions. In this work, we present simple and effective black-box adversarial attacks on deep convolutional neural networks. We make the following main contributions in this paper. + +(1) The first question we investigate is the influence of perturbing a single pixel on the prediction. To do so, we devise a simple scheme, based on randomly selecting a single pixel and applying a strong perturbation to it. Somewhat surprisingly, we noticed that a few trails of this random experiment is already quite enough in generating adversarial images for low resolution image sets. In fact, in many cases, for misclassification, the amount of perturbation needed to be applied to the selected pixel is also quite small. For high-resolution images, a similar phenomena holds, except our scheme now picks a random set of around 50 pixels. These simple experiments show the ease of generating adversarial images for modern deep CNNs without knowledge of either the network architecture or its parameters. There is however one shortcoming in these approaches in that the perturbed image might have pixel values that are outside some expected range. + +(2) We overcome this above shortcoming by showing that lower perturbation suffices if we carefully select the pixels for perturbation. The approach is based the idea of greedy local search, an iterative search procedure, where in each round a local neighborhood is used to refine the current image and in process minimizing the probability of the network assigning high confidence scores to the true class label. Again while the algorithm is quite simple, it is rather effective in generating adversarial images with quite small perturbations. We also show an interesting connection between the pixels chosen for perturbation by our approach and the saliency map of an image, as defined by Simonyan et al. (2014), that ranks pixels based on their influence on the output score. In effect our approach identifies pixels with high saliency scores but without explicitly using any gradient information (as needed in the definition of saliency map (Simonyan et al., 2014)). Intuitively, in each round, our local-search based approach computes an implicit approximation to the gradient of the current image by understanding the influence of a few pixels on the output, which is then used to update the current image. + +(3) We perform extensive experimental evaluations, and show that our local-search based approach reliably generates adversarial examples with little perturbation (even when compared to a recent elegant adversarial attack proposed by Goodfellow et al. (2015) which needs perfect knowledge of the network). Another feature of our attack is that, by design, our approach only perturbs a very small fraction of the pixels during the adversarial image generation process (e.g., on the ImageNet1000 dataset we on average perturb only about $0 . 5 \%$ of the pixels per image). Most previous attacks require the ability to perturb all the pixels in the image. + +(4) Our approaches naturally extend to a stronger notion of misclassification (that we refer to as $k$ -misclassification), where the goal is to ensure that the true label of the image does not even appear in the top- $k$ predictions of the network (obtained by sorting the confidence score vector). This notion especially captures the fact that many modern systems (e.g., ImageNet competition entrants) are evaluated based on top- $k$ predictions. To the best of our knowledge, these are the first adversarial attacks on deep neural networks achieving $k$ -misclassification. + +# 2 RELATED WORK + +Starting with the seminal paper by Szegedy et al. (2014), which showed that the state-of-the-art neural networks are vulnerable to adversarial attacks, there has been significant attention focused on this problem. The research has led to investigation of different adversarial threat models and scenarios (Papernot et al., 2016c;b; Grosse et al., 2016; Kurakin et al., 2016; Fawzi et al., 2016), computationally efficient attacks (Goodfellow et al., 2015), perturbation efficient attacks (MoosaviDezfooli et al., 2016), etc. + +Szegedy et al. (2014) used a box-constrained L-BFGS technique to generate adversarial examples. They also showed a transferability (or generalization) property for adversarial examples, in that adversarial examples generated for one network might also be misclassified by a related network with possibly different hyper-parameters (number of layers, initial weights, etc.). However, the need for a solving a series of costly penalized optimization problems makes this technique computationally expensive for generating adversarial examples. This issue was fixed by Goodfellow et al. (2015) who motivated by the underlying linearity of the components used to build a network proposed an elegant scheme based on adding perturbation proportional to sign of the network’s cost function gradient. Recently, Moosavi-Dezfooli et al. (2016) used an iterative linearization procedure to generate adversarial examples with lesser perturbation. Another recent attack proposed by Papernot et al. (2016c) uses a notion of adversarial saliency maps (based on the saliency maps introduced by (Simonyan et al., 2014)) to select the most sensitive input components for perturbation. This attack has been adapted by Grosse et al. (2016) for generating adversarial samples for neural networks used as malware classifiers. However, all these above described attacks require perfect knowledge of the target network’s architecture and parameters which limits their applicability to strong adversaries with the capability of gaining insider knowledge of the target system. + +Our focus in this paper is the setting of black-box attacks, where we assume that an adversary has only the ability to use the network as an oracle. The adversary can obtain output from supplied inputs, and use the observed input-output relationship to craft adversarial images.1 In the context of deep neural networks, a black-box attack was first proposed by Papernot et al. (2016b) with the motivation of constructing an attack on a remotely hosted system.2 Their general idea is to first approximate the target network by querying it for output labels, which is used to train a substitute network, which is then used to craft adversarial examples for the original network. The success of the attack crucially depends on the transferability property to hold between the original and the substitute network. Our black-box attack is more direct, and completely avoids the transferability assumption, making it far more applicable. We also avoid the overhead of gathering data and training a substitute network. Additionally, our techniques can be adapted to a stronger notion of misclassification. + +A complementary line of work has focused on building defenses against adversarial attacks. Although designing defenses is beyond scope of this paper, it is possible that adapting the previous suggested defense solutions such as Jacobian-based regularization (Gu & Rigazio, 2015) and distillation (Papernot et al., 2016d) can reduce the efficacy of our proposed attacks. Moreover, the recently proposed technique of differentially private training (Abadi et al., 2016) can also prove beneficial here. + +The study of adversarial instability have led to development of solutions that seeks to improve training to in return increase the robustness and classification performance of the network. In some case, adding adversarial examples to the training (adversarial training) set can act like a regularizer (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016). The phenomenon of adversarial instability has also been theoretically investigated for certain families of classifiers under various models of (semi) random noise (Fawzi et al., 2015; 2016). However, as we discuss later, due to peculiar nature of adversarial images generated by our approaches, a simple adversarial training is only mildly effective in preventing future similar adversarial attacks. + +The security of machine learning in settings distinct from deep neural networks is also an area of active research with various known attacks under different threat models. We refer the reader to a recent survey by McDaniel et al. (2016) and references therein. + +# 3 PRELIMINARIES + +Notation and Normalization. We denote by $[ n ]$ the set $\{ 1 , \ldots , n \}$ . The dataset of images is partitioned into train and test (or validation) subsets. An element of a dataset is a pair $( I , c ( I ) )$ for an image $I$ and a ground truth label $c ( I )$ of this image. We assume that the class labels are drawn from the set $\{ 1 , \ldots , C \}$ , i.e., we have a set of $C \in \mathbb { N }$ possible labels. We assume that images have $\ell$ channels (in experiments we use the RGB format) and are of width $w \in \mathbb { N }$ and height $h \in \mathbb { N }$ . We say that $( b , x , y )$ is a coordinate of an image for channel $b$ and location $( x , y )$ , and $( \star , x , y )$ is a pixel of an image where $( \star , x , y )$ represents all the $\ell$ coordinates corresponding to different channels at location $( x , { \bar { y } } )$ . $I ( b , x , y ) \in \mathbb { R }$ is the value of $I$ at the $( b , x , y )$ coordinate, and similarly $I ( \star , x , y ) \in \mathbb { R } ^ { \ell }$ represents the vector of values of $I$ at the $( \star , x , y )$ pixel. + +It is a common practice to normalize the image before passing it to the network. A normalized image has the same dimension as the original image, but differs in the coordinate values. In this work we treat the normalization procedure as an external procedure and assume that all images are normalized. As we always work with normalized images, in the following, a reference to image means a normalized input image. We denote by LB and UB two constants such that all the coordinates of all the normalized images fall in the range [LB, UB]. Generally, $\mathrm { L B } < 0$ and $\mathrm { U B > 0 }$ . We denote by $\mathbb { I } \subset \mathbb { R } ^ { \ell \times w \times h }$ the space of all (valid) images which satisfy the following property: for every $I \in \mathbb { I }$ , for all coordinates $( \bar { b , \boldsymbol { x } } , \boldsymbol { y } ) \in [ \ell ] \times [ w ] \times [ \bar { h ] } , I ( b , \boldsymbol { x } , \boldsymbol { y } ) \in [ \mathrm { \bar { L } B , U B } ] .$ . + +We denote by NN a trained neural network (trained on some set of training images). NN takes an image $I$ as an input and outputs a vector $\mathbf { N N } ( I ) = ( o _ { 1 } , \dots , o _ { C } )$ , where $o _ { j }$ denotes the probability as determined by NN that image $I$ belongs to class $j$ . We denote $\pi ( \mathrm { N N } ( I ) , k )$ a function that returns a set of indices that are the top- $k$ predictions (ranked by decreasing probability scores with ties broken arbitrarily) of the network NN. For example, if $\mathbf { N N } ( I ) = ( 0 . 2 5 , 0 . 1 , 0 . 2 , 0 . 4 5 )$ , then $\pi ( \operatorname { N N } ( I ) , 1 ) = \{ 4 \}$ (corresponding to the location of the entry 0.45). Similarly, $\pi ( { \mathrm { N N } } ( I ) , 2 ) =$ $\{ 4 , 1 \}$ , $\pi ( \mathbf { N N } ( I ) , 3 ) = \{ 4 , 1 , 3 \}$ , etc. + +Adversarial Goal. Before we define the goal of black-box adversarial attacks, we define misclassification for a NN. In this paper, we use a stronger notion of misclassification, which we refer to as $k$ -misclassification for $k \in \mathbb N$ . + +Definition 1 ( $k$ -misclassification) A neural network NN $k$ -misclassifies an image $I$ with true label $c ( I )$ iff the output $\mathrm { N N } ( I )$ of the network satisfies $c ( I ) \not \in \pi ( { \mathrm { N N } } ( I ) , k )$ . + +In other words, $k$ -misclassification means that the network ranks the true label below at least $k$ other labels. Traditionally the literature on adversarial attacks have only considered the case where $k = 1$ . Note that an adversary that achieves a $k$ -misclassification for $k > 1$ is a stronger adversary than one achieving an 1-misclassification ( $k$ -misclassification implies $k ^ { \prime }$ -misclassification for all $1 \overset { \cdot } { \leq } k ^ { \prime } \leq k ,$ ). If $k = 1$ , we simply say that NN misclassifies the image. + +In our setting, an adversary ADV is a function that takes in image $I$ as input and whose output is another image $\mathrm { A D V } ( I )$ (with same number of coordinates as $I$ ). We define an adversarial image as one that fools a network into $k$ -misclassification. + +Definition 2 (Adversarial Image) Given access to an image $I$ , we say that an $\mathrm { A D V } ( I )$ is a $k$ - adversarial image (resp. adversarial image) if $c ( I ) \in \pi ( \mathrm { N N } ( I ) , k )$ and $c ( I ) \notin \pi ( \mathbf { N N } ( \mathbf { A D V } ( I ) ) , k )$ (resp. $c ( I ) \in \pi ( \mathrm { N N } ( I ) , 1 )$ and $c ( I ) \notin \pi ( \mathbf { N N } ( \mathbf { A D V } ( I ) ) , 1 ) .$ ). + +The goal of adversarial attacks is to design this function ADV that succeeds in fooling the network for a large set of images. Ideally, we would like to achieve this misclassification3 by adding only some small perturbation (under some metric) to the image. The presence of adversarial images shows that there exist small perturbations in input that produce large perturbations at the output of the last layer. + +Adversarial threat models can be divided into two broad classes.4 The first class of models roughly assumes that the adversary has a total knowledge of the network architecture and the parameters resulting from training (or access to the labeled training set). The second class of threat models, as considered in this paper, make no assumptions about the adversary having access to the network architecture, network parameters, or the training set. In this case, the adversary has only a black-box (oracle) access to the network, in that it can query the network NN on an image $I$ and observe the output $\mathrm { N N } ( I )$ . In our experimental section (Section 6), we also consider a slight weakening of this black-box model where the adversary has only the ability to use a proxy of the network NN as an oracle. + +A black-box threat model in the context of deep neural networks was first considered by Papernot et al. (2016b). There is however one subtle difference between the threat model considered here and that considered by Papernot et al. (2016b) in what the adversary can access as an output. While the adversary presented in (Papernot et al., 2016b) requires access to the class label assigned by the network which is the same level of access needed by our simple randomized adversary (presented in Section 4), our local-search adversary (presented in Section 5) requires access to $o _ { c ( I ) }$ (the probability assigned to the true label $c ( I )$ by the network on input $I$ ) and the $\pi$ vector (for checking whether $k$ -misclassification has been achieved). Our adversarial approaches does not require access to the complete probability vector $( \mathrm { N N } ( I ) )$ . Also as pointed out earlier, compared to (Papernot et al., 2016b), our approach is more direct (needs no transferability assumption), requires no retraining, and can be adapted to achieve $k$ -misclassification rather than just 1-misclassification. + +# 4 BLACK-BOX GENERATION: A FIRST ATTEMPT + +In this section, we present a simple black-box adversary that operates by perturbing a single pixel (or a small set of pixels) selected at random. In the next section, we build upon this idea to construct an adversary that achieves better success by making adaptive choices. + +Power of One Pixel. Starting point of our investigation is to understand the influence of a single pixel in an adversarial setting. Most existing adversarial attacks operate by applying the same perturbation on each individual pixel while minimizing the overall perturbation (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016), while recent research have yielded attacks that perturb only a fraction of the pixels (Papernot et al., 2016c;b; Grosse et al., 2016). However, in all these cases, no explicit restriction is placed on the number of pixels that can be perturbed. Therefore, it is natural to ask: whether it is possible to force the network to misclassify an image by modifying a single pixel? If so, how strong should this perturbation be? We run several experiments to shed light on these questions. For simplicity, in this section, we focus the case of 1-misclassification, even though all discussions easily extend to the case of $k$ -misclassification for $k > 1$ . We begin with a useful definition. + +Definition 3 (Critical Pixel) 5 Given a trained neural network NN and an image $I$ , a pixel $( \star , x , y )$ in $I$ is a critical pixel if a perturbation of this pixel generates an image that is misclassified by the network NN. In other words, $( \star , x , y )$ is a critical pixel in $I$ if there exists another neighboring image $I _ { p }$ which differs from $I$ only in values at the pixel location $( x , y )$ such that $c ( I ) \notin \pi ( \mathbf { N N } ( I _ { p } ) , 1 )$ . + +The image $I _ { p }$ can be generated in multiple ways, here we consider a class of sign-preserving perturbation functions defined as follows. Let $\mathtt { P E R T } ( I , p , x , y )$ be a function that takes as input an image $I$ , a perturbation parameter $p \in \mathbb { R }$ , and a location $( x , y )$ , and outputs an image $I _ { p } ^ { ( x , y ) } \in$ $\mathbb { R } ^ { \ell \times w \times h }$ , defined as: + +$$ +I _ { p } ^ { ( x , y ) } ( b , u , v ) \stackrel { \mathrm { \tiny ~ d e f n } } { = } \left\{ { \begin{array} { l l } { { \scriptstyle I ( b , u , v ) } } & { { \mathrm { i f ~ } } x \neq u { \mathrm { ~ o r ~ } } y \neq v } \\ { { p \times \mathrm { s i g n } } ( I ( b , u , v ) ) } & { { \mathrm { o t h e r w i s e } } } \end{array} } \right. +$$ + +In other words, the image $I _ { p } ^ { ( x , y ) } = \mathtt { P E R T } ( I , p , x , y )$ has same values as image $I$ at all pixels except the pixel $( \star , x , y )$ . The value of the image $I _ { p } ^ { ( x , y ) }$ at pixel $( \star , x , y )$ is just $p \times \mathrm { s i g n } ( I ( \star , x , y ) )$ . Note that unlike some of the previous adversarial attacks (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c; Grosse et al., 2016), our threat model does not assume access to the true network gradient factors, hence the construction of the perturbed image has to be oblivious to the network parameters. + +In the following, we say a pixel $( \star , x , y )$ in image $I$ is critical iff $c ( I ) \notin \pi ( \mathbf { N N } ( I _ { p } ^ { ( x , y ) } ) , 1 )$ + +Critical Pixels are Common. Our first experiment is to investigate existence of critical pixels in the considered dataset of images. To do so, we perform a simple procedure that picks a location $( x , y )$ in the image $I$ and applies the PERT function to this pixel to obtain a perturbed image $I _ { p } ^ { ( x , y ) }$ . Then the perturbed image is run through the trained network, and we check whether it was misclassified or not. If the perturbed image $I _ { p } ^ { ( x , y ) }$ is misclassified then we have identified a critical pixel. While we can exhaustively repeat this procedure for all pixels in an image, for computational efficiency we instead perform it only on a fraction of randomly chosen pixels, and our results somewhat surprisingly suggest that in many cases this is sufficient to generate an adversarial image. Algorithm RANDADV presents the pseudo-code for this experiment. Algorithm RANDADV, selects $U$ random pixels (with replacement) and performs checks whether the pixel is critical or not. The algorithm output is an unbiased estimate for the fraction of critical pixels in the input image $I$ . Note that the algorithm can fail in generating an adversarial image (i.e., in finding any critical pixel for an image). The following definition will be useful for our ensuing discussion. + +Definition 4 (Good Image) We say that an image $I$ with true label $c ( I )$ is good for a network NN iff $c ( I ) \in \pi ( N N ( I ) , 1 )$ (i.e., NN predicts $c ( I )$ as the most likely label for $I$ ). + +Our first observation is that sometimes even small perturbation to a pixel can be sufficient to obtain an adversarial image. Table 2 shows two images and their adversarial counterparts, with $p = 1$ . Often, original and adversarial images are indistinguishable to the human eye, but sometimes the critical pixel is visible (Table 2). + +![](images/35f61844a2fc87ca9ef855e0bf44a533ed34704baca6fd233c93e79e87e3e179.jpg) +Table 2: The row contains original images followed by misclassified images where only one pixel (pointed using a black arrow) was perturbed with perturbation parameter $p = 1$ . After perturbation, in the first case (images (a) and (b)) an automobile gets misclassified as a truck, and in the second case (images (c) and (d)) a cat gets misclassified as a dog. + +We also tried to understand the effect of larger perturbation parameter values. We set $U$ to half the number of pixels in each image. After usual training of the neural network using the training set (see Section 6 for more details about training), we ran Algorithm RANDADV on 1000 randomly drawn images from the test set of the corresponding dataset. In our experiments, we varied perturbation parameter in the range $\{ 1 , 5 , 1 0 , 1 0 0 \}$ . Before we consider our results, we note some of the perturbation values that we use to construct the adversarial image might construct images that are not in the original image space.6 However, these results are still somewhat surprising, because even though we allow large (even out-of-range) perturbation, it is applied to exactly one pixel in the image, and it appears that it suffices to even pick the pixel at random. + +Figures 1 and 2 show results for 4 datasets (more details about the datasets and the networks are presented in Section 6). On the $\mathbf { X }$ -axis we show the perturbation parameter $p$ . In Figure 1, the y-axis represents the output of Algorithm RANDADV averaged over good images for the network.7 The first observation that we can make is that the critical pixels are common, and in fact, as $p$ grows the fraction of critical pixels increases. For example, in CIFAR10, with $p = 1 0 0$ , almost $80 \%$ (on average) of the pixels randomly selected are critical. In Figure 2, the y-axis represents the fraction of successful adversarial images generated by Algorithm RANDADV, i.e., fraction of inputs where Algorithm RANDADV is successful in finding at least one critical pixel. Again we notice that as $p$ grows it gets easier for Algorithm RANDADV to construct an adversarial image. + +# Algorithm 1 RANDADV (NN) + +1: Input: Image $I$ with true label $c ( I ) \in \{ 1 , \ldots , C \}$ , perturbation factor $p \in \mathbb R$ , and a budget $U \in \mathbb { N }$ on the +number of trials +2: Output: A randomized estimate on the fraction of critical pixels in the input image $I$ +3: $i = 1$ , critical $= 0$ +4: while $i \leq U$ do +5: randomly pick a pixel $( \star , x , y )$ +6: compute a perturbed image I(x,y)p = $I _ { p } ^ { ( x , y ) } = \mathtt { P E R T } ( I , p , x , y )$ +7: if $c ( I ) \notin \pi ( \mathbf { N N } ( I _ { p } ^ { ( x , y ) } ) , 1 )$ then +8: critical critical $+ 1$ +9: end if +10: $i \gets i + 1$ +11: end while +12: {The algorithm succeeds in generating an adversarial image if it finds at least one critical pixel} +13: return $\frac { \mathrm { \bar { c r i t i c a l } } } { U }$ + +Another observation is that for the MNIST and STL10 datasets, Algorithm RANDADV succeeds in finding fewer critical pixels as compared to SVHN and CIFAR10 datasets. We give the following explanation for this observation. The majority of pixels in an MNIST image belong to the background, hence, these pixels are less likely to be critical. On the other hand, STL10 contains high resolution images, $9 6 \times 9 6$ , where perhaps a single pixel has less of an impact on the output prediction. The latter observation motivated us to generalize the notion of a critical pixel to a critical set. + +![](images/7e8864ae3078d8172ab7a9829d78956096dcab004436e610d6380e7219120e9d.jpg) +Figure 1: Output of Algorithm RANDADV (averaged over good images). The results are for two networks: a) Network-in-Network and b) VGG. The perturbation parameter $p$ is varied from $\{ 1 , 5 , 1 0 , 1 0 0 \}$ . + +![](images/e6719714440c8cfd2fc901d4cd2fef26915fc30b8d7a9db8a45e0c6471206f23.jpg) +Figure 2: Fraction of images where Algorithm RANDADV succeeds in finding at least one critical pixel. Again we only start with only good images. + +Definition 5 (Critical Set) Given a trained neural network NN and an image I, a critical set of I is a set of pixels $\textstyle \bigcup _ { ( x , y ) } \{ ( \star , x , y ) \}$ in $I$ such that a perturbation of these pixels generates an image that is misclassified by the network NN. + +The general goal will be to find critical sets of small size in an image. With this notion of critical set, we considered constructing adversarial images on the high-resolution ImageNet1000 dataset. We can modify the definition of $I _ { p } ^ { ( x , y ) }$ (from (1)) where instead of a single pixel we perturb all the pixels in a set. Similarly, we can devise a simple extension to Algorithm RANDADV to operate with a set of pixels and to output an unbiased estimate for the fraction of critical sets of some fixed size (50 in our case) in the input image.8 Note that a set size of 50 pixels is still a tiny fraction of all the pixels in a standard (center) crop of size $2 2 4 \times 2 2 4$ , namely just $0 . 0 9 \%$ . We use a larger perturbation parameter $p$ than before, and set $( U )$ the budget on the number of trials on an image as 5000. Figure 3 shows our results. Overall, we note that we can draw similar conclusions as before, i.e., increasing the perturbation parameter creates more critical sets making them easier to find and relatively small perturbations are sufficient to construct adversarial images. + +# 5 BLACK-BOX GENERATION: A GREEDY APPROACH + +The results from Section 4 show that most images have critical pixels such that modifying these pixels significantly leads to a failure of NN to classify the image correctly. However, one shortcoming of Algorithm RANDADV was that to build adversarial images, we sometimes had to apply a large perturbation to a single pixel (or a small set of pixels). Hence, there might exist a pixel (or a set of pixels) in the adversarial image whose coordinate value could lie outside the valid range [LB, UB]. To overcome this issue, we need to redesign the search procedure to generate adversarial images that still belong to the original image space I (defined in Section 3). Here a brute-force approach is generally not feasible because of computational reasons, especially in high-resolution images. Hence, we need to develop an efficient heuristic procedure to find the right small set of pixels to be perturbed. Our solution presented in this section is based on performing a greedy local search over the image space. + +![](images/83372eea17b2e701cefc45d156d3d9f66be0989cf1e8cc78c4eecbe71d0ecc36.jpg) +Figure 3: Experiments in Figures 1 and 2 for the high-resolution ImageNet1000 dataset. The results are again for good images from a set of 1000 randomly selected images. We use a slightly modified version of Algorithm RANDADV that perturbs a set of 50 pixels. + +We consider the general $k$ -misclassification problem (Definition 1) where an adversarial attack ensures that the true label does not appear in the top- $k$ predictions of the network. We utilize a local-search procedure, which is an incomplete search procedure that is widely used for solving combinatorial problems appearing in diverse domains such as graph clustering, scheduling, logistics, and verification (Lenstra, 1997). For a general optimization problem it works as follows. Consider an objective function $f ( \mathbf { z } ) : \mathbb { R } ^ { n } \to \mathbb { R }$ where the goal is to minimize $f ( \mathbf { z } )$ . The local-search procedure works in rounds, where each round consists of two steps. Let $\mathbf { z } _ { i - 1 }$ be the solution iterate after round $i - 1$ . Consider round $i$ . The first step is to select a small subset of points $Z = \{ \hat { \mathbf { z } } _ { 1 } , \hdots , \hat { \mathbf { z } } _ { n } \}$ , a so called local neighborhood, and evaluate $f ( \hat { \mathbf { z } } _ { j } )$ for every $\hat { \mathbf { z } } _ { j } \in Z$ . Usually, the set $Z$ consist of points that are close to current $\mathbf { z } _ { i - 1 }$ for some measure of distance which is domain specific. The second step selects a new solution $\mathbf { z } _ { i }$ taking into account the previous solution $\mathbf { z } _ { i - 1 }$ and the points in $Z$ . Hence, $\mathbf { z } _ { i } = g ( f ( \mathbf { z } _ { i - 1 } ) , f ( \hat { \mathbf { z } } _ { 1 } ) , \hdots , f ( \hat { \mathbf { z } } _ { n } ) )$ , where $g$ is some pre-defined transformation function. + +We adapt this general procedure to search critical sets efficiently as explained below. Our optimization problem will try to minimize the probability that the network determines an perturbed image has the class label of the original image, and by using a local-search procedure we generate perturbed images which differ from the original image in only few pixels. Intuitively, in each round, our local-search procedure computes an implicit approximation to the gradient of the current image by understanding the influence of a few pixels on the output, which is then used to update the current image. + +(a) First, we need to define the cost function $f$ . Let $I$ be the image (with true label $c ( I ) ,$ ) whose adversarial image we want to generate for a target neural network NN. For some input image $\hat { I }$ , we use the objective function $f _ { c ( I ) } ( \hat { I } )$ which equals the probability assigned by the network NN that the input image $\hat { I }$ belongs to class $c ( I )$ . More formally, + +$$ +f _ { c ( I ) } ( \hat { I } ) = o _ { c ( I ) } \mathrm { w h e r e } \mathrm { N N } ( \hat { I } ) = ( o _ { 1 } , \dots , o _ { C } ) , +$$ + +with $o _ { j }$ denoting the probability as determined by NN that image $\hat { I }$ belongs to class $j$ . Our local-search procedure aims to minimize this function. + +(b) Second, we consider how to form a neighborhood set of images. As mentioned above, the localsearch procedure operates in rounds. Let $\hat { I } _ { i - 1 }$ be the image after round $i - 1$ . Our neighborhood will consist of images that are different in one pixel from the image $\hat { I } _ { i - 1 }$ . In other words, if we measure the distance between $\hat { I } _ { i - 1 }$ and any image in the neighborhood as the number of perturbed pixels, then this distance is the same (equal to one) for all of them. Therefore, we can define the neighborhood in terms of a set of pixel locations. Let $( P _ { X } , P _ { Y } ) _ { i }$ be a set of pixel locations. For the first round $( P _ { X } , P _ { Y } ) _ { 0 }$ is randomly generated. At each subsequent round, it is formed based on a set of pixel locations which were perturbed in the previous round. Let $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 }$ denote the pixel locations that were perturbed in round $i - 1$ (formally defined below). Then + +$$ +( P _ { X } , P _ { Y } ) _ { i } = \bigcup _ { \{ ( a , b ) \in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 } \} } \bigcup _ { \{ x \in [ a - d , a + d ] , y \in [ b - d , b + d ] \} } ( x , y ) , +$$ + +where $d$ is a parameter. In other words, we consider pixels that were perturbed in the previous round, and for each such pixel we consider all pixels in a small square with the side length $2 d$ centered at that pixel. This defines the neighborhood considered in round $i$ . + +(c) Third, we describe the transformation function $g$ of a set of pixel locations. The function $g$ takes as input an image $\hat { I }$ , a set of pixel locations $( P _ { X } , P _ { Y } )$ , a parameter $t$ that defines how many pixels will be perturbed by $g$ , and two perturbation parameters $p$ and $r$ . In round $i$ of the local-search procedure, the function $g ( \hat { I } _ { i - 1 } , ( P _ { X } , P _ { Y } ) _ { i - 1 } , t , p , r )$ outputs a new image, such that exactly $t$ pixels of $\hat { I } _ { i - 1 }$ are perturbed, and an auxiliary set of pixel locations $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i }$ to record which $t$ pixels where perturbed at this round, so we have $( \hat { I } _ { i } , ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i } ) = g ( \hat { I } _ { i - 1 } , ( P _ { X } , P _ { Y } ) _ { i - 1 } , t , p , r )$ . Next we describe transformations that $g$ performs in round $i$ . As the first step, $g$ constructs a set of perturbed images based on $( P _ { X } , P _ { Y } ) _ { i - 1 }$ : + +$$ +\mathcal { T } = \bigcup _ { ( x , y ) \in ( P _ { X } , P _ { Y } ) _ { i - 1 } } \{ \operatorname { P E R T } ( \hat { I } _ { i - 1 } , p , ( x , y ) ) \} , +$$ + +where PERT is the perturbation function defined through (1). Then it computes the score of each image in $\mathcal { T }$ as + +$$ +\forall \tilde { I } \in \mathcal { T } : \mathrm { s c o r e } ( \tilde { I } ) = f _ { c ( I ) } ( \tilde { I } ) , +$$ + +and it sorts (in decreasing order) images in $\mathcal { T }$ based on the above score function to construct sorted $( \mathcal { T } )$ . Pixels whose perturbation lead to a larger decrease of $f$ are more likely useful in constructing an adversarial candidate. From sorted $( \mathcal { T } )$ , it records a set of pixel locations $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i }$ based on the first $t$ elements of sorted $( \mathcal { T } )$ , where the parameter $t$ regulates the number of pixels perturbed in each round. Formally, + +$$ +( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i } = \{ ( x , y ) : { \mathrm { P E R T } } ( { \hat { I } } _ { i - 1 } , p , ( x , y ) ) \in { \mathrm { s o r t e d } } ( { \mathcal { T } } ) [ 1 : t ] \} , +$$ + +where sorted $\left( \mathcal { T } \right) \left[ 1 : t \right]$ represents the first $t$ sorted images in sorted $( \mathcal { T } )$ . Finally, $\hat { I } _ { i }$ is constructed from $\hat { I } _ { i - 1 }$ by perturbing each pixel in location $( x , y ) \in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i }$ with a perturbation value $r$ . The perturbation is performed in a cyclic way (as explained in Algorithm CYCLIC) so that we make sure that all coordinate values in $\hat { I } _ { i }$ are within the valid bounds of LB and UB. Note that at the end of every round $i$ , $\hat { I } _ { i }$ is a valid image from the image space $\mathbb { I }$ . + +We want to point out that the function $g$ uses two perturbation parameters, $p$ and $r$ . The value of $r$ is kept small in the range [0, 2]. On the other hand, we do not put any explicit restrictions on the value of $p$ . The best choice of $p$ will be one that facilitates the identification of the “best” pixels to perturb in each round. In our experiments, we adjust the value of $p$ automatically during the search. We defer this discussion to the experimental section. + +Algorithm LOCSEARCHADV shows the complete pseudocode of our local-search procedure. At the high level, the algorithm takes an image as input, and in each round, finds some pixel locations to perturb using the above defined objective function and then applies the above defined transformation function to these selected pixels to construct a new (perturbed) image. It terminates if it succeeds to push the true label below the $k$ th place in the confidence score vector at any round. Otherwise, it proceeds to the next round (for a maximum of $R$ rounds). Note that the number of pixels in an image perturbed by Algorithm LOCSEARCHADV is at most $t \times R$ and in practice (see Tables 4, 5,and 6 in Section 6) it is much less. In round $i$ , we query the network at most the number of times as the number of pixels in $( P _ { X } , P _ { Y } ) _ { i }$ which after the first round is at most $2 d \times 2 d \times t$ (again in practice this is much less because of the overlaps in the neighborhood squares). + +In Section 6, we demonstrate the efficacy of Algorithm LOCSEARCHADV in constructing adversarial images. We first highlight an interesting connection between the pixels perturbed and their influences measured by a notion of called saliency map. + +A Relation to Saliency Maps. Simonyan et al. (2014) introduced the notion of saliency map as a way to rank pixels of the original images w.r.t. their influence on the output of the network. The intuition is that influential pixels in the saliency map are more likely to be important pixels that represent objects and, for example, can be used for weakly supervised object localization. Formally, let $\mathrm { N } \mathrm { \bar { N } } _ { c ( I ) } ( I )$ denote the probability assigned to true class $c ( I )$ by the network NN on input $I \in \mathbb { R } ^ { \ell \times \mathbf { \bar { w } } \times h }$ . Let $\dot { W _ { c ( I ) } } \in R ^ { \ell \times w \times h }$ denote the derivative of $\mathrm { N N } _ { c ( I ) }$ with respect to the input evaluated at image $I$ . + +
Algorithm 2 CYCLIC (r,b,x,y)
Assumptions:Perturbation parameter r ∈ [O,2] and LB≤O≤UB
Output:Perturbed image value at the coordinate (b,x,y) which lies in the range [LB,UB]
if rI(b,x,y)<LB then return rI(b,x,y)+(UB-LB)
else if rI(b,x,y)>UB then
return rI(b,x,y)-(UB-LB)
else
return rI(b,x,y)
+ +# Algorithm 3 LOCSEARCHADV (NN) + +Input: Image $I$ with true label $c ( I ) \in \{ 1 , \ldots , C \}$ , two perturbation parameters $p \in \mathbb R$ and $r \in [ 0 , 2 ]$ , and four other parameters: the half side length of the neighborhood square $d \in \mathbb { N }$ , the number of pixels perturbed at each round $t \in \mathbb { N }$ , the threshold $k \in \mathbb N$ for $k$ -misclassification, and an upper bound on the number of rounds $R \in \mathbb N$ . + +Output: Success/Failure depending on whether the algorithm finds an adversarial image or not +$\hat { I } _ { 0 } = I , i = 1$ +Pick $1 0 \%$ of pixel locations from $I$ at random to form $( P _ { X } , P _ { Y } ) _ { 0 }$ +while $i \leq R$ do {Computing the function $g$ using the neighborhood} $\begin{array} { r } { \mathcal { T } \bigcup _ { ( x , y ) \in ( P _ { X } , P _ { Y } ) _ { i - 1 } } \{ \mathrm { P E R T } \big ( \hat { I } _ { i - 1 } , p , x , y ) \} } \end{array}$ Compute $\mathrm { s c o r e } ( \tilde { I } ) = f _ { c ( I ) } ( \tilde { I } )$ for each $\tilde { I } \in \mathcal { I }$ (where $f _ { c ( I ) } ( \tilde { I } ) = o _ { c ( I ) }$ with $\mathbf { N N } ( \tilde { I } ) = ( o _ { 1 } , \dots , o _ { C } ) )$ sorted $( { \mathcal { T } } ) \gets$ images in $\mathcal { T }$ sorted by descending order of score $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i } \gets \{ ( x , y ) : \operatorname { P E R T } ( \hat { I } _ { i - 1 } , p , x , y ) \in \operatorname { s o r t e d } ( \mathcal { D } ) [ 1 : t ] \}$ (with ties broken arbitrarily) {Generation of the perturbed image $\hat { I } _ { i }$ } for $( x , y ) \in ( P _ { X } ^ { * } , \bar { P } _ { Y } ^ { * } ) _ { i }$ and each channel $^ { b }$ do $\hat { I } _ { i } ( b , x , y ) \mathrm { C Y C l }$ LIC $( r , b , x , y )$ end for {Check whether the perturbed image $\hat { I } _ { i }$ is an adversarial image} if $c ( I ) \notin \pi ( \mathbf { N N } ( \hat { I } _ { i } ) , \bar { k } )$ then return Success end if {Update a neighborhood of pixel locations for the next round} $\begin{array} { r } { ( \hat { P _ { X } } , P _ { Y } ) _ { i } \longleftarrow \bigcup _ { \{ ( a , b ) \in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 } \} } \bigcup _ { \{ x \in [ a - d , a + d ] , y \in [ b - d , b + d ] \} } ( x , y ) } \end{array}$ i ← i + 1 +end while +return Failure + +The saliency map of $I$ is the matrix $M \in \mathbb { R } ^ { w \times h }$ such that $\begin{array} { r } { M _ { i , j } = \operatorname* { m a x } _ { b \in [ \ell ] } \ | W _ { c ( I ) } ( b , x , y ) | } \end{array}$ , where $W _ { c ( I ) } ( b , x , y )$ is the element of $W _ { c ( I ) }$ corresponding to channel $b$ and location $( x , y )$ . Pixels with higher scores are considered more influential. In subsequent works, this notion has been extended to adversarial saliency maps that can be useful in generating adversarial perturbations (Papernot et al., 2016c). + +Computing the exact saliency scores for an image requires complete access to the network NN, which we do not assume. However, a natural hypothesis is that the pixels selected by Algorithm LOCSEARCHADV for perturbation are related to pixels with large saliency scores. We use the ImageNet1000 dataset to test this hypothesis. In Figure 3, we present some qualitative results. As can be seen from the pictures, the pixels perturbed by Algorithm LOCSEARCHADV appear correlated with pixels with high saliency scores. Quantitatively, we observed that the pixels that occupy top- $10 \%$ of the saliency map, on average contain more than $23 \%$ of the pixels chosen by Algorithm LOCSEARCHADV for perturbation (and this overlap only grows when we consider a bigger chunk of pixels picked by their saliency scores). Note that this is correlation is not though a random occurrence. For an image $I$ , let $S _ { I }$ denote the set of pixels in $I$ that rank among the top- $10 \%$ in the saliency map. If we pick a random set of around 200 pixels (this is on average number of pixels perturbed per image by Algorithm LOCSEARCHADV perturbs, see Table 5), we expect only about $10 \%$ t o them to intersect with $S _ { I }$ and standard tail bounds show that the probability that at least $23 \%$ of the pixels of this random set intersects with $S _ { I }$ is extremely small.9 Therefore, it appears that Algorithm LOCSEARCHADV rediscovers part of the high salient score pixels but without explicitly computing the gradients. + +# 6 EXPERIMENTAL EVALUATION + +We start by describing our experimental setup. We used Caffe and Torch machine learning frameworks to train the networks. All algorithms to generate adversarial images were implemented in Lua within Torch 7. All experiments were performed on a cluster of GPUs using a single GPU for each run. + +Datasets. We use 5 popular datasets: MNIST (handwritten digits recognition dataset), CIFAR10 (objects recognition dataset), SVHN (digits recognition dataset), STL10 (objects recognition dataset), and ImageNet1000 (objects recognition dataset). + +Models. We trained Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman, 2014) for MNIST, CIFAR, SVHN, STL10, with minor adjustments for the corresponding image sizes. Network-in-Network is a building block of the commonly used GoogLeNet architecture that has demonstrated very good performance on medium size datasets, e.g. CIFAR10 (Zagoruyko, 2015). VGG is another powerful network that proved to be useful in many applications beyond image classification, like object localization (Ren et al., 2015). We trained each model in two variants: with and without batch normalization (Ioffe & Szegedy, 2015). Batch normalization was placed before a ReLU layer in all networks. For the ImageNet1000 dataset, we used pre-trained VGG models from (Chatfield et al., 2014b) (we did not train them from scratch due to limited resources). All Caffe VGG models were converted to Torch models using the loadcaffe package (Zagoruyko, 2016a). These models use different normalization procedures which we reproduced for each model based on provided descriptions. Tables 4 and 5 (the second column ERRTOP-1) show the top-1 (base) error for all datasets and models that we considered. The results are comparable with the known state-of-the-art results on these datasets (Benenson, 2016). + +Related Techniques. There are quite a few approaches for generating adversarial images (as discussed in Section 2). Most of these approaches require access to the network architecture and its parameter values (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c).10 The general idea behind these attacks is based on the evaluating the network’s sensitivity to the input components in order to determine a perturbation that achieves the adversarial misclassification goal. Among these approaches, the attack approach (known as the “fast-gradient sign method”) suggested by Goodfellow et al. (2015) stands out for being able to efficiently generate adversarial images. Here we compare the performance of our local-search based attack against this fast-gradient sign method.11 + +![](images/6a4d9a530eb9e0cd10fe492164133a18e40afd6afa68fbfcd68a866b16882a02.jpg) +Table 3: Results on ImageNet1000 using VGG CNN-S (Caffe) network (Chatfield et al., 2014a). Columns from left to right: the original image, top 150 pixels chosen according to their saliency scores (in white), the absolute difference between the perturbed image and the true image (the pixels that are perturbed appear in white), and the perturbed image. Adversarial misclassification (rows from top to bottom): a ruffed grouse misclassified as a frilled lizard, an artichoke misclassified as a sleeping bag, a bubble misclassified as a fountain, and a hare misclassified as a cheetah. + +For completeness, we now briefly explain the fast-gradient sign method of Goodfellow et al. (2015). Given an image $I _ { 0 }$ , a label $a \in \{ 1 , \ldots , C \}$ , and a network NN, the fast-gradient sign method perturbs $I _ { 0 }$ using the following update rule: $I _ { 0 } ^ { \mathrm { p e r t } } = I _ { 0 } + \epsilon \cdot \mathrm { s i g n } ( \nabla _ { I = I _ { 0 } } \mathrm { L o s s } ( \mathbf { N N } ( I ) , a ) )$ where $\operatorname { s i g n } ( \nabla _ { I = I _ { 0 } } \operatorname { L o s s } ( \mathbf { N N } ( I ) , a ) )$ is the sign of the network’s cost function gradient (here $\mathrm { L o s s } ( \mathbf { N N } ( I ) , a )$ denotes the loss function of the network NN given input $I$ and class $a$ ). We vary $a$ over all possible labels in the dataset and choose the best result where this procedure is successful in generating an adversarial image. Without general guidelines for setting $\epsilon$ , we experimented with several values of $\epsilon$ starting from 0.07 and increasing this number. We found that the value $\epsilon = 0 . 2 ^ { 1 2 }$ was the smallest value where the fast-gradient sign method started to yield competitive performance compared to our algorithm. Smaller values of $\epsilon$ leads to generation of fewer adversarial images, e.g., at $\epsilon = 0 . 1$ , the percentage of generated adversarial images is reduced by around $10 \%$ as compared to the value at $\epsilon = 0 . 2$ for the CIFAR10 dataset on the Network-in-Network model. Larger values of $\epsilon$ tends to generate more adversarial images, but this comes at the cost of an increase in the perturbation. As we discuss later, our local-search based approach yields better results than the fast-gradient sign method in both the volume of adversarial images generated and the amount of perturbation applied. Another important point to remember is that unlike the fast-gradient sign method, our approach is based on a weaker and more realistic assumption on the adversarial power, making our attacks more widely applicable. + +Implementing Algorithm LOCSEARCHADV. For each image $I$ , we ran Algorithm LOCSEARCHADV for at most 150 rounds, perturbing 5 pixels at each round, and use squares of side length 10 to form the neighborhood (i.e., $R = 1 5 0 , t = 5 , d = 5 )$ . With this setting of parameters, we perturb a maximum of $t \times R = 7 5 0$ pixels in an image. The perturbation parameter $p$ was adaptively adjusted during the search. This helps in faster determination of the most helpful pixels in generating the adversarial image. Let $I$ be the original image. For some round $i$ of the algorithm, define $\bar { o } _ { c ( I ) } = \arg _ { ( x , y ) } \{ o _ { c ( I ) } : ( x , y ) \in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 } \} .$ , where ${ { O } _ { c ( I ) } }$ is the probability assigned to class label $c ( I )$ in $\mathrm { N N } \big ( \mathrm { P E R T } \big ( \hat { I } _ { i - 1 } , p , x , y \big ) \big )$ (here $\bar { o } _ { c ( I ) }$ provides an approximation of the average confidence of the network NN in predicting the true label over perturbed images). At each round, we increase the value of $p$ if $\bar { o } _ { c ( I ) }$ is close to one and decrease $p$ if $\bar { o } _ { c ( I ) }$ is low, e.g., below 0.3. For Algorithm CYCLIC, we set $r = 3 / 2$ . To avoid perturbing the most sensitive pixels frequently, we make sure that if a pixel is perturbed in a round then we exclude it from consideration for the next 30 rounds. + +Experimental Observations. For ease of comparison with the fast-gradient sign method (Goodfellow et al., 2015), we set $k = 1$ and focus on achieving 1-misclassification. Tables 4 and 5 show the results of our experiments on the test sets. The first column shows the dataset name. The second column (ERRTOP-1) presents the top-1 misclassification rate on the corresponding test dataset without any perturbation (base error). ERRTOP-1(ADV) is the top-1 misclassification rate where each original image in the test set was replaced with an generated perturbed image (using either our approach or the fast-gradient sign method (Goodfellow et al., 2015) which is denoted as FGSM).13 + +In the following, we say an adversarial generation technique ADV, given an input image $I$ , succeeds in generating an adversarial image $\mathrm { A D V } ( I )$ for a network NN iff $c ( I ) \in \pi ( \mathrm { N N } ( I ) , 1 )$ and $c ( I ) \notin$ $\pi ( \bar { \mathrm { N N } } ( \mathrm { A D V } ( I ) ) , 1 )$ . The CONF column shows the average confidence over all successful adversarial images for the corresponding technique. The PTB column shows the average (absolute) perturbation added per coordinate in cases of successful adversarial generation. More formally, let $\tau$ denote the test set and $\tau _ { \mathrm { A D V } } \subseteq \tau$ denote the set of images in $\tau$ on which ADV is successful. Then, + +$$ +\mathsf { P T B } = \frac { 1 } { | \mathcal T _ { \mathrm { A D V } } | } \sum _ { I \in \mathcal T _ { \mathrm { A D V } } } \frac { 1 } { \ell \times w \times h } \sum _ { b , x , y } | I ( b , x , y ) - \mathrm { A D V } ( I ) ( b , x , y ) | , +$$ + +where $I \in \mathbb { R } ^ { \ell \times w \times h }$ is the original image and $\mathrm { A D V } ( I ) \in \mathbb { R } ^ { \ell \times w \times h }$ is the corresponding adversarial image. Note that the inner summation is measuring the $L _ { 1 }$ -distance between $I$ and $\mathrm { A D V } ( I )$ . The #PTBPIXELS column shows the average percentage of perturbed pixels in the successful adversarial images. Similarly, TIME column shows the average time (in seconds) to generate a successful adversarial image. Finally, the last column indicates the type of network architecture. + +As is quite evident from these results, Algorithm LOCSEARCHADV is more effective than the fastgradient sign method in generating adversarial images, even without having access to the network architecture and its parameter values. The difference is quite prominent for networks trained with batch normalization as here we noticed that the fast-gradient sign method has difficulties producing adversarial images.14 Another advantage with our approach is that it modifies a very tiny fraction of pixels as compared to all the pixels perturbed by the fast-gradient sign method, and also in many cases with far less average perturbation. Putting these points together demonstrates that + +Algorithm LOCSEARCHADV is successful in generating more adversarial images than the fastgradient sign method, while modifying far fewer pixels and adding less noise per image. On the other side, the fast-gradient sign method takes lesser time in the generation process and generally seems to produce higher confidence scores for the adversarial (misclassified) images. + +Table 5 shows the results for several variants of VGG network trained on the ImageNet1000 dataset. These networks do not have batch normalization layers (Chatfield et al., 2014b; Zagoruyko, 2016a). We set $\epsilon = 1$ for the fast-gradient sign method as a different pre-processing technique was used for this network (we converted these networks from pre-trained Caffe models). Results are similar to that observed on the smaller datasets. In most cases, our proposed local-search based approach is more successful in generating adversarial images while on average perturbing less than $0 . 5 5 \%$ of the pixels. + +Case of Larger $k$ ’s. We now consider achieving $k$ -misclassification for $k \geq 1$ using Algorithm LOCSEARCHADV. In Table 6, we present the results as we change the goal from 1- misclassification to 4-misclassification on the CIFAR10 dataset. We use the same parameters as before for Algorithm LOCSEARCHADV. As one would expect, as we increase the value of $k$ , the effectiveness of the attack decreases, perturbation and time needed increases. But overall our local-search procedure is still able to generate a large fraction of adversarial images at even $k = 4$ with a small perturbation and computation time, meaning that these images will fool even a system that is evaluated on a top-4 classification criteria. We are not aware of a straightforward extension of the fast-gradient sign method (Goodfellow et al., 2015) to achieve $k$ -misclassification. + +Even Weaker Adversarial Models. We also consider a weaker model where the adversary does not even have a black-box (oracle) access to the network (NN) of interest, and has to rely on a black-box access to somewhat of a “similar” (proxy) network as NN. For example, the adversary might want to evade a spam filter A, but might have to develop adversarial images by utilizing the output of a spam filter B, which might share properties similar to A. + +We trained several modifications of Network-in-Network model for the CIFAR10 dataset, varying the initial value of the learning rate, the size of filters, and the number of layers in the network. We observed that between $2 5 \%$ to $43 \%$ of adversarial images generated by Algorithm LOCSEARCHADV using the original network were also adversarial for these modified networks (at $k = 1$ ). The transferability of adversarial images that we observe here has also been observed with other attacks too (Szegedy et al., 2014; Goodfellow et al., 2015; Papernot et al., 2016b;a) and demonstrates the wider applicability of all these attacks. + +# 7 CONCLUSION + +We investigate the inherent vulnerabilities in modern CNNs to practical black-box adversarial attacks. We present approaches that can efficiently locate a small set of pixels, without using any gradient information, which when perturbed lead to misclassification by a deep neural network. Our extensive experimental results, somewhat surprisingly, demonstrates the effectiveness of our simple approaches in generating adversarial examples. + +Defenses against these attacks is an interesting research direction. However, we note that here that by limiting the perturbation to some pixels (being localized) the adversarial images generated by our local-search based approach do not represent the distribution of the original data. This means for these adversarial images, the use of adversarial training (or fine-tuning), a technique of training (or fine-tuning) networks on adversarial images to build more robust classifiers, is not very effective. In fact, even with adversarial training we noticed that the networks ability to resist new local-search based adversarial attack improves only marginally (on average between $1 - 2 \%$ ). On the other hand, we suspect that one possible counter-measure to these localized adversarial attacks could be based on performing a careful analysis of the oracle queries to thwart the attempts to generate an adversarial image. + +Finally, we believe that our local-search approach can also be used for attacks against other machine learning systems and can serve as an useful tool in measuring the robustness of these systems. + +Table 4: Results for four datasets: CIFAR10, STL10, SVHN, and MNIST. The entries denote by denoted by “– ” are the cases where the fast-gradient sign method fails to produce any adversarial image in our experimental setup. + +
DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)TechniqueNetwork
NNs trained with batch normalization
CIFAR1011.6597.630.47 0.550.04 0.203.750.68LOCSEARCHADV (Ours)NinN
CIFAR1070.69
11.6297.510.74100.000.01FGSM(Goodfellow et al., 2015)NinN
CIFAR100.043.160.78LOCSEARCHADV (Ours)VGG
CIFAR1011.621111FGSM(Goodfellow et al.,2015)VGG
STL1058.170.420.021.207.15LOCSEARCHADV (Ours)NinN
STL1029.8154.850.530.20100.000.03FGSM(Goodfellow et al.,2015)NinN
STL1065.760.470.021.1113.90LOCSEARCHADV (Ours)VGG
STL1026.5026.50二1 0.05二二FGSM(Goodfellow et al., 2015)VGG
SVHN SVHN9.7197.06 48.620.47 0.490.204.51 100.001.02 0.02LOCSEARCHADV (Ours) FGSM(Goodfellow et al.,2015)NinN NinN
SVHN81.100.660.07 15.432.15LOCSEARCHADV (Ours)VGG VGG
4.77
SVHN4.771 0.54 0.201 2.241 0.64FGSM(Goodfellow et al.,2015)NinN
MNIST91.42
MNIST0.331.650.580.20100.000.02LOCSEARCHADV (Ours) FGSM (Goodfellow et al., 2015)
93.480.630.21 2.200.64LOCSEARCHADV(Ours)NinN VGG
MNIST MNIST0.440.44111 NNs trained without batch normalization1FGSM(Goodfellow et al., 2015)VGG
97.8916.540.720.043.240.58LOCSEARCHADV (Ours)NinN
CIFAR10 CIFAR1093.670.930.20100.000.02FGSM(Goodfellow et al., 2015)NinN
CIFAR1019.7997.980.770.042.990.72LOCSEARCHADV (Ours)VGG
90.930.900.20100.000.04FGSM(Goodfellow et al.,015)VGG
CIFAR1035.470.021.176.42LOCSEARCHADV (Ours)
STL1052.650.56 0.940.200.04NinN
STL1043.9187.160.520.01100.00FGSM(Goodfellow et al.,2015)NinN
STL1059.380.930.201.09 100.0019.65 0.10LOCSEARCHADV (Ours) FGSM(Goodfellow et al., 2015)VGG VGG
STL10SVHN 6.1573.9791.36 92.310.680.05
+ +Table 5: Results for the ImageNet1000 dataset using a center crop of size $2 2 4 \times 2 2 4$ for each image. + +
DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%）TIME (in sec)TechniqueNetwork
ImageNet100058.2793.590.290.290.4312.72LOCSEARCHADV (Ours)VGG CNN-S (Caffe)
ImageNet100085.510.491.00100.004.74FGSM(Goodfellow et al.,2015)VGG CNN-S (Caffe)
ImageNet100058.9691.360.280.290.4010.01LOCSEARCHADV (Ours)VGG CNN-M (Caffe)
ImageNet100087.850.481.00100.004.36FGSM(Goodfellow etal.,015)VGG CNN-M (Caffe)
ImageNet100058.8092.820.290.300.4111.09LOCSEARCHADV (Ours)VGG CNN-M 2048 (Caffe)
ImageNet100088.430.521.00100.004.42FGSM(Goodfellow et al.,2015)VGG CNN-M 2048 (Caffe)
ImageNet100046.4072.070.300.540.5573.64LOCSEARCHADV (Ours)VGG ILSVRC 19 (Caffe)
ImageNet100085.050.521.00100.0023.94FGSM(Goodfellow et al., 2015)VGG ILSVRC 19 (Caffe)
+ +
DatasetkERRTOP-kERRTOP-k(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)Network
CIFAR1016.5497.890.720.043.240.58NinN
CIFAR106.8876.650.880.075.501.02NinN
CIFAR1012343.5859.020.900.087.091.85NinN
CIFAR101.8448.890.900.097.632.12NinN
+ +Table 6: Effect of increasing $k$ on the performance of Algorithm LOCSEARCHADV (without batch normalization). + +# ACKNOWLEDGMENTS + +The authors would like to thank Hamid Maei for helpful initial discussions. + +# REFERENCES + +Martín Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In ACM CCS, 2016. + +R. Benenson. What is the class of this image ? http://rodrigob.github.io/are_we_ there_yet/build/classification_datasets_results.html, 2016. + +Eli Biham and Adi Shamir. Differential cryptanalysis of des-like cryptosystems. Journal of CRYPTOLOGY, 4(1):3–72, 1991. + +K. Chatfield, K. Simonyan, A. Vedaldi, and A. Zisserman. Caffe model description: VGG_CNN_S. https://gist.github.com/ksimonyan/fd8800eeb36e276cd6f9, 2014a. + +Ken Chatfield, Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Return of the devil in the details: Delving deep into convolutional nets. In BMVC, 2014b. + +Alhussein Fawzi, Omar Fawzi, and Pascal Frossard. Analysis of classifiers’ robustness to adversarial perturbations. CoRR, abs/1502.02590, 2015. + +Alhussein Fawzi, Seyed-Mohsen Moosavi-Dezfooli, and Pascal Frossard. Robustness of classifiers: from adversarial to random noise. arXiv preprint arXiv:1608.08967, 2016. + +Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. Book in preparation for MIT Press, 2016. URL http://www.deeplearningbook.org. + +Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. In ICLR, 2015. + +Kathrin Grosse, Nicolas Papernot, Praveen Manoharan, Michael Backes, and Patrick McDaniel. Adversarial perturbations against deep neural networks for malware classification. arXiv preprint arXiv:1606.04435, 2016. + +Shixiang Gu and Luca Rigazio. Towards deep neural network architectures robust to adversarial examples. In ICLR Workshop, 2015. + +Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, pp. 448–456, 2015. + +Alexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial examples in the physical world. arXiv preprint arXiv:1607.02533, 2016. + +Jan Karel Lenstra. Local search in combinatorial optimization. Princeton University Press, 1997. + +Min Lin, Qiang Chen, and Shuicheng Yan. Network in network. In ICLR, 2014. + +Patrick McDaniel, Nicolas Papernot, and Z. Berkay Celik. Machine learning in adversarial settings. IEEE Security & Privacy, 14(3):68–72, 2016. + +Seyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi, and Pascal Frossard. Deepfool: a simple and accurate method to fool deep neural networks. In CVPR, 2016. + +Nicolas Papernot, Patrick McDaniel, and Ian Goodfellow. Transferability in machine learning: from phenomena to black-box attacks using adversarial samples. arXiv preprint arXiv:1605.07277, 2016a. + +Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against deep learning systems using adversarial examples. arXiv preprint arXiv:1602.02697, 2016b. + +Nicolas Papernot, Patrick McDaniel, Somesh Jha, Matt Fredrikson, Z. Berkay Celik, and Ananthram Swami. The limitations of deep learning in adversarial settings. In 2016 IEEE European Symposium on Security and Privacy (EuroS&P), pp. 372–387. IEEE, 2016c. + +Nicolas Papernot, Patrick McDaniel, Xi Wu, Somesh Jha, and Ananthram Swami. Distillation as a defense to adversarial perturbations against deep neural networks. In IEEE Symposium on Security and Privacy, 2016d. + +Shaoqing Ren, Kaiming He, Ross B. Girshick, and Jian Sun. Faster R-CNN: towards real-time object detection with region proposal networks. In NIPS, pp. 91–99, 2015. + +Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. + +Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps. In ICLR Workshop, 2014. + +Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. In ICLR, 2014. + +S. Zagoruyko. Cifar-10 in torch. http://torch.ch/blog/2015/07/30/cifar.html, 2015. + +S. Zagoruyko. Loadcaffe. https://github.com/szagoruyko/loadcaffe, 2016a. + +S. Zagoruyko. Are deep learning algorithms easily hackable? http://coxlab.github.io/ ostrichinator, 2016b. \ No newline at end of file diff --git a/parse/train/SJCscQcge/SJCscQcge_content_list.json b/parse/train/SJCscQcge/SJCscQcge_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..624e75060168ddae21ed850bbab4f4a417ed2760 --- /dev/null +++ b/parse/train/SJCscQcge/SJCscQcge_content_list.json @@ -0,0 +1,1666 @@ +[ + { + "type": "text", + "text": "SIMPLE BLACK-BOX ADVERSARIAL PERTURBATIONS FOR DEEP NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nina Narodytska Shiva Kasiviswanathan Samsung Research America Mountain View, CA 94043, USA {n.narodytska,kasivisw}@gmail.com ", + "bbox": [ + 184, + 169, + 509, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 262, + 544, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks are powerful and popular learning models that achieve stateof-the-art pattern recognition performance on many computer vision, speech, and language processing tasks. However, these networks have also been shown susceptible to carefully crafted adversarial perturbations which force misclassification of the inputs. Adversarial examples enable adversaries to subvert the expected system behavior leading to undesired consequences and could pose a security risk when these systems are deployed in the real world. ", + "bbox": [ + 233, + 295, + 766, + 392 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we focus on deep convolutional neural networks and demonstrate that adversaries can easily craft adversarial examples even without any internal knowledge of the target network. Our attacks treat the network as an oracle (blackbox) and only assume that the output of the network can be observed on the probed inputs. Our first attack is based on a simple idea of adding perturbation to a randomly selected single pixel or a small set of them. We then improve the effectiveness of this attack by carefully constructing a small set of pixels to perturb by using the idea of greedy local-search. Our proposed attacks also naturally extend to a stronger notion of misclassification. Our extensive experimental results illustrate that even these elementary attacks can reveal a deep neural network’s vulnerabilities. The simplicity and effectiveness of our proposed schemes mean that they could serve as a litmus test for designing robust networks. ", + "bbox": [ + 233, + 395, + 766, + 560 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 582, + 336, + 598 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Convolutional neural networks (CNNs) are among the most popular techniques employed for computer vision tasks, including but not limited to image recognition, localization, video tracking, and image and video segmentation (Goodfellow et al., 2016). Though these deep networks have exhibited good performances for these tasks, they have recently been shown to be particularly susceptible to adversarial perturbations to the input images (Szegedy et al., 2014; Goodfellow et al., 2015; MoosaviDezfooli et al., 2016; Papernot et al., 2016c;b; Kurakin et al., 2016; Grosse et al., 2016; Zagoruyko, 2016b). Vulnerability of these networks to adversarial attacks can lead to undesirable consequences in many practical applications using them. For example, adversarial attacks can be used to subvert fraud detection, malware detection, or mislead autonomous navigation systems (Papernot et al., 2016c; Grosse et al., 2016). Further strengthening these results is a recent observation by Kurakin et al. (2016) who showed that a significant fraction of adversarial images crafted using the original network are misclassified even when fed to the classifier through a physical world system (such as a camera). ", + "bbox": [ + 174, + 611, + 825, + 777 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we investigate the problem of robustness of state-of-the-art convolutional neural networks (CNNs) to simple black-box adversarial attacks. The rough goal of adversarial attacks is as follows: Given an image $I$ that is correctly classified by a machine learning system (say, a CNN), is it possible to construct a transformation of $I$ (say, by adding a small perturbation to some or all the pixels) that now leads to misclassification by the system. Since large perturbations can trivially lead to misclassification, the attacks seek to limit the amount of perturbation applied under some chosen metric. More often than not, in these attacks, the modification done to the image is so subtle that the changes are imperceptible to a human eye. Our proposed attacks also share this property, in addition to being practical and simplistic, thus highlighting a worrying aspect about lack of robustness prevalent in these modern vision techniques. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There are two main research directions in the literature on adversarial attacks based on different assumptions about the adversarial knowledge of the target network. The first line of work assumes that the adversary has detailed knowledge of the network architecture and the parameters resulting from training (or access to the labeled training set) (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c). Using this information, an adversary constructs a perturbation for a given image. The most effective methods are gradient-based: a small perturbation is constructed based on the gradients of the loss function w.r.t. the input image and a target label. Often, adding this small perturbation to the original image leads to a misclassification. In the second line of work an adversary has restricted knowledge about the network from being able to only observe the network’s output on some probed inputs (Papernot et al., 2016b). Our work falls into this category. While this black-box model is a much more realistic and applicable threat model, it is also more challenging because it considers weak adversaries without knowledge of the network architecture, parameters, or training data. Interestingly, our results suggest that this level of access and a small number of queries provide sufficient information to construct an adversarial image. ", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/8f8bd387fd79bbd926de42b8e1bd85f7e2cc447025bdada2d104ff3603f7faa7.jpg", + "image_caption": [ + "Table 1: The top row shows the original images and the bottom row shows the perturbed images. The misclassification is as follows: (a) a stingray misclassified as a sea lion, (b) an ostrich misclassified as a goose, (c) a jay misclassified as a junco, and (d) a water ouzel misclassified as a redshank. " + ], + "image_footnote": [], + "bbox": [ + 187, + 311, + 810, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As we operate in a black-box setting, we use a gradient-free approach to adversarial image generation. Papernot et al. (2016b) were the first to discuss a black-box attack against deep learning systems. Their attack crucially relies on the observation that there is a transferability (generalization) property in adversarial examples, i.e., adversarial examples form one model transfers to another. Our proposed attacks on the other hand is much more simple and direct, does not require this transferability property, and hence is more effective in constructing adversarial images, in addition to having some other computational advantages. We demonstrate that our method is capable of constructing adversarial images for several network architectures trained on different datasets. In particular in this paper, we consider the CIFAR10, MNIST, SVHN, STL10, and ImageNet1000 datasets, and two popular network architectures, Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman, 2014). In Table 1, we show four images from the ImageNet1000 dataset. The original images are in the upper row. The bottom row shows the corresponding perturbed images produced by our algorithm which are misclassified by a VGG CNN-S network (Chatfield et al., 2014a). ", + "bbox": [ + 174, + 619, + 826, + 799 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our Contributions. In this work, we present simple and effective black-box adversarial attacks on deep convolutional neural networks. We make the following main contributions in this paper. ", + "bbox": [ + 171, + 813, + 823, + 840 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(1) The first question we investigate is the influence of perturbing a single pixel on the prediction. To do so, we devise a simple scheme, based on randomly selecting a single pixel and applying a strong perturbation to it. Somewhat surprisingly, we noticed that a few trails of this random experiment is already quite enough in generating adversarial images for low resolution image sets. In fact, in many cases, for misclassification, the amount of perturbation needed to be applied to the selected pixel is also quite small. For high-resolution images, a similar phenomena holds, except our scheme now picks a random set of around 50 pixels. These simple experiments show the ease of generating adversarial images for modern deep CNNs without knowledge of either the network architecture or its parameters. There is however one shortcoming in these approaches in that the perturbed image might have pixel values that are outside some expected range. ", + "bbox": [ + 163, + 854, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 187, + 103, + 823, + 172 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(2) We overcome this above shortcoming by showing that lower perturbation suffices if we carefully select the pixels for perturbation. The approach is based the idea of greedy local search, an iterative search procedure, where in each round a local neighborhood is used to refine the current image and in process minimizing the probability of the network assigning high confidence scores to the true class label. Again while the algorithm is quite simple, it is rather effective in generating adversarial images with quite small perturbations. We also show an interesting connection between the pixels chosen for perturbation by our approach and the saliency map of an image, as defined by Simonyan et al. (2014), that ranks pixels based on their influence on the output score. In effect our approach identifies pixels with high saliency scores but without explicitly using any gradient information (as needed in the definition of saliency map (Simonyan et al., 2014)). Intuitively, in each round, our local-search based approach computes an implicit approximation to the gradient of the current image by understanding the influence of a few pixels on the output, which is then used to update the current image. ", + "bbox": [ + 161, + 181, + 825, + 362 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(3) We perform extensive experimental evaluations, and show that our local-search based approach reliably generates adversarial examples with little perturbation (even when compared to a recent elegant adversarial attack proposed by Goodfellow et al. (2015) which needs perfect knowledge of the network). Another feature of our attack is that, by design, our approach only perturbs a very small fraction of the pixels during the adversarial image generation process (e.g., on the ImageNet1000 dataset we on average perturb only about $0 . 5 \\%$ of the pixels per image). Most previous attacks require the ability to perturb all the pixels in the image. ", + "bbox": [ + 161, + 369, + 825, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(4) Our approaches naturally extend to a stronger notion of misclassification (that we refer to as $k$ -misclassification), where the goal is to ensure that the true label of the image does not even appear in the top- $k$ predictions of the network (obtained by sorting the confidence score vector). This notion especially captures the fact that many modern systems (e.g., ImageNet competition entrants) are evaluated based on top- $k$ predictions. To the best of our knowledge, these are the first adversarial attacks on deep neural networks achieving $k$ -misclassification. ", + "bbox": [ + 161, + 474, + 825, + 559 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 580, + 341, + 597 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Starting with the seminal paper by Szegedy et al. (2014), which showed that the state-of-the-art neural networks are vulnerable to adversarial attacks, there has been significant attention focused on this problem. The research has led to investigation of different adversarial threat models and scenarios (Papernot et al., 2016c;b; Grosse et al., 2016; Kurakin et al., 2016; Fawzi et al., 2016), computationally efficient attacks (Goodfellow et al., 2015), perturbation efficient attacks (MoosaviDezfooli et al., 2016), etc. ", + "bbox": [ + 174, + 611, + 825, + 694 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Szegedy et al. (2014) used a box-constrained L-BFGS technique to generate adversarial examples. They also showed a transferability (or generalization) property for adversarial examples, in that adversarial examples generated for one network might also be misclassified by a related network with possibly different hyper-parameters (number of layers, initial weights, etc.). However, the need for a solving a series of costly penalized optimization problems makes this technique computationally expensive for generating adversarial examples. This issue was fixed by Goodfellow et al. (2015) who motivated by the underlying linearity of the components used to build a network proposed an elegant scheme based on adding perturbation proportional to sign of the network’s cost function gradient. Recently, Moosavi-Dezfooli et al. (2016) used an iterative linearization procedure to generate adversarial examples with lesser perturbation. Another recent attack proposed by Papernot et al. (2016c) uses a notion of adversarial saliency maps (based on the saliency maps introduced by (Simonyan et al., 2014)) to select the most sensitive input components for perturbation. This attack has been adapted by Grosse et al. (2016) for generating adversarial samples for neural networks used as malware classifiers. However, all these above described attacks require perfect knowledge of the target network’s architecture and parameters which limits their applicability to strong adversaries with the capability of gaining insider knowledge of the target system. ", + "bbox": [ + 174, + 702, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our focus in this paper is the setting of black-box attacks, where we assume that an adversary has only the ability to use the network as an oracle. The adversary can obtain output from supplied inputs, and use the observed input-output relationship to craft adversarial images.1 In the context of deep neural networks, a black-box attack was first proposed by Papernot et al. (2016b) with the motivation of constructing an attack on a remotely hosted system.2 Their general idea is to first approximate the target network by querying it for output labels, which is used to train a substitute network, which is then used to craft adversarial examples for the original network. The success of the attack crucially depends on the transferability property to hold between the original and the substitute network. Our black-box attack is more direct, and completely avoids the transferability assumption, making it far more applicable. We also avoid the overhead of gathering data and training a substitute network. Additionally, our techniques can be adapted to a stronger notion of misclassification. ", + "bbox": [ + 174, + 103, + 825, + 256 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A complementary line of work has focused on building defenses against adversarial attacks. Although designing defenses is beyond scope of this paper, it is possible that adapting the previous suggested defense solutions such as Jacobian-based regularization (Gu & Rigazio, 2015) and distillation (Papernot et al., 2016d) can reduce the efficacy of our proposed attacks. Moreover, the recently proposed technique of differentially private training (Abadi et al., 2016) can also prove beneficial here. ", + "bbox": [ + 174, + 263, + 825, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The study of adversarial instability have led to development of solutions that seeks to improve training to in return increase the robustness and classification performance of the network. In some case, adding adversarial examples to the training (adversarial training) set can act like a regularizer (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016). The phenomenon of adversarial instability has also been theoretically investigated for certain families of classifiers under various models of (semi) random noise (Fawzi et al., 2015; 2016). However, as we discuss later, due to peculiar nature of adversarial images generated by our approaches, a simple adversarial training is only mildly effective in preventing future similar adversarial attacks. ", + "bbox": [ + 174, + 339, + 825, + 452 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The security of machine learning in settings distinct from deep neural networks is also an area of active research with various known attacks under different threat models. We refer the reader to a recent survey by McDaniel et al. (2016) and references therein. ", + "bbox": [ + 176, + 458, + 825, + 501 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 517, + 339, + 532 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Notation and Normalization. We denote by $[ n ]$ the set $\\{ 1 , \\ldots , n \\}$ . The dataset of images is partitioned into train and test (or validation) subsets. An element of a dataset is a pair $( I , c ( I ) )$ for an image $I$ and a ground truth label $c ( I )$ of this image. We assume that the class labels are drawn from the set $\\{ 1 , \\ldots , C \\}$ , i.e., we have a set of $C \\in \\mathbb { N }$ possible labels. We assume that images have $\\ell$ channels (in experiments we use the RGB format) and are of width $w \\in \\mathbb { N }$ and height $h \\in \\mathbb { N }$ . We say that $( b , x , y )$ is a coordinate of an image for channel $b$ and location $( x , y )$ , and $( \\star , x , y )$ is a pixel of an image where $( \\star , x , y )$ represents all the $\\ell$ coordinates corresponding to different channels at location $( x , { \\bar { y } } )$ . $I ( b , x , y ) \\in \\mathbb { R }$ is the value of $I$ at the $( b , x , y )$ coordinate, and similarly $I ( \\star , x , y ) \\in \\mathbb { R } ^ { \\ell }$ represents the vector of values of $I$ at the $( \\star , x , y )$ pixel. ", + "bbox": [ + 173, + 547, + 825, + 674 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is a common practice to normalize the image before passing it to the network. A normalized image has the same dimension as the original image, but differs in the coordinate values. In this work we treat the normalization procedure as an external procedure and assume that all images are normalized. As we always work with normalized images, in the following, a reference to image means a normalized input image. We denote by LB and UB two constants such that all the coordinates of all the normalized images fall in the range [LB, UB]. Generally, $\\mathrm { L B } < 0$ and $\\mathrm { U B > 0 }$ . We denote by $\\mathbb { I } \\subset \\mathbb { R } ^ { \\ell \\times w \\times h }$ the space of all (valid) images which satisfy the following property: for every $I \\in \\mathbb { I }$ , for all coordinates $( \\bar { b , \\boldsymbol { x } } , \\boldsymbol { y } ) \\in [ \\ell ] \\times [ w ] \\times [ \\bar { h ] } , I ( b , \\boldsymbol { x } , \\boldsymbol { y } ) \\in [ \\mathrm { \\bar { L } B , U B } ] .$ . ", + "bbox": [ + 173, + 680, + 825, + 794 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We denote by NN a trained neural network (trained on some set of training images). NN takes an image $I$ as an input and outputs a vector $\\mathbf { N N } ( I ) = ( o _ { 1 } , \\dots , o _ { C } )$ , where $o _ { j }$ denotes the probability as determined by NN that image $I$ belongs to class $j$ . We denote $\\pi ( \\mathrm { N N } ( I ) , k )$ a function that returns a set of indices that are the top- $k$ predictions (ranked by decreasing probability scores with ties broken arbitrarily) of the network NN. For example, if $\\mathbf { N N } ( I ) = ( 0 . 2 5 , 0 . 1 , 0 . 2 , 0 . 4 5 )$ , then $\\pi ( \\operatorname { N N } ( I ) , 1 ) = \\{ 4 \\}$ (corresponding to the location of the entry 0.45). Similarly, $\\pi ( { \\mathrm { N N } } ( I ) , 2 ) =$ $\\{ 4 , 1 \\}$ , $\\pi ( \\mathbf { N N } ( I ) , 3 ) = \\{ 4 , 1 , 3 \\}$ , etc. ", + "bbox": [ + 173, + 799, + 825, + 854 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Adversarial Goal. Before we define the goal of black-box adversarial attacks, we define misclassification for a NN. In this paper, we use a stronger notion of misclassification, which we refer to as $k$ -misclassification for $k \\in \\mathbb N$ . ", + "bbox": [ + 174, + 157, + 825, + 199 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 1 ( $k$ -misclassification) A neural network NN $k$ -misclassifies an image $I$ with true label $c ( I )$ iff the output $\\mathrm { N N } ( I )$ of the network satisfies $c ( I ) \\not \\in \\pi ( { \\mathrm { N N } } ( I ) , k )$ . ", + "bbox": [ + 173, + 212, + 823, + 241 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In other words, $k$ -misclassification means that the network ranks the true label below at least $k$ other labels. Traditionally the literature on adversarial attacks have only considered the case where $k = 1$ . Note that an adversary that achieves a $k$ -misclassification for $k > 1$ is a stronger adversary than one achieving an 1-misclassification ( $k$ -misclassification implies $k ^ { \\prime }$ -misclassification for all $1 \\overset { \\cdot } { \\leq } k ^ { \\prime } \\leq k ,$ ). If $k = 1$ , we simply say that NN misclassifies the image. ", + "bbox": [ + 174, + 252, + 825, + 323 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In our setting, an adversary ADV is a function that takes in image $I$ as input and whose output is another image $\\mathrm { A D V } ( I )$ (with same number of coordinates as $I$ ). We define an adversarial image as one that fools a network into $k$ -misclassification. ", + "bbox": [ + 176, + 330, + 823, + 371 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 2 (Adversarial Image) Given access to an image $I$ , we say that an $\\mathrm { A D V } ( I )$ is a $k$ - adversarial image (resp. adversarial image) if $c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , k )$ and $c ( I ) \\notin \\pi ( \\mathbf { N N } ( \\mathbf { A D V } ( I ) ) , k )$ (resp. $c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , 1 )$ and $c ( I ) \\notin \\pi ( \\mathbf { N N } ( \\mathbf { A D V } ( I ) ) , 1 ) .$ ). ", + "bbox": [ + 174, + 383, + 825, + 428 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The goal of adversarial attacks is to design this function ADV that succeeds in fooling the network for a large set of images. Ideally, we would like to achieve this misclassification3 by adding only some small perturbation (under some metric) to the image. The presence of adversarial images shows that there exist small perturbations in input that produce large perturbations at the output of the last layer. ", + "bbox": [ + 174, + 438, + 825, + 494 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Adversarial threat models can be divided into two broad classes.4 The first class of models roughly assumes that the adversary has a total knowledge of the network architecture and the parameters resulting from training (or access to the labeled training set). The second class of threat models, as considered in this paper, make no assumptions about the adversary having access to the network architecture, network parameters, or the training set. In this case, the adversary has only a black-box (oracle) access to the network, in that it can query the network NN on an image $I$ and observe the output $\\mathrm { N N } ( I )$ . In our experimental section (Section 6), we also consider a slight weakening of this black-box model where the adversary has only the ability to use a proxy of the network NN as an oracle. ", + "bbox": [ + 173, + 501, + 825, + 626 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A black-box threat model in the context of deep neural networks was first considered by Papernot et al. (2016b). There is however one subtle difference between the threat model considered here and that considered by Papernot et al. (2016b) in what the adversary can access as an output. While the adversary presented in (Papernot et al., 2016b) requires access to the class label assigned by the network which is the same level of access needed by our simple randomized adversary (presented in Section 4), our local-search adversary (presented in Section 5) requires access to $o _ { c ( I ) }$ (the probability assigned to the true label $c ( I )$ by the network on input $I$ ) and the $\\pi$ vector (for checking whether $k$ -misclassification has been achieved). Our adversarial approaches does not require access to the complete probability vector $( \\mathrm { N N } ( I ) )$ . Also as pointed out earlier, compared to (Papernot et al., 2016b), our approach is more direct (needs no transferability assumption), requires no retraining, and can be adapted to achieve $k$ -misclassification rather than just 1-misclassification. ", + "bbox": [ + 173, + 632, + 825, + 787 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 BLACK-BOX GENERATION: A FIRST ATTEMPT ", + "text_level": 1, + "bbox": [ + 173, + 801, + 591, + 818 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we present a simple black-box adversary that operates by perturbing a single pixel (or a small set of pixels) selected at random. In the next section, we build upon this idea to construct an adversary that achieves better success by making adaptive choices. ", + "bbox": [ + 176, + 830, + 823, + 872 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Power of One Pixel. Starting point of our investigation is to understand the influence of a single pixel in an adversarial setting. Most existing adversarial attacks operate by applying the same perturbation on each individual pixel while minimizing the overall perturbation (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016), while recent research have yielded attacks that perturb only a fraction of the pixels (Papernot et al., 2016c;b; Grosse et al., 2016). However, in all these cases, no explicit restriction is placed on the number of pixels that can be perturbed. Therefore, it is natural to ask: whether it is possible to force the network to misclassify an image by modifying a single pixel? If so, how strong should this perturbation be? We run several experiments to shed light on these questions. For simplicity, in this section, we focus the case of 1-misclassification, even though all discussions easily extend to the case of $k$ -misclassification for $k > 1$ . We begin with a useful definition. ", + "bbox": [ + 173, + 103, + 825, + 256 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition 3 (Critical Pixel) 5 Given a trained neural network NN and an image $I$ , a pixel $( \\star , x , y )$ in $I$ is a critical pixel if a perturbation of this pixel generates an image that is misclassified by the network NN. In other words, $( \\star , x , y )$ is a critical pixel in $I$ if there exists another neighboring image $I _ { p }$ which differs from $I$ only in values at the pixel location $( x , y )$ such that $c ( I ) \\notin \\pi ( \\mathbf { N N } ( I _ { p } ) , 1 )$ . ", + "bbox": [ + 174, + 265, + 825, + 324 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The image $I _ { p }$ can be generated in multiple ways, here we consider a class of sign-preserving perturbation functions defined as follows. Let $\\mathtt { P E R T } ( I , p , x , y )$ be a function that takes as input an image $I$ , a perturbation parameter $p \\in \\mathbb { R }$ , and a location $( x , y )$ , and outputs an image $I _ { p } ^ { ( x , y ) } \\in$ $\\mathbb { R } ^ { \\ell \\times w \\times h }$ , defined as: ", + "bbox": [ + 174, + 334, + 825, + 395 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/e270b4e8d34f4bc1b3e96bff2bc2a041fad51c462455c9ffddcaf5c5f1b751f2.jpg", + "text": "$$\nI _ { p } ^ { ( x , y ) } ( b , u , v ) \\stackrel { \\mathrm { \\tiny ~ d e f n } } { = } \\left\\{ { \\begin{array} { l l } { { \\scriptstyle I ( b , u , v ) } } & { { \\mathrm { i f ~ } } x \\neq u { \\mathrm { ~ o r ~ } } y \\neq v } \\\\ { { p \\times \\mathrm { s i g n } } ( I ( b , u , v ) ) } & { { \\mathrm { o t h e r w i s e } } } \\end{array} } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 297, + 398, + 699, + 434 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In other words, the image $I _ { p } ^ { ( x , y ) } = \\mathtt { P E R T } ( I , p , x , y )$ has same values as image $I$ at all pixels except the pixel $( \\star , x , y )$ . The value of the image $I _ { p } ^ { ( x , y ) }$ at pixel $( \\star , x , y )$ is just $p \\times \\mathrm { s i g n } ( I ( \\star , x , y ) )$ . Note that unlike some of the previous adversarial attacks (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c; Grosse et al., 2016), our threat model does not assume access to the true network gradient factors, hence the construction of the perturbed image has to be oblivious to the network parameters. ", + "bbox": [ + 173, + 440, + 826, + 530 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the following, we say a pixel $( \\star , x , y )$ in image $I$ is critical iff $c ( I ) \\notin \\pi ( \\mathbf { N N } ( I _ { p } ^ { ( x , y ) } ) , 1 )$ ", + "bbox": [ + 171, + 537, + 763, + 556 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Critical Pixels are Common. Our first experiment is to investigate existence of critical pixels in the considered dataset of images. To do so, we perform a simple procedure that picks a location $( x , y )$ in the image $I$ and applies the PERT function to this pixel to obtain a perturbed image $I _ { p } ^ { ( x , y ) }$ . Then the perturbed image is run through the trained network, and we check whether it was misclassified or not. If the perturbed image $I _ { p } ^ { ( x , y ) }$ is misclassified then we have identified a critical pixel. While we can exhaustively repeat this procedure for all pixels in an image, for computational efficiency we instead perform it only on a fraction of randomly chosen pixels, and our results somewhat surprisingly suggest that in many cases this is sufficient to generate an adversarial image. Algorithm RANDADV presents the pseudo-code for this experiment. Algorithm RANDADV, selects $U$ random pixels (with replacement) and performs checks whether the pixel is critical or not. The algorithm output is an unbiased estimate for the fraction of critical pixels in the input image $I$ . Note that the algorithm can fail in generating an adversarial image (i.e., in finding any critical pixel for an image). The following definition will be useful for our ensuing discussion. ", + "bbox": [ + 173, + 566, + 825, + 753 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Definition 4 (Good Image) We say that an image $I$ with true label $c ( I )$ is good for a network NN iff $c ( I ) \\in \\pi ( N N ( I ) , 1 )$ (i.e., NN predicts $c ( I )$ as the most likely label for $I$ ). ", + "bbox": [ + 174, + 767, + 821, + 795 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our first observation is that sometimes even small perturbation to a pixel can be sufficient to obtain an adversarial image. Table 2 shows two images and their adversarial counterparts, with $p = 1$ . Often, original and adversarial images are indistinguishable to the human eye, but sometimes the critical pixel is visible (Table 2). ", + "bbox": [ + 174, + 806, + 825, + 863 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/35f61844a2fc87ca9ef855e0bf44a533ed34704baca6fd233c93e79e87e3e179.jpg", + "image_caption": [ + "Table 2: The row contains original images followed by misclassified images where only one pixel (pointed using a black arrow) was perturbed with perturbation parameter $p = 1$ . After perturbation, in the first case (images (a) and (b)) an automobile gets misclassified as a truck, and in the second case (images (c) and (d)) a cat gets misclassified as a dog. " + ], + "image_footnote": [], + "bbox": [ + 187, + 101, + 812, + 227 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We also tried to understand the effect of larger perturbation parameter values. We set $U$ to half the number of pixels in each image. After usual training of the neural network using the training set (see Section 6 for more details about training), we ran Algorithm RANDADV on 1000 randomly drawn images from the test set of the corresponding dataset. In our experiments, we varied perturbation parameter in the range $\\{ 1 , 5 , 1 0 , 1 0 0 \\}$ . Before we consider our results, we note some of the perturbation values that we use to construct the adversarial image might construct images that are not in the original image space.6 However, these results are still somewhat surprising, because even though we allow large (even out-of-range) perturbation, it is applied to exactly one pixel in the image, and it appears that it suffices to even pick the pixel at random. ", + "bbox": [ + 173, + 323, + 825, + 448 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figures 1 and 2 show results for 4 datasets (more details about the datasets and the networks are presented in Section 6). On the $\\mathbf { X }$ -axis we show the perturbation parameter $p$ . In Figure 1, the y-axis represents the output of Algorithm RANDADV averaged over good images for the network.7 The first observation that we can make is that the critical pixels are common, and in fact, as $p$ grows the fraction of critical pixels increases. For example, in CIFAR10, with $p = 1 0 0$ , almost $80 \\%$ (on average) of the pixels randomly selected are critical. In Figure 2, the y-axis represents the fraction of successful adversarial images generated by Algorithm RANDADV, i.e., fraction of inputs where Algorithm RANDADV is successful in finding at least one critical pixel. Again we notice that as $p$ grows it gets easier for Algorithm RANDADV to construct an adversarial image. ", + "bbox": [ + 173, + 455, + 825, + 580 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Algorithm 1 RANDADV (NN) ", + "text_level": 1, + "bbox": [ + 174, + 598, + 380, + 613 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1: Input: Image $I$ with true label $c ( I ) \\in \\{ 1 , \\ldots , C \\}$ , perturbation factor $p \\in \\mathbb R$ , and a budget $U \\in \\mathbb { N }$ on the \nnumber of trials \n2: Output: A randomized estimate on the fraction of critical pixels in the input image $I$ \n3: $i = 1$ , critical $= 0$ \n4: while $i \\leq U$ do \n5: randomly pick a pixel $( \\star , x , y )$ \n6: compute a perturbed image I(x,y)p = $I _ { p } ^ { ( x , y ) } = \\mathtt { P E R T } ( I , p , x , y )$ \n7: if $c ( I ) \\notin \\pi ( \\mathbf { N N } ( I _ { p } ^ { ( x , y ) } ) , 1 )$ then \n8: critical critical $+ 1$ \n9: end if \n10: $i \\gets i + 1$ \n11: end while \n12: {The algorithm succeeds in generating an adversarial image if it finds at least one critical pixel} \n13: return $\\frac { \\mathrm { \\bar { c r i t i c a l } } } { U }$ ", + "bbox": [ + 176, + 617, + 826, + 809 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Another observation is that for the MNIST and STL10 datasets, Algorithm RANDADV succeeds in finding fewer critical pixels as compared to SVHN and CIFAR10 datasets. We give the following explanation for this observation. The majority of pixels in an MNIST image belong to the background, hence, these pixels are less likely to be critical. On the other hand, STL10 contains high resolution images, $9 6 \\times 9 6$ , where perhaps a single pixel has less of an impact on the output prediction. The latter observation motivated us to generalize the notion of a critical pixel to a critical set. ", + "bbox": [ + 176, + 828, + 825, + 871 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/7e8864ae3078d8172ab7a9829d78956096dcab004436e610d6380e7219120e9d.jpg", + "image_caption": [ + "Figure 1: Output of Algorithm RANDADV (averaged over good images). The results are for two networks: a) Network-in-Network and b) VGG. The perturbation parameter $p$ is varied from $\\{ 1 , 5 , 1 0 , 1 0 0 \\}$ . " + ], + "image_footnote": [], + "bbox": [ + 171, + 102, + 810, + 207 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/e6719714440c8cfd2fc901d4cd2fef26915fc30b8d7a9db8a45e0c6471206f23.jpg", + "image_caption": [ + "Figure 2: Fraction of images where Algorithm RANDADV succeeds in finding at least one critical pixel. Again we only start with only good images. " + ], + "image_footnote": [], + "bbox": [ + 171, + 257, + 810, + 363 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 426, + 825, + 468 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Definition 5 (Critical Set) Given a trained neural network NN and an image I, a critical set of I is a set of pixels $\\textstyle \\bigcup _ { ( x , y ) } \\{ ( \\star , x , y ) \\}$ in $I$ such that a perturbation of these pixels generates an image that is misclassified by the network NN. ", + "bbox": [ + 174, + 479, + 825, + 523 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The general goal will be to find critical sets of small size in an image. With this notion of critical set, we considered constructing adversarial images on the high-resolution ImageNet1000 dataset. We can modify the definition of $I _ { p } ^ { ( x , y ) }$ (from (1)) where instead of a single pixel we perturb all the pixels in a set. Similarly, we can devise a simple extension to Algorithm RANDADV to operate with a set of pixels and to output an unbiased estimate for the fraction of critical sets of some fixed size (50 in our case) in the input image.8 Note that a set size of 50 pixels is still a tiny fraction of all the pixels in a standard (center) crop of size $2 2 4 \\times 2 2 4$ , namely just $0 . 0 9 \\%$ . We use a larger perturbation parameter $p$ than before, and set $( U )$ the budget on the number of trials on an image as 5000. Figure 3 shows our results. Overall, we note that we can draw similar conclusions as before, i.e., increasing the perturbation parameter creates more critical sets making them easier to find and relatively small perturbations are sufficient to construct adversarial images. ", + "bbox": [ + 173, + 536, + 825, + 693 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 BLACK-BOX GENERATION: A GREEDY APPROACH", + "text_level": 1, + "bbox": [ + 174, + 707, + 625, + 723 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The results from Section 4 show that most images have critical pixels such that modifying these pixels significantly leads to a failure of NN to classify the image correctly. However, one shortcoming of Algorithm RANDADV was that to build adversarial images, we sometimes had to apply a large perturbation to a single pixel (or a small set of pixels). Hence, there might exist a pixel (or a set of pixels) in the adversarial image whose coordinate value could lie outside the valid range [LB, UB]. To overcome this issue, we need to redesign the search procedure to generate adversarial images that still belong to the original image space I (defined in Section 3). Here a brute-force approach is generally not feasible because of computational reasons, especially in high-resolution images. Hence, we need to develop an efficient heuristic procedure to find the right small set of pixels to be perturbed. Our solution presented in this section is based on performing a greedy local search over the image space. ", + "bbox": [ + 173, + 734, + 825, + 887 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/83372eea17b2e701cefc45d156d3d9f66be0989cf1e8cc78c4eecbe71d0ecc36.jpg", + "image_caption": [ + "Figure 3: Experiments in Figures 1 and 2 for the high-resolution ImageNet1000 dataset. The results are again for good images from a set of 1000 randomly selected images. We use a slightly modified version of Algorithm RANDADV that perturbs a set of 50 pixels. " + ], + "image_footnote": [], + "bbox": [ + 217, + 102, + 777, + 270 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We consider the general $k$ -misclassification problem (Definition 1) where an adversarial attack ensures that the true label does not appear in the top- $k$ predictions of the network. We utilize a local-search procedure, which is an incomplete search procedure that is widely used for solving combinatorial problems appearing in diverse domains such as graph clustering, scheduling, logistics, and verification (Lenstra, 1997). For a general optimization problem it works as follows. Consider an objective function $f ( \\mathbf { z } ) : \\mathbb { R } ^ { n } \\to \\mathbb { R }$ where the goal is to minimize $f ( \\mathbf { z } )$ . The local-search procedure works in rounds, where each round consists of two steps. Let $\\mathbf { z } _ { i - 1 }$ be the solution iterate after round $i - 1$ . Consider round $i$ . The first step is to select a small subset of points $Z = \\{ \\hat { \\mathbf { z } } _ { 1 } , \\hdots , \\hat { \\mathbf { z } } _ { n } \\}$ , a so called local neighborhood, and evaluate $f ( \\hat { \\mathbf { z } } _ { j } )$ for every $\\hat { \\mathbf { z } } _ { j } \\in Z$ . Usually, the set $Z$ consist of points that are close to current $\\mathbf { z } _ { i - 1 }$ for some measure of distance which is domain specific. The second step selects a new solution $\\mathbf { z } _ { i }$ taking into account the previous solution $\\mathbf { z } _ { i - 1 }$ and the points in $Z$ . Hence, $\\mathbf { z } _ { i } = g ( f ( \\mathbf { z } _ { i - 1 } ) , f ( \\hat { \\mathbf { z } } _ { 1 } ) , \\hdots , f ( \\hat { \\mathbf { z } } _ { n } ) )$ , where $g$ is some pre-defined transformation function. ", + "bbox": [ + 173, + 348, + 826, + 517 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We adapt this general procedure to search critical sets efficiently as explained below. Our optimization problem will try to minimize the probability that the network determines an perturbed image has the class label of the original image, and by using a local-search procedure we generate perturbed images which differ from the original image in only few pixels. Intuitively, in each round, our local-search procedure computes an implicit approximation to the gradient of the current image by understanding the influence of a few pixels on the output, which is then used to update the current image. ", + "bbox": [ + 173, + 522, + 825, + 607 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) First, we need to define the cost function $f$ . Let $I$ be the image (with true label $c ( I ) ,$ ) whose adversarial image we want to generate for a target neural network NN. For some input image $\\hat { I }$ , we use the objective function $f _ { c ( I ) } ( \\hat { I } )$ which equals the probability assigned by the network NN that the input image $\\hat { I }$ belongs to class $c ( I )$ . More formally, ", + "bbox": [ + 161, + 619, + 825, + 684 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/d3beeb6ee5e9e695c0fde6d8d5c55aac1602ffb5eb31d7d45e0c13de7a87b8dc.jpg", + "text": "$$\nf _ { c ( I ) } ( \\hat { I } ) = o _ { c ( I ) } \\mathrm { w h e r e } \\mathrm { N N } ( \\hat { I } ) = ( o _ { 1 } , \\dots , o _ { C } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 690, + 660, + 712 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "with $o _ { j }$ denoting the probability as determined by NN that image $\\hat { I }$ belongs to class $j$ . Our local-search procedure aims to minimize this function. ", + "bbox": [ + 183, + 719, + 821, + 750 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(b) Second, we consider how to form a neighborhood set of images. As mentioned above, the localsearch procedure operates in rounds. Let $\\hat { I } _ { i - 1 }$ be the image after round $i - 1$ . Our neighborhood will consist of images that are different in one pixel from the image $\\hat { I } _ { i - 1 }$ . In other words, if we measure the distance between $\\hat { I } _ { i - 1 }$ and any image in the neighborhood as the number of perturbed pixels, then this distance is the same (equal to one) for all of them. Therefore, we can define the neighborhood in terms of a set of pixel locations. Let $( P _ { X } , P _ { Y } ) _ { i }$ be a set of pixel locations. For the first round $( P _ { X } , P _ { Y } ) _ { 0 }$ is randomly generated. At each subsequent round, it is formed based on a set of pixel locations which were perturbed in the previous round. Let $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 }$ denote the pixel locations that were perturbed in round $i - 1$ (formally defined below). Then ", + "bbox": [ + 161, + 752, + 825, + 885 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/8556cc2aaaa13de8089d58b7a0f06d5b8ca23543f5df4a7db287dce4ea7852d5.jpg", + "text": "$$\n( P _ { X } , P _ { Y } ) _ { i } = \\bigcup _ { \\{ ( a , b ) \\in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 } \\} } \\bigcup _ { \\{ x \\in [ a - d , a + d ] , y \\in [ b - d , b + d ] \\} } ( x , y ) ,\n$$", + "text_format": "latex", + "bbox": [ + 294, + 891, + 718, + 928 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $d$ is a parameter. In other words, we consider pixels that were perturbed in the previous round, and for each such pixel we consider all pixels in a small square with the side length $2 d$ centered at that pixel. This defines the neighborhood considered in round $i$ . ", + "bbox": [ + 186, + 103, + 823, + 146 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(c) Third, we describe the transformation function $g$ of a set of pixel locations. The function $g$ takes as input an image $\\hat { I }$ , a set of pixel locations $( P _ { X } , P _ { Y } )$ , a parameter $t$ that defines how many pixels will be perturbed by $g$ , and two perturbation parameters $p$ and $r$ . In round $i$ of the local-search procedure, the function $g ( \\hat { I } _ { i - 1 } , ( P _ { X } , P _ { Y } ) _ { i - 1 } , t , p , r )$ outputs a new image, such that exactly $t$ pixels of $\\hat { I } _ { i - 1 }$ are perturbed, and an auxiliary set of pixel locations $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i }$ to record which $t$ pixels where perturbed at this round, so we have $( \\hat { I } _ { i } , ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i } ) = g ( \\hat { I } _ { i - 1 } , ( P _ { X } , P _ { Y } ) _ { i - 1 } , t , p , r )$ . Next we describe transformations that $g$ performs in round $i$ . As the first step, $g$ constructs a set of perturbed images based on $( P _ { X } , P _ { Y } ) _ { i - 1 }$ : ", + "bbox": [ + 163, + 148, + 826, + 271 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/698f75ce5d2122fffa2c4a5f7b717a00fb6be04709d25e2aeb0207ee6a34c361.jpg", + "text": "$$\n\\mathcal { T } = \\bigcup _ { ( x , y ) \\in ( P _ { X } , P _ { Y } ) _ { i - 1 } } \\{ \\operatorname { P E R T } ( \\hat { I } _ { i - 1 } , p , ( x , y ) ) \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 277, + 656, + 314 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "where PERT is the perturbation function defined through (1). Then it computes the score of each image in $\\mathcal { T }$ as ", + "bbox": [ + 186, + 320, + 823, + 348 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/02368759a659cbe749500f508e7ebcca53af59bc6fee1144ec7cde5c3491314f.jpg", + "text": "$$\n\\forall \\tilde { I } \\in \\mathcal { T } : \\mathrm { s c o r e } ( \\tilde { I } ) = f _ { c ( I ) } ( \\tilde { I } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 348, + 609, + 369 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "and it sorts (in decreasing order) images in $\\mathcal { T }$ based on the above score function to construct sorted $( \\mathcal { T } )$ . Pixels whose perturbation lead to a larger decrease of $f$ are more likely useful in constructing an adversarial candidate. From sorted $( \\mathcal { T } )$ , it records a set of pixel locations $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i }$ based on the first $t$ elements of sorted $( \\mathcal { T } )$ , where the parameter $t$ regulates the number of pixels perturbed in each round. Formally, ", + "bbox": [ + 187, + 371, + 825, + 441 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/9766d5de09e7bcf0e339ba9bf6b5ea73d0d4d16cc22d665fa739ddf1ff2f2d79.jpg", + "text": "$$\n( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i } = \\{ ( x , y ) : { \\mathrm { P E R T } } ( { \\hat { I } } _ { i - 1 } , p , ( x , y ) ) \\in { \\mathrm { s o r t e d } } ( { \\mathcal { T } } ) [ 1 : t ] \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 289, + 449, + 722, + 468 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "where sorted $\\left( \\mathcal { T } \\right) \\left[ 1 : t \\right]$ represents the first $t$ sorted images in sorted $( \\mathcal { T } )$ . Finally, $\\hat { I } _ { i }$ is constructed from $\\hat { I } _ { i - 1 }$ by perturbing each pixel in location $( x , y ) \\in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i }$ with a perturbation value $r$ . The perturbation is performed in a cyclic way (as explained in Algorithm CYCLIC) so that we make sure that all coordinate values in $\\hat { I } _ { i }$ are within the valid bounds of LB and UB. Note that at the end of every round $i$ , $\\hat { I } _ { i }$ is a valid image from the image space $\\mathbb { I }$ . ", + "bbox": [ + 187, + 477, + 825, + 553 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We want to point out that the function $g$ uses two perturbation parameters, $p$ and $r$ . The value of $r$ is kept small in the range [0, 2]. On the other hand, we do not put any explicit restrictions on the value of $p$ . The best choice of $p$ will be one that facilitates the identification of the “best” pixels to perturb in each round. In our experiments, we adjust the value of $p$ automatically during the search. We defer this discussion to the experimental section. ", + "bbox": [ + 187, + 554, + 825, + 623 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Algorithm LOCSEARCHADV shows the complete pseudocode of our local-search procedure. At the high level, the algorithm takes an image as input, and in each round, finds some pixel locations to perturb using the above defined objective function and then applies the above defined transformation function to these selected pixels to construct a new (perturbed) image. It terminates if it succeeds to push the true label below the $k$ th place in the confidence score vector at any round. Otherwise, it proceeds to the next round (for a maximum of $R$ rounds). Note that the number of pixels in an image perturbed by Algorithm LOCSEARCHADV is at most $t \\times R$ and in practice (see Tables 4, 5,and 6 in Section 6) it is much less. In round $i$ , we query the network at most the number of times as the number of pixels in $( P _ { X } , P _ { Y } ) _ { i }$ which after the first round is at most $2 d \\times 2 d \\times t$ (again in practice this is much less because of the overlaps in the neighborhood squares). ", + "bbox": [ + 173, + 636, + 825, + 776 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In Section 6, we demonstrate the efficacy of Algorithm LOCSEARCHADV in constructing adversarial images. We first highlight an interesting connection between the pixels perturbed and their influences measured by a notion of called saliency map. ", + "bbox": [ + 174, + 782, + 825, + 825 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A Relation to Saliency Maps. Simonyan et al. (2014) introduced the notion of saliency map as a way to rank pixels of the original images w.r.t. their influence on the output of the network. The intuition is that influential pixels in the saliency map are more likely to be important pixels that represent objects and, for example, can be used for weakly supervised object localization. Formally, let $\\mathrm { N } \\mathrm { \\bar { N } } _ { c ( I ) } ( I )$ denote the probability assigned to true class $c ( I )$ by the network NN on input $I \\in \\mathbb { R } ^ { \\ell \\times \\mathbf { \\bar { w } } \\times h }$ . Let $\\dot { W _ { c ( I ) } } \\in R ^ { \\ell \\times w \\times h }$ denote the derivative of $\\mathrm { N N } _ { c ( I ) }$ with respect to the input evaluated at image $I$ . ", + "bbox": [ + 173, + 837, + 826, + 925 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/2e2833620c0ddc909411f8b5855a75c4fc2c0c61d9d118c075113a29089f6c07.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm 2 CYCLIC (r,b,x,y)
Assumptions:Perturbation parameter r ∈ [O,2] and LB≤O≤UB
Output:Perturbed image value at the coordinate (b,x,y) which lies in the range [LB,UB]
if rI(b,x,y)<LB then return rI(b,x,y)+(UB-LB)
else if rI(b,x,y)>UB then
return rI(b,x,y)-(UB-LB)
else
return rI(b,x,y)
", + "bbox": [ + 169, + 172, + 823, + 318 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Algorithm 3 LOCSEARCHADV (NN) ", + "text_level": 1, + "bbox": [ + 176, + 473, + 426, + 488 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Input: Image $I$ with true label $c ( I ) \\in \\{ 1 , \\ldots , C \\}$ , two perturbation parameters $p \\in \\mathbb R$ and $r \\in [ 0 , 2 ]$ , and four other parameters: the half side length of the neighborhood square $d \\in \\mathbb { N }$ , the number of pixels perturbed at each round $t \\in \\mathbb { N }$ , the threshold $k \\in \\mathbb N$ for $k$ -misclassification, and an upper bound on the number of rounds $R \\in \\mathbb N$ . ", + "bbox": [ + 186, + 493, + 825, + 544 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Output: Success/Failure depending on whether the algorithm finds an adversarial image or not \n$\\hat { I } _ { 0 } = I , i = 1$ \nPick $1 0 \\%$ of pixel locations from $I$ at random to form $( P _ { X } , P _ { Y } ) _ { 0 }$ \nwhile $i \\leq R$ do {Computing the function $g$ using the neighborhood} $\\begin{array} { r } { \\mathcal { T } \\bigcup _ { ( x , y ) \\in ( P _ { X } , P _ { Y } ) _ { i - 1 } } \\{ \\mathrm { P E R T } \\big ( \\hat { I } _ { i - 1 } , p , x , y ) \\} } \\end{array}$ Compute $\\mathrm { s c o r e } ( \\tilde { I } ) = f _ { c ( I ) } ( \\tilde { I } )$ for each $\\tilde { I } \\in \\mathcal { I }$ (where $f _ { c ( I ) } ( \\tilde { I } ) = o _ { c ( I ) }$ with $\\mathbf { N N } ( \\tilde { I } ) = ( o _ { 1 } , \\dots , o _ { C } ) )$ sorted $( { \\mathcal { T } } ) \\gets$ images in $\\mathcal { T }$ sorted by descending order of score $( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i } \\gets \\{ ( x , y ) : \\operatorname { P E R T } ( \\hat { I } _ { i - 1 } , p , x , y ) \\in \\operatorname { s o r t e d } ( \\mathcal { D } ) [ 1 : t ] \\}$ (with ties broken arbitrarily) {Generation of the perturbed image $\\hat { I } _ { i }$ } for $( x , y ) \\in ( P _ { X } ^ { * } , \\bar { P } _ { Y } ^ { * } ) _ { i }$ and each channel $^ { b }$ do $\\hat { I } _ { i } ( b , x , y ) \\mathrm { C Y C l }$ LIC $( r , b , x , y )$ end for {Check whether the perturbed image $\\hat { I } _ { i }$ is an adversarial image} if $c ( I ) \\notin \\pi ( \\mathbf { N N } ( \\hat { I } _ { i } ) , \\bar { k } )$ then return Success end if {Update a neighborhood of pixel locations for the next round} $\\begin{array} { r } { ( \\hat { P _ { X } } , P _ { Y } ) _ { i } \\longleftarrow \\bigcup _ { \\{ ( a , b ) \\in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 } \\} } \\bigcup _ { \\{ x \\in [ a - d , a + d ] , y \\in [ b - d , b + d ] \\} } ( x , y ) } \\end{array}$ i ← i + 1 \nend while \nreturn Failure ", + "bbox": [ + 179, + 545, + 802, + 848 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The saliency map of $I$ is the matrix $M \\in \\mathbb { R } ^ { w \\times h }$ such that $\\begin{array} { r } { M _ { i , j } = \\operatorname* { m a x } _ { b \\in [ \\ell ] } \\ | W _ { c ( I ) } ( b , x , y ) | } \\end{array}$ , where $W _ { c ( I ) } ( b , x , y )$ is the element of $W _ { c ( I ) }$ corresponding to channel $b$ and location $( x , y )$ . Pixels with higher scores are considered more influential. In subsequent works, this notion has been extended to adversarial saliency maps that can be useful in generating adversarial perturbations (Papernot et al., 2016c). ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Computing the exact saliency scores for an image requires complete access to the network NN, which we do not assume. However, a natural hypothesis is that the pixels selected by Algorithm LOCSEARCHADV for perturbation are related to pixels with large saliency scores. We use the ImageNet1000 dataset to test this hypothesis. In Figure 3, we present some qualitative results. As can be seen from the pictures, the pixels perturbed by Algorithm LOCSEARCHADV appear correlated with pixels with high saliency scores. Quantitatively, we observed that the pixels that occupy top- $10 \\%$ of the saliency map, on average contain more than $23 \\%$ of the pixels chosen by Algorithm LOCSEARCHADV for perturbation (and this overlap only grows when we consider a bigger chunk of pixels picked by their saliency scores). Note that this is correlation is not though a random occurrence. For an image $I$ , let $S _ { I }$ denote the set of pixels in $I$ that rank among the top- $10 \\%$ in the saliency map. If we pick a random set of around 200 pixels (this is on average number of pixels perturbed per image by Algorithm LOCSEARCHADV perturbs, see Table 5), we expect only about $10 \\%$ t o them to intersect with $S _ { I }$ and standard tail bounds show that the probability that at least $23 \\%$ of the pixels of this random set intersects with $S _ { I }$ is extremely small.9 Therefore, it appears that Algorithm LOCSEARCHADV rediscovers part of the high salient score pixels but without explicitly computing the gradients. ", + "bbox": [ + 174, + 183, + 826, + 402 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "6 EXPERIMENTAL EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 419, + 450, + 434 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We start by describing our experimental setup. We used Caffe and Torch machine learning frameworks to train the networks. All algorithms to generate adversarial images were implemented in Lua within Torch 7. All experiments were performed on a cluster of GPUs using a single GPU for each run. ", + "bbox": [ + 174, + 446, + 825, + 488 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Datasets. We use 5 popular datasets: MNIST (handwritten digits recognition dataset), CIFAR10 (objects recognition dataset), SVHN (digits recognition dataset), STL10 (objects recognition dataset), and ImageNet1000 (objects recognition dataset). ", + "bbox": [ + 176, + 501, + 821, + 542 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Models. We trained Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman, 2014) for MNIST, CIFAR, SVHN, STL10, with minor adjustments for the corresponding image sizes. Network-in-Network is a building block of the commonly used GoogLeNet architecture that has demonstrated very good performance on medium size datasets, e.g. CIFAR10 (Zagoruyko, 2015). VGG is another powerful network that proved to be useful in many applications beyond image classification, like object localization (Ren et al., 2015). We trained each model in two variants: with and without batch normalization (Ioffe & Szegedy, 2015). Batch normalization was placed before a ReLU layer in all networks. For the ImageNet1000 dataset, we used pre-trained VGG models from (Chatfield et al., 2014b) (we did not train them from scratch due to limited resources). All Caffe VGG models were converted to Torch models using the loadcaffe package (Zagoruyko, 2016a). These models use different normalization procedures which we reproduced for each model based on provided descriptions. Tables 4 and 5 (the second column ERRTOP-1) show the top-1 (base) error for all datasets and models that we considered. The results are comparable with the known state-of-the-art results on these datasets (Benenson, 2016). ", + "bbox": [ + 174, + 556, + 825, + 750 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Related Techniques. There are quite a few approaches for generating adversarial images (as discussed in Section 2). Most of these approaches require access to the network architecture and its parameter values (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c).10 The general idea behind these attacks is based on the evaluating the network’s sensitivity to the input components in order to determine a perturbation that achieves the adversarial misclassification goal. Among these approaches, the attack approach (known as the “fast-gradient sign method”) suggested by Goodfellow et al. (2015) stands out for being able to efficiently generate adversarial images. Here we compare the performance of our local-search based attack against this fast-gradient sign method.11 ", + "bbox": [ + 174, + 762, + 825, + 847 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/6a4d9a530eb9e0cd10fe492164133a18e40afd6afa68fbfcd68a866b16882a02.jpg", + "image_caption": [ + "Table 3: Results on ImageNet1000 using VGG CNN-S (Caffe) network (Chatfield et al., 2014a). Columns from left to right: the original image, top 150 pixels chosen according to their saliency scores (in white), the absolute difference between the perturbed image and the true image (the pixels that are perturbed appear in white), and the perturbed image. Adversarial misclassification (rows from top to bottom): a ruffed grouse misclassified as a frilled lizard, an artichoke misclassified as a sleeping bag, a bubble misclassified as a fountain, and a hare misclassified as a cheetah. " + ], + "image_footnote": [], + "bbox": [ + 186, + 102, + 812, + 520 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 642, + 825, + 685 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For completeness, we now briefly explain the fast-gradient sign method of Goodfellow et al. (2015). Given an image $I _ { 0 }$ , a label $a \\in \\{ 1 , \\ldots , C \\}$ , and a network NN, the fast-gradient sign method perturbs $I _ { 0 }$ using the following update rule: $I _ { 0 } ^ { \\mathrm { p e r t } } = I _ { 0 } + \\epsilon \\cdot \\mathrm { s i g n } ( \\nabla _ { I = I _ { 0 } } \\mathrm { L o s s } ( \\mathbf { N N } ( I ) , a ) )$ where $\\operatorname { s i g n } ( \\nabla _ { I = I _ { 0 } } \\operatorname { L o s s } ( \\mathbf { N N } ( I ) , a ) )$ is the sign of the network’s cost function gradient (here $\\mathrm { L o s s } ( \\mathbf { N N } ( I ) , a )$ denotes the loss function of the network NN given input $I$ and class $a$ ). We vary $a$ over all possible labels in the dataset and choose the best result where this procedure is successful in generating an adversarial image. Without general guidelines for setting $\\epsilon$ , we experimented with several values of $\\epsilon$ starting from 0.07 and increasing this number. We found that the value $\\epsilon = 0 . 2 ^ { 1 2 }$ was the smallest value where the fast-gradient sign method started to yield competitive performance compared to our algorithm. Smaller values of $\\epsilon$ leads to generation of fewer adversarial images, e.g., at $\\epsilon = 0 . 1$ , the percentage of generated adversarial images is reduced by around $10 \\%$ as compared to the value at $\\epsilon = 0 . 2$ for the CIFAR10 dataset on the Network-in-Network model. Larger values of $\\epsilon$ tends to generate more adversarial images, but this comes at the cost of an increase in the perturbation. As we discuss later, our local-search based approach yields better results than the fast-gradient sign method in both the volume of adversarial images generated and the amount of perturbation applied. Another important point to remember is that unlike the fast-gradient sign method, our approach is based on a weaker and more realistic assumption on the adversarial power, making our attacks more widely applicable. ", + "bbox": [ + 173, + 691, + 825, + 859 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Implementing Algorithm LOCSEARCHADV. For each image $I$ , we ran Algorithm LOCSEARCHADV for at most 150 rounds, perturbing 5 pixels at each round, and use squares of side length 10 to form the neighborhood (i.e., $R = 1 5 0 , t = 5 , d = 5 )$ . With this setting of parameters, we perturb a maximum of $t \\times R = 7 5 0$ pixels in an image. The perturbation parameter $p$ was adaptively adjusted during the search. This helps in faster determination of the most helpful pixels in generating the adversarial image. Let $I$ be the original image. For some round $i$ of the algorithm, define $\\bar { o } _ { c ( I ) } = \\arg _ { ( x , y ) } \\{ o _ { c ( I ) } : ( x , y ) \\in ( P _ { X } ^ { * } , P _ { Y } ^ { * } ) _ { i - 1 } \\} .$ , where ${ { O } _ { c ( I ) } }$ is the probability assigned to class label $c ( I )$ in $\\mathrm { N N } \\big ( \\mathrm { P E R T } \\big ( \\hat { I } _ { i - 1 } , p , x , y \\big ) \\big )$ (here $\\bar { o } _ { c ( I ) }$ provides an approximation of the average confidence of the network NN in predicting the true label over perturbed images). At each round, we increase the value of $p$ if $\\bar { o } _ { c ( I ) }$ is close to one and decrease $p$ if $\\bar { o } _ { c ( I ) }$ is low, e.g., below 0.3. For Algorithm CYCLIC, we set $r = 3 / 2$ . To avoid perturbing the most sensitive pixels frequently, we make sure that if a pixel is perturbed in a round then we exclude it from consideration for the next 30 rounds. ", + "bbox": [ + 173, + 199, + 826, + 386 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Experimental Observations. For ease of comparison with the fast-gradient sign method (Goodfellow et al., 2015), we set $k = 1$ and focus on achieving 1-misclassification. Tables 4 and 5 show the results of our experiments on the test sets. The first column shows the dataset name. The second column (ERRTOP-1) presents the top-1 misclassification rate on the corresponding test dataset without any perturbation (base error). ERRTOP-1(ADV) is the top-1 misclassification rate where each original image in the test set was replaced with an generated perturbed image (using either our approach or the fast-gradient sign method (Goodfellow et al., 2015) which is denoted as FGSM).13 ", + "bbox": [ + 173, + 398, + 825, + 497 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In the following, we say an adversarial generation technique ADV, given an input image $I$ , succeeds in generating an adversarial image $\\mathrm { A D V } ( I )$ for a network NN iff $c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , 1 )$ and $c ( I ) \\notin$ $\\pi ( \\bar { \\mathrm { N N } } ( \\mathrm { A D V } ( I ) ) , 1 )$ . The CONF column shows the average confidence over all successful adversarial images for the corresponding technique. The PTB column shows the average (absolute) perturbation added per coordinate in cases of successful adversarial generation. More formally, let $\\tau$ denote the test set and $\\tau _ { \\mathrm { A D V } } \\subseteq \\tau$ denote the set of images in $\\tau$ on which ADV is successful. Then, ", + "bbox": [ + 174, + 503, + 825, + 588 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0639db17f6e794d32ccc2d1ad748aef9531b0ba73cfff3789896c974ee4f1d9b.jpg", + "text": "$$\n\\mathsf { P T B } = \\frac { 1 } { | \\mathcal T _ { \\mathrm { A D V } } | } \\sum _ { I \\in \\mathcal T _ { \\mathrm { A D V } } } \\frac { 1 } { \\ell \\times w \\times h } \\sum _ { b , x , y } | I ( b , x , y ) - \\mathrm { A D V } ( I ) ( b , x , y ) | ,\n$$", + "text_format": "latex", + "bbox": [ + 259, + 594, + 735, + 638 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $I \\in \\mathbb { R } ^ { \\ell \\times w \\times h }$ is the original image and $\\mathrm { A D V } ( I ) \\in \\mathbb { R } ^ { \\ell \\times w \\times h }$ is the corresponding adversarial image. Note that the inner summation is measuring the $L _ { 1 }$ -distance between $I$ and $\\mathrm { A D V } ( I )$ . The #PTBPIXELS column shows the average percentage of perturbed pixels in the successful adversarial images. Similarly, TIME column shows the average time (in seconds) to generate a successful adversarial image. Finally, the last column indicates the type of network architecture. ", + "bbox": [ + 174, + 646, + 825, + 718 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "As is quite evident from these results, Algorithm LOCSEARCHADV is more effective than the fastgradient sign method in generating adversarial images, even without having access to the network architecture and its parameter values. The difference is quite prominent for networks trained with batch normalization as here we noticed that the fast-gradient sign method has difficulties producing adversarial images.14 Another advantage with our approach is that it modifies a very tiny fraction of pixels as compared to all the pixels perturbed by the fast-gradient sign method, and also in many cases with far less average perturbation. Putting these points together demonstrates that ", + "bbox": [ + 173, + 724, + 825, + 821 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Algorithm LOCSEARCHADV is successful in generating more adversarial images than the fastgradient sign method, while modifying far fewer pixels and adding less noise per image. On the other side, the fast-gradient sign method takes lesser time in the generation process and generally seems to produce higher confidence scores for the adversarial (misclassified) images. ", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table 5 shows the results for several variants of VGG network trained on the ImageNet1000 dataset. These networks do not have batch normalization layers (Chatfield et al., 2014b; Zagoruyko, 2016a). We set $\\epsilon = 1$ for the fast-gradient sign method as a different pre-processing technique was used for this network (we converted these networks from pre-trained Caffe models). Results are similar to that observed on the smaller datasets. In most cases, our proposed local-search based approach is more successful in generating adversarial images while on average perturbing less than $0 . 5 5 \\%$ of the pixels. ", + "bbox": [ + 174, + 166, + 825, + 263 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Case of Larger $k$ ’s. We now consider achieving $k$ -misclassification for $k \\geq 1$ using Algorithm LOCSEARCHADV. In Table 6, we present the results as we change the goal from 1- misclassification to 4-misclassification on the CIFAR10 dataset. We use the same parameters as before for Algorithm LOCSEARCHADV. As one would expect, as we increase the value of $k$ , the effectiveness of the attack decreases, perturbation and time needed increases. But overall our local-search procedure is still able to generate a large fraction of adversarial images at even $k = 4$ with a small perturbation and computation time, meaning that these images will fool even a system that is evaluated on a top-4 classification criteria. We are not aware of a straightforward extension of the fast-gradient sign method (Goodfellow et al., 2015) to achieve $k$ -misclassification. ", + "bbox": [ + 174, + 289, + 826, + 414 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Even Weaker Adversarial Models. We also consider a weaker model where the adversary does not even have a black-box (oracle) access to the network (NN) of interest, and has to rely on a black-box access to somewhat of a “similar” (proxy) network as NN. For example, the adversary might want to evade a spam filter A, but might have to develop adversarial images by utilizing the output of a spam filter B, which might share properties similar to A. ", + "bbox": [ + 174, + 439, + 823, + 508 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We trained several modifications of Network-in-Network model for the CIFAR10 dataset, varying the initial value of the learning rate, the size of filters, and the number of layers in the network. We observed that between $2 5 \\%$ to $43 \\%$ of adversarial images generated by Algorithm LOCSEARCHADV using the original network were also adversarial for these modified networks (at $k = 1$ ). The transferability of adversarial images that we observe here has also been observed with other attacks too (Szegedy et al., 2014; Goodfellow et al., 2015; Papernot et al., 2016b;a) and demonstrates the wider applicability of all these attacks. ", + "bbox": [ + 174, + 515, + 825, + 613 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 640, + 318, + 656 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We investigate the inherent vulnerabilities in modern CNNs to practical black-box adversarial attacks. We present approaches that can efficiently locate a small set of pixels, without using any gradient information, which when perturbed lead to misclassification by a deep neural network. Our extensive experimental results, somewhat surprisingly, demonstrates the effectiveness of our simple approaches in generating adversarial examples. ", + "bbox": [ + 174, + 672, + 825, + 742 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Defenses against these attacks is an interesting research direction. However, we note that here that by limiting the perturbation to some pixels (being localized) the adversarial images generated by our local-search based approach do not represent the distribution of the original data. This means for these adversarial images, the use of adversarial training (or fine-tuning), a technique of training (or fine-tuning) networks on adversarial images to build more robust classifiers, is not very effective. In fact, even with adversarial training we noticed that the networks ability to resist new local-search based adversarial attack improves only marginally (on average between $1 - 2 \\%$ ). On the other hand, we suspect that one possible counter-measure to these localized adversarial attacks could be based on performing a careful analysis of the oracle queries to thwart the attempts to generate an adversarial image. ", + "bbox": [ + 174, + 750, + 825, + 888 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Finally, we believe that our local-search approach can also be used for attacks against other machine learning systems and can serve as an useful tool in measuring the robustness of these systems. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/660d338a8eec33d2d4a2b53fdad858405796b2c15edb97d6d235a0304b61349b.jpg", + "table_caption": [ + "Table 4: Results for four datasets: CIFAR10, STL10, SVHN, and MNIST. The entries denote by denoted by “– ” are the cases where the fast-gradient sign method fails to produce any adversarial image in our experimental setup. " + ], + "table_footnote": [], + "table_body": "
DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)TechniqueNetwork
NNs trained with batch normalization
CIFAR1011.6597.630.47 0.550.04 0.203.750.68LOCSEARCHADV (Ours)NinN
CIFAR1070.69
11.6297.510.74100.000.01FGSM(Goodfellow et al., 2015)NinN
CIFAR100.043.160.78LOCSEARCHADV (Ours)VGG
CIFAR1011.621111FGSM(Goodfellow et al.,2015)VGG
STL1058.170.420.021.207.15LOCSEARCHADV (Ours)NinN
STL1029.8154.850.530.20100.000.03FGSM(Goodfellow et al.,2015)NinN
STL1065.760.470.021.1113.90LOCSEARCHADV (Ours)VGG
STL1026.5026.50二1 0.05二二FGSM(Goodfellow et al., 2015)VGG
SVHN SVHN9.7197.06 48.620.47 0.490.204.51 100.001.02 0.02LOCSEARCHADV (Ours) FGSM(Goodfellow et al.,2015)NinN NinN
SVHN81.100.660.07 15.432.15LOCSEARCHADV (Ours)VGG VGG
4.77
SVHN4.771 0.54 0.201 2.241 0.64FGSM(Goodfellow et al.,2015)NinN
MNIST91.42
MNIST0.331.650.580.20100.000.02LOCSEARCHADV (Ours) FGSM (Goodfellow et al., 2015)
93.480.630.21 2.200.64LOCSEARCHADV(Ours)NinN VGG
MNIST MNIST0.440.44111 NNs trained without batch normalization1FGSM(Goodfellow et al., 2015)VGG
97.8916.540.720.043.240.58LOCSEARCHADV (Ours)NinN
CIFAR10 CIFAR1093.670.930.20100.000.02FGSM(Goodfellow et al., 2015)NinN
CIFAR1019.7997.980.770.042.990.72LOCSEARCHADV (Ours)VGG
90.930.900.20100.000.04FGSM(Goodfellow et al.,015)VGG
CIFAR1035.470.021.176.42LOCSEARCHADV (Ours)
STL1052.650.56 0.940.200.04NinN
STL1043.9187.160.520.01100.00FGSM(Goodfellow et al.,2015)NinN
STL1059.380.930.201.09 100.0019.65 0.10LOCSEARCHADV (Ours) FGSM(Goodfellow et al., 2015)VGG VGG
STL10SVHN 6.1573.9791.36 92.310.680.05
", + "bbox": [ + 173, + 127, + 867, + 476 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/60e0030664f0b9477ce857e7bc5b4ddeb425f917de26cbcbff0c21922b7d5cfc.jpg", + "table_caption": [ + "Table 5: Results for the ImageNet1000 dataset using a center crop of size $2 2 4 \\times 2 2 4$ for each image. " + ], + "table_footnote": [], + "table_body": "
DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%）TIME (in sec)TechniqueNetwork
ImageNet100058.2793.590.290.290.4312.72LOCSEARCHADV (Ours)VGG CNN-S (Caffe)
ImageNet100085.510.491.00100.004.74FGSM(Goodfellow et al.,2015)VGG CNN-S (Caffe)
ImageNet100058.9691.360.280.290.4010.01LOCSEARCHADV (Ours)VGG CNN-M (Caffe)
ImageNet100087.850.481.00100.004.36FGSM(Goodfellow etal.,015)VGG CNN-M (Caffe)
ImageNet100058.8092.820.290.300.4111.09LOCSEARCHADV (Ours)VGG CNN-M 2048 (Caffe)
ImageNet100088.430.521.00100.004.42FGSM(Goodfellow et al.,2015)VGG CNN-M 2048 (Caffe)
ImageNet100046.4072.070.300.540.5573.64LOCSEARCHADV (Ours)VGG ILSVRC 19 (Caffe)
ImageNet100085.050.521.00100.0023.94FGSM(Goodfellow et al., 2015)VGG ILSVRC 19 (Caffe)
", + "bbox": [ + 96, + 585, + 900, + 702 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/766f3353e07a3f7b08d6d6b6404f6aae7e7035c75e13b8b696c05fe6876edb1a.jpg", + "table_caption": [], + "table_footnote": [ + "Table 6: Effect of increasing $k$ on the performance of Algorithm LOCSEARCHADV (without batch normalization). " + ], + "table_body": "
DatasetkERRTOP-kERRTOP-k(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)Network
CIFAR1016.5497.890.720.043.240.58NinN
CIFAR106.8876.650.880.075.501.02NinN
CIFAR1012343.5859.020.900.087.091.85NinN
CIFAR101.8448.890.900.097.632.12NinN
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Adversarial examples enable adversaries to subvert the expected system", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 288, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 288, + 469, + 301 + ], + "score": 1.0, + "content": "behavior leading to undesired consequences and could pose a security risk when", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 324, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 324, + 313 + ], + "score": 1.0, + "content": "these systems are deployed in the real world.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 141, + 234, + 471, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 313, + 469, + 444 + ], + "lines": [ + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "score": 1.0, + "content": "In this work, we focus on deep convolutional neural networks and demonstrate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 324, + 469, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 469, + 336 + ], + "score": 1.0, + "content": "that adversaries can easily craft adversarial examples even without any internal", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 335, + 470, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 470, + 347 + ], + "score": 1.0, + "content": "knowledge of the target network. Our attacks treat the network as an oracle (black-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 346, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 469, + 357 + ], + "score": 1.0, + "content": "box) and only assume that the output of the network can be observed on the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 357, + 470, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 470, + 369 + ], + "score": 1.0, + "content": "probed inputs. Our first attack is based on a simple idea of adding perturbation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 368, + 470, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 368, + 470, + 380 + ], + "score": 1.0, + "content": "to a randomly selected single pixel or a small set of them. We then improve the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 378, + 470, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 470, + 391 + ], + "score": 1.0, + "content": "effectiveness of this attack by carefully constructing a small set of pixels to perturb", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 390, + 469, + 402 + ], + "spans": [ + { + "bbox": [ + 141, + 390, + 469, + 402 + ], + "score": 1.0, + "content": "by using the idea of greedy local-search. Our proposed attacks also naturally", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 401, + 469, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 401, + 469, + 412 + ], + "score": 1.0, + "content": "extend to a stronger notion of misclassification. Our extensive experimental results", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 411, + 470, + 424 + ], + "spans": [ + { + "bbox": [ + 141, + 411, + 470, + 424 + ], + "score": 1.0, + "content": "illustrate that even these elementary attacks can reveal a deep neural network’s", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 422, + 469, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 422, + 469, + 434 + ], + "score": 1.0, + "content": "vulnerabilities. The simplicity and effectiveness of our proposed schemes mean", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 434, + 412, + 445 + ], + "spans": [ + { + "bbox": [ + 142, + 434, + 412, + 445 + ], + "score": 1.0, + "content": "that they could serve as a litmus test for designing robust networks.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5, + "bbox_fs": [ + 141, + 313, + 470, + 445 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 461, + 206, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 208, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 208, + 477 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "Convolutional neural networks (CNNs) are among the most popular techniques employed for com-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "puter vision tasks, including but not limited to image recognition, localization, video tracking, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "image and video segmentation (Goodfellow et al., 2016). Though these deep networks have exhibited", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "good performances for these tasks, they have recently been shown to be particularly susceptible to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 507, + 541 + ], + "score": 1.0, + "content": "adversarial perturbations to the input images (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "Dezfooli et al., 2016; Papernot et al., 2016c;b; Kurakin et al., 2016; Grosse et al., 2016; Zagoruyko,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "2016b). Vulnerability of these networks to adversarial attacks can lead to undesirable consequences in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 562, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 573 + ], + "score": 1.0, + "content": "many practical applications using them. For example, adversarial attacks can be used to subvert fraud", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "detection, malware detection, or mislead autonomous navigation systems (Papernot et al., 2016c;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "Grosse et al., 2016). Further strengthening these results is a recent observation by Kurakin et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 595, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 606 + ], + "score": 1.0, + "content": "(2016) who showed that a significant fraction of adversarial images crafted using the original network", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "are misclassified even when fed to the classifier through a physical world system (such as a camera).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 484, + 507, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "In this paper, we investigate the problem of robustness of state-of-the-art convolutional neural", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "networks (CNNs) to simple black-box adversarial attacks. The rough goal of adversarial attacks", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 233, + 657 + ], + "score": 1.0, + "content": "is as follows: Given an image", + "type": "text" + }, + { + "bbox": [ + 233, + 644, + 240, + 654 + ], + "score": 0.59, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "that is correctly classified by a machine learning system (say, a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 317, + 668 + ], + "score": 1.0, + "content": "CNN), is it possible to construct a transformation of", + "type": "text" + }, + { + "bbox": [ + 317, + 655, + 324, + 665 + ], + "score": 0.71, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "(say, by adding a small perturbation to some", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "or all the pixels) that now leads to misclassification by the system. Since large perturbations can", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "trivially lead to misclassification, the attacks seek to limit the amount of perturbation applied under", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "some chosen metric. More often than not, in these attacks, the modification done to the image is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "so subtle that the changes are imperceptible to a human eye. Our proposed attacks also share this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "property, in addition to being practical and simplistic, thus highlighting a worrying aspect about lack", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 341, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 341, + 733 + ], + "score": 1.0, + "content": "of robustness prevalent in these modern vision techniques.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 621, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "There are two main research directions in the literature on adversarial attacks based on different", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "assumptions about the adversarial knowledge of the target network. The first line of work assumes", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "that the adversary has detailed knowledge of the network architecture and the parameters resulting", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "from training (or access to the labeled training set) (Szegedy et al., 2014; Goodfellow et al., 2015;", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c). Using this information, an adversary constructs", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "a perturbation for a given image. The most effective methods are gradient-based: a small perturbation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 160 + ], + "score": 1.0, + "content": "is constructed based on the gradients of the loss function w.r.t. the input image and a target label.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "Often, adding this small perturbation to the original image leads to a misclassification. In the second", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "line of work an adversary has restricted knowledge about the network from being able to only observe", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 179, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 507, + 195 + ], + "score": 1.0, + "content": "the network’s output on some probed inputs (Papernot et al., 2016b). Our work falls into this category.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "While this black-box model is a much more realistic and applicable threat model, it is also more", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "challenging because it considers weak adversaries without knowledge of the network architecture,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "parameters, or training data. Interestingly, our results suggest that this level of access and a small", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 440, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 440, + 239 + ], + "score": 1.0, + "content": "number of queries provide sufficient information to construct an adversarial image.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "image", + "bbox": [ + 115, + 247, + 496, + 429 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 247, + 496, + 429 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 247, + 496, + 429 + ], + "spans": [ + { + "bbox": [ + 115, + 247, + 496, + 429 + ], + "score": 0.976, + "type": "image", + "image_path": "8f8bd387fd79bbd926de42b8e1bd85f7e2cc447025bdada2d104ff3603f7faa7.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 115, + 247, + 496, + 307.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 115, + 307.6666666666667, + 496, + 368.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 115, + 368.33333333333337, + 496, + 429.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 438, + 505, + 472 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "Table 1: The top row shows the original images and the bottom row shows the perturbed images. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "misclassification is as follows: (a) a stingray misclassified as a sea lion, (b) an ostrich misclassified as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 461, + 474, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 474, + 472 + ], + "score": 1.0, + "content": "a goose, (c) a jay misclassified as a junco, and (d) a water ouzel misclassified as a redshank.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "As we operate in a black-box setting, we use a gradient-free approach to adversarial image generation.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 507, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 507, + 516 + ], + "score": 1.0, + "content": "Papernot et al. (2016b) were the first to discuss a black-box attack against deep learning systems.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 511, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 527 + ], + "score": 1.0, + "content": "Their attack crucially relies on the observation that there is a transferability (generalization) property", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "in adversarial examples, i.e., adversarial examples form one model transfers to another. Our proposed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "score": 1.0, + "content": "attacks on the other hand is much more simple and direct, does not require this transferability property,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "and hence is more effective in constructing adversarial images, in addition to having some other", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "computational advantages. We demonstrate that our method is capable of constructing adversarial", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "images for several network architectures trained on different datasets. In particular in this paper,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 577, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 592 + ], + "score": 1.0, + "content": "we consider the CIFAR10, MNIST, SVHN, STL10, and ImageNet1000 datasets, and two popular", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "network architectures, Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "2014). In Table 1, we show four images from the ImageNet1000 dataset. The original images are in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "the upper row. The bottom row shows the corresponding perturbed images produced by our algorithm", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 622, + 411, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 411, + 635 + ], + "score": 1.0, + "content": "which are misclassified by a VGG CNN-S network (Chatfield et al., 2014a).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 105, + 644, + 504, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "Our Contributions. In this work, we present simple and effective black-box adversarial attacks on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 654, + 479, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 479, + 667 + ], + "score": 1.0, + "content": "deep convolutional neural networks. We make the following main contributions in this paper.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 100, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 99, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 99, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "(1) The first question we investigate is the influence of perturbing a single pixel on the prediction.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 114, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "To do so, we devise a simple scheme, based on randomly selecting a single pixel and applying", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 114, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 114, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "a strong perturbation to it. Somewhat surprisingly, we noticed that a few trails of this random", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 115, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 115, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "experiment is already quite enough in generating adversarial images for low resolution image sets.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 115, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "In fact, in many cases, for misclassification, the amount of perturbation needed to be applied to the", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "There are two main research directions in the literature on adversarial attacks based on different", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "assumptions about the adversarial knowledge of the target network. The first line of work assumes", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "that the adversary has detailed knowledge of the network architecture and the parameters resulting", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "from training (or access to the labeled training set) (Szegedy et al., 2014; Goodfellow et al., 2015;", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c). Using this information, an adversary constructs", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "a perturbation for a given image. The most effective methods are gradient-based: a small perturbation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 160 + ], + "score": 1.0, + "content": "is constructed based on the gradients of the loss function w.r.t. the input image and a target label.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "Often, adding this small perturbation to the original image leads to a misclassification. In the second", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "line of work an adversary has restricted knowledge about the network from being able to only observe", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 179, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 507, + 195 + ], + "score": 1.0, + "content": "the network’s output on some probed inputs (Papernot et al., 2016b). Our work falls into this category.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "While this black-box model is a much more realistic and applicable threat model, it is also more", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "challenging because it considers weak adversaries without knowledge of the network architecture,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "parameters, or training data. Interestingly, our results suggest that this level of access and a small", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 440, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 440, + 239 + ], + "score": 1.0, + "content": "number of queries provide sufficient information to construct an adversarial image.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 83, + 507, + 239 + ] + }, + { + "type": "image", + "bbox": [ + 115, + 247, + 496, + 429 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 247, + 496, + 429 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 247, + 496, + 429 + ], + "spans": [ + { + "bbox": [ + 115, + 247, + 496, + 429 + ], + "score": 0.976, + "type": "image", + "image_path": "8f8bd387fd79bbd926de42b8e1bd85f7e2cc447025bdada2d104ff3603f7faa7.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 115, + 247, + 496, + 307.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 115, + 307.6666666666667, + 496, + 368.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 115, + 368.33333333333337, + 496, + 429.00000000000006 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 438, + 505, + 472 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "Table 1: The top row shows the original images and the bottom row shows the perturbed images. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "misclassification is as follows: (a) a stingray misclassified as a sea lion, (b) an ostrich misclassified as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 461, + 474, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 474, + 472 + ], + "score": 1.0, + "content": "a goose, (c) a jay misclassified as a junco, and (d) a water ouzel misclassified as a redshank.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "As we operate in a black-box setting, we use a gradient-free approach to adversarial image generation.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 507, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 507, + 516 + ], + "score": 1.0, + "content": "Papernot et al. (2016b) were the first to discuss a black-box attack against deep learning systems.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 511, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 527 + ], + "score": 1.0, + "content": "Their attack crucially relies on the observation that there is a transferability (generalization) property", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "in adversarial examples, i.e., adversarial examples form one model transfers to another. Our proposed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "score": 1.0, + "content": "attacks on the other hand is much more simple and direct, does not require this transferability property,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "and hence is more effective in constructing adversarial images, in addition to having some other", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "computational advantages. We demonstrate that our method is capable of constructing adversarial", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "score": 1.0, + "content": "images for several network architectures trained on different datasets. In particular in this paper,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 577, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 592 + ], + "score": 1.0, + "content": "we consider the CIFAR10, MNIST, SVHN, STL10, and ImageNet1000 datasets, and two popular", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "network architectures, Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "2014). In Table 1, we show four images from the ImageNet1000 dataset. The original images are in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "the upper row. The bottom row shows the corresponding perturbed images produced by our algorithm", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 622, + 411, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 411, + 635 + ], + "score": 1.0, + "content": "which are misclassified by a VGG CNN-S network (Chatfield et al., 2014a).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 491, + 507, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 644, + 504, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "Our Contributions. In this work, we present simple and effective black-box adversarial attacks on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 654, + 479, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 479, + 667 + ], + "score": 1.0, + "content": "deep convolutional neural networks. We make the following main contributions in this paper.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 643, + 505, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 99, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 99, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "(1) The first question we investigate is the influence of perturbing a single pixel on the prediction.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 114, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 114, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "To do so, we devise a simple scheme, based on randomly selecting a single pixel and applying", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 114, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 114, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "a strong perturbation to it. Somewhat surprisingly, we noticed that a few trails of this random", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 115, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 115, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "experiment is already quite enough in generating adversarial images for low resolution image sets.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 115, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "In fact, in many cases, for misclassification, the amount of perturbation needed to be applied to the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 114, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 114, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "selected pixel is also quite small. For high-resolution images, a similar phenomena holds, except", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 115, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 115, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "our scheme now picks a random set of around 50 pixels. These simple experiments show the ease", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 114, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 114, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "of generating adversarial images for modern deep CNNs without knowledge of either the network", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 114, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 114, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "architecture or its parameters. There is however one shortcoming in these approaches in that the", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 114, + 126, + 432, + 140 + ], + "spans": [ + { + "bbox": [ + 114, + 126, + 432, + 140 + ], + "score": 1.0, + "content": "perturbed image might have pixel values that are outside some expected range.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 37, + "bbox_fs": [ + 99, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 115, + 82, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 114, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 114, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "selected pixel is also quite small. For high-resolution images, a similar phenomena holds, except", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 115, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 115, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "our scheme now picks a random set of around 50 pixels. These simple experiments show the ease", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 114, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 114, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "of generating adversarial images for modern deep CNNs without knowledge of either the network", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 114, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 114, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "architecture or its parameters. There is however one shortcoming in these approaches in that the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 114, + 126, + 432, + 140 + ], + "spans": [ + { + "bbox": [ + 114, + 126, + 432, + 140 + ], + "score": 1.0, + "content": "perturbed image might have pixel values that are outside some expected range.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 99, + 144, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 98, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 98, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "(2) We overcome this above shortcoming by showing that lower perturbation suffices if we carefully", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 115, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 115, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "select the pixels for perturbation. The approach is based the idea of greedy local search, an iterative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 114, + 166, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 114, + 166, + 506, + 179 + ], + "score": 1.0, + "content": "search procedure, where in each round a local neighborhood is used to refine the current image and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 114, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 114, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "in process minimizing the probability of the network assigning high confidence scores to the true", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 114, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 114, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "class label. Again while the algorithm is quite simple, it is rather effective in generating adversarial", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 114, + 199, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 114, + 199, + 505, + 211 + ], + "score": 1.0, + "content": "images with quite small perturbations. We also show an interesting connection between the pixels", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 114, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 114, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "chosen for perturbation by our approach and the saliency map of an image, as defined by Simonyan", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 114, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 114, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "et al. (2014), that ranks pixels based on their influence on the output score. In effect our approach", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 114, + 231, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 114, + 231, + 505, + 245 + ], + "score": 1.0, + "content": "identifies pixels with high saliency scores but without explicitly using any gradient information", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 114, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "(as needed in the definition of saliency map (Simonyan et al., 2014)). Intuitively, in each round,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 114, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "our local-search based approach computes an implicit approximation to the gradient of the current", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 115, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 115, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "image by understanding the influence of a few pixels on the output, which is then used to update", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 114, + 275, + 190, + 289 + ], + "spans": [ + { + "bbox": [ + 114, + 275, + 190, + 289 + ], + "score": 1.0, + "content": "the current image.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 99, + 293, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 98, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 98, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "(3) We perform extensive experimental evaluations, and show that our local-search based approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 115, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "reliably generates adversarial examples with little perturbation (even when compared to a recent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 113, + 314, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 113, + 314, + 505, + 329 + ], + "score": 1.0, + "content": "elegant adversarial attack proposed by Goodfellow et al. (2015) which needs perfect knowledge", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 113, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 113, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "of the network). Another feature of our attack is that, by design, our approach only perturbs a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 114, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 114, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "very small fraction of the pixels during the adversarial image generation process (e.g., on the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 114, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 114, + 347, + 352, + 361 + ], + "score": 1.0, + "content": "ImageNet1000 dataset we on average perturb only about", + "type": "text" + }, + { + "bbox": [ + 352, + 348, + 374, + 358 + ], + "score": 0.86, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "of the pixels per image). Most", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 114, + 358, + 405, + 373 + ], + "spans": [ + { + "bbox": [ + 114, + 358, + 405, + 373 + ], + "score": 1.0, + "content": "previous attacks require the ability to perturb all the pixels in the image.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 99, + 376, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 98, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 98, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "(4) Our approaches naturally extend to a stronger notion of misclassification (that we refer to as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 116, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 116, + 388, + 122, + 397 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "-misclassification), where the goal is to ensure that the true label of the image does not even", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 113, + 398, + 507, + 411 + ], + "spans": [ + { + "bbox": [ + 113, + 398, + 186, + 411 + ], + "score": 1.0, + "content": "appear in the top-", + "type": "text" + }, + { + "bbox": [ + 187, + 399, + 194, + 408 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 398, + 507, + 411 + ], + "score": 1.0, + "content": "predictions of the network (obtained by sorting the confidence score vector).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 408, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 114, + 408, + 505, + 423 + ], + "score": 1.0, + "content": "This notion especially captures the fact that many modern systems (e.g., ImageNet competition", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 420, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 114, + 420, + 259, + 432 + ], + "score": 1.0, + "content": "entrants) are evaluated based on top-", + "type": "text" + }, + { + "bbox": [ + 259, + 421, + 266, + 430 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 420, + 506, + 432 + ], + "score": 1.0, + "content": "predictions. To the best of our knowledge, these are the first", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 432, + 412, + 443 + ], + "spans": [ + { + "bbox": [ + 115, + 432, + 332, + 443 + ], + "score": 1.0, + "content": "adversarial attacks on deep neural networks achieving", + "type": "text" + }, + { + "bbox": [ + 333, + 432, + 339, + 442 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 432, + 412, + 443 + ], + "score": 1.0, + "content": "-misclassification.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 209, + 473 + ], + "lines": [ + { + "bbox": [ + 104, + 459, + 210, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 210, + 475 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "Starting with the seminal paper by Szegedy et al. (2014), which showed that the state-of-the-art", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "neural networks are vulnerable to adversarial attacks, there has been significant attention focused", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "on this problem. The research has led to investigation of different adversarial threat models and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "scenarios (Papernot et al., 2016c;b; Grosse et al., 2016; Kurakin et al., 2016; Fawzi et al., 2016),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 507, + 541 + ], + "score": 1.0, + "content": "computationally efficient attacks (Goodfellow et al., 2015), perturbation efficient attacks (Moosavi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 538, + 213, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 213, + 552 + ], + "score": 1.0, + "content": "Dezfooli et al., 2016), etc.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "Szegedy et al. (2014) used a box-constrained L-BFGS technique to generate adversarial examples.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 567, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 579 + ], + "score": 1.0, + "content": "They also showed a transferability (or generalization) property for adversarial examples, in that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "adversarial examples generated for one network might also be misclassified by a related network with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "possibly different hyper-parameters (number of layers, initial weights, etc.). However, the need for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "a solving a series of costly penalized optimization problems makes this technique computationally", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "expensive for generating adversarial examples. This issue was fixed by Goodfellow et al. (2015)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "who motivated by the underlying linearity of the components used to build a network proposed an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "elegant scheme based on adding perturbation proportional to sign of the network’s cost function", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "gradient. Recently, Moosavi-Dezfooli et al. (2016) used an iterative linearization procedure to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "generate adversarial examples with lesser perturbation. Another recent attack proposed by Papernot", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "et al. (2016c) uses a notion of adversarial saliency maps (based on the saliency maps introduced", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "by (Simonyan et al., 2014)) to select the most sensitive input components for perturbation. This attack", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "has been adapted by Grosse et al. (2016) for generating adversarial samples for neural networks used", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "as malware classifiers. However, all these above described attacks require perfect knowledge of the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "target network’s architecture and parameters which limits their applicability to strong adversaries", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 721, + 385, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 385, + 733 + ], + "score": 1.0, + "content": "with the capability of gaining insider knowledge of the target system.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 45.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 115, + 82, + 504, + 137 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 114, + 81, + 505, + 140 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 99, + 144, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 98, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 98, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "(2) We overcome this above shortcoming by showing that lower perturbation suffices if we carefully", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 115, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 115, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "select the pixels for perturbation. The approach is based the idea of greedy local search, an iterative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 114, + 166, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 114, + 166, + 506, + 179 + ], + "score": 1.0, + "content": "search procedure, where in each round a local neighborhood is used to refine the current image and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 114, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 114, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "in process minimizing the probability of the network assigning high confidence scores to the true", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 114, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 114, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "class label. Again while the algorithm is quite simple, it is rather effective in generating adversarial", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 114, + 199, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 114, + 199, + 505, + 211 + ], + "score": 1.0, + "content": "images with quite small perturbations. We also show an interesting connection between the pixels", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 114, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 114, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "chosen for perturbation by our approach and the saliency map of an image, as defined by Simonyan", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 114, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 114, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "et al. (2014), that ranks pixels based on their influence on the output score. In effect our approach", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 114, + 231, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 114, + 231, + 505, + 245 + ], + "score": 1.0, + "content": "identifies pixels with high saliency scores but without explicitly using any gradient information", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 114, + 243, + 506, + 255 + ], + "score": 1.0, + "content": "(as needed in the definition of saliency map (Simonyan et al., 2014)). Intuitively, in each round,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 114, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "our local-search based approach computes an implicit approximation to the gradient of the current", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 115, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 115, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "image by understanding the influence of a few pixels on the output, which is then used to update", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 114, + 275, + 190, + 289 + ], + "spans": [ + { + "bbox": [ + 114, + 275, + 190, + 289 + ], + "score": 1.0, + "content": "the current image.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 98, + 144, + 506, + 289 + ] + }, + { + "type": "text", + "bbox": [ + 99, + 293, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 98, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 98, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "(3) We perform extensive experimental evaluations, and show that our local-search based approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 115, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "reliably generates adversarial examples with little perturbation (even when compared to a recent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 113, + 314, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 113, + 314, + 505, + 329 + ], + "score": 1.0, + "content": "elegant adversarial attack proposed by Goodfellow et al. (2015) which needs perfect knowledge", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 113, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 113, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "of the network). Another feature of our attack is that, by design, our approach only perturbs a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 114, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 114, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "very small fraction of the pixels during the adversarial image generation process (e.g., on the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 114, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 114, + 347, + 352, + 361 + ], + "score": 1.0, + "content": "ImageNet1000 dataset we on average perturb only about", + "type": "text" + }, + { + "bbox": [ + 352, + 348, + 374, + 358 + ], + "score": 0.86, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "of the pixels per image). Most", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 114, + 358, + 405, + 373 + ], + "spans": [ + { + "bbox": [ + 114, + 358, + 405, + 373 + ], + "score": 1.0, + "content": "previous attacks require the ability to perturb all the pixels in the image.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 98, + 293, + 506, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 99, + 376, + 505, + 443 + ], + "lines": [ + { + "bbox": [ + 98, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 98, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "(4) Our approaches naturally extend to a stronger notion of misclassification (that we refer to as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 116, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 116, + 388, + 122, + 397 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "-misclassification), where the goal is to ensure that the true label of the image does not even", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 113, + 398, + 507, + 411 + ], + "spans": [ + { + "bbox": [ + 113, + 398, + 186, + 411 + ], + "score": 1.0, + "content": "appear in the top-", + "type": "text" + }, + { + "bbox": [ + 187, + 399, + 194, + 408 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 398, + 507, + 411 + ], + "score": 1.0, + "content": "predictions of the network (obtained by sorting the confidence score vector).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 114, + 408, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 114, + 408, + 505, + 423 + ], + "score": 1.0, + "content": "This notion especially captures the fact that many modern systems (e.g., ImageNet competition", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 420, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 114, + 420, + 259, + 432 + ], + "score": 1.0, + "content": "entrants) are evaluated based on top-", + "type": "text" + }, + { + "bbox": [ + 259, + 421, + 266, + 430 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 420, + 506, + 432 + ], + "score": 1.0, + "content": "predictions. To the best of our knowledge, these are the first", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 432, + 412, + 443 + ], + "spans": [ + { + "bbox": [ + 115, + 432, + 332, + 443 + ], + "score": 1.0, + "content": "adversarial attacks on deep neural networks achieving", + "type": "text" + }, + { + "bbox": [ + 333, + 432, + 339, + 442 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 432, + 412, + 443 + ], + "score": 1.0, + "content": "-misclassification.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 98, + 377, + 507, + 443 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 209, + 473 + ], + "lines": [ + { + "bbox": [ + 104, + 459, + 210, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 210, + 475 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "Starting with the seminal paper by Szegedy et al. (2014), which showed that the state-of-the-art", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "neural networks are vulnerable to adversarial attacks, there has been significant attention focused", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "on this problem. The research has led to investigation of different adversarial threat models and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "scenarios (Papernot et al., 2016c;b; Grosse et al., 2016; Kurakin et al., 2016; Fawzi et al., 2016),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 507, + 541 + ], + "score": 1.0, + "content": "computationally efficient attacks (Goodfellow et al., 2015), perturbation efficient attacks (Moosavi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 538, + 213, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 213, + 552 + ], + "score": 1.0, + "content": "Dezfooli et al., 2016), etc.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 484, + 507, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "Szegedy et al. (2014) used a box-constrained L-BFGS technique to generate adversarial examples.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 567, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 506, + 579 + ], + "score": 1.0, + "content": "They also showed a transferability (or generalization) property for adversarial examples, in that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "adversarial examples generated for one network might also be misclassified by a related network with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "possibly different hyper-parameters (number of layers, initial weights, etc.). However, the need for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "a solving a series of costly penalized optimization problems makes this technique computationally", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "expensive for generating adversarial examples. This issue was fixed by Goodfellow et al. (2015)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "who motivated by the underlying linearity of the components used to build a network proposed an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "elegant scheme based on adding perturbation proportional to sign of the network’s cost function", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "gradient. Recently, Moosavi-Dezfooli et al. (2016) used an iterative linearization procedure to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "generate adversarial examples with lesser perturbation. Another recent attack proposed by Papernot", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "et al. (2016c) uses a notion of adversarial saliency maps (based on the saliency maps introduced", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "by (Simonyan et al., 2014)) to select the most sensitive input components for perturbation. This attack", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "has been adapted by Grosse et al. (2016) for generating adversarial samples for neural networks used", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "as malware classifiers. However, all these above described attacks require perfect knowledge of the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "target network’s architecture and parameters which limits their applicability to strong adversaries", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 721, + 385, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 385, + 733 + ], + "score": 1.0, + "content": "with the capability of gaining insider knowledge of the target system.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 556, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Our focus in this paper is the setting of black-box attacks, where we assume that an adversary has", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 107 + ], + "score": 1.0, + "content": "only the ability to use the network as an oracle. The adversary can obtain output from supplied inputs,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 102, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 102, + 506, + 119 + ], + "score": 1.0, + "content": "and use the observed input-output relationship to craft adversarial images.1 In the context of deep", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "neural networks, a black-box attack was first proposed by Papernot et al. (2016b) with the motivation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "of constructing an attack on a remotely hosted system.2 Their general idea is to first approximate the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "target network by querying it for output labels, which is used to train a substitute network, which is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "then used to craft adversarial examples for the original network. The success of the attack crucially", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "score": 1.0, + "content": "depends on the transferability property to hold between the original and the substitute network. Our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "black-box attack is more direct, and completely avoids the transferability assumption, making it far", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "more applicable. We also avoid the overhead of gathering data and training a substitute network.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 446, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 446, + 205 + ], + "score": 1.0, + "content": "Additionally, our techniques can be adapted to a stronger notion of misclassification.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "A complementary line of work has focused on building defenses against adversarial attacks. Although", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "designing defenses is beyond scope of this paper, it is possible that adapting the previous suggested", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "defense solutions such as Jacobian-based regularization (Gu & Rigazio, 2015) and distillation (Pa-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "pernot et al., 2016d) can reduce the efficacy of our proposed attacks. Moreover, the recently proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 480, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 480, + 266 + ], + "score": 1.0, + "content": "technique of differentially private training (Abadi et al., 2016) can also prove beneficial here.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "The study of adversarial instability have led to development of solutions that seeks to improve", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "training to in return increase the robustness and classification performance of the network. In", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "some case, adding adversarial examples to the training (adversarial training) set can act like a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "regularizer (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016). The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "phenomenon of adversarial instability has also been theoretically investigated for certain families", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "of classifiers under various models of (semi) random noise (Fawzi et al., 2015; 2016). However, as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 504, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 504, + 348 + ], + "score": 1.0, + "content": "we discuss later, due to peculiar nature of adversarial images generated by our approaches, a simple", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 470, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 470, + 358 + ], + "score": 1.0, + "content": "adversarial training is only mildly effective in preventing future similar adversarial attacks.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 108, + 363, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "The security of machine learning in settings distinct from deep neural networks is also an area of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "active research with various known attacks under different threat models. We refer the reader to a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 360, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 360, + 398 + ], + "score": 1.0, + "content": "recent survey by McDaniel et al. (2016) and references therein.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 410, + 208, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 209, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 209, + 425 + ], + "score": 1.0, + "content": "3 PRELIMINARIES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 302, + 447 + ], + "score": 1.0, + "content": "Notation and Normalization. We denote by", + "type": "text" + }, + { + "bbox": [ + 302, + 434, + 315, + 446 + ], + "score": 0.88, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 433, + 348, + 447 + ], + "score": 1.0, + "content": "the set", + "type": "text" + }, + { + "bbox": [ + 349, + 434, + 392, + 447 + ], + "score": 0.93, + "content": "\\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 433, + 505, + 447 + ], + "score": 1.0, + "content": ". The dataset of images is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 454, + 459 + ], + "score": 1.0, + "content": "partitioned into train and test (or validation) subsets. An element of a dataset is a pair", + "type": "text" + }, + { + "bbox": [ + 455, + 446, + 489, + 458 + ], + "score": 0.93, + "content": "( I , c ( I ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 146, + 469 + ], + "score": 1.0, + "content": "an image", + "type": "text" + }, + { + "bbox": [ + 146, + 457, + 153, + 466 + ], + "score": 0.71, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 456, + 254, + 469 + ], + "score": 1.0, + "content": "and a ground truth label", + "type": "text" + }, + { + "bbox": [ + 254, + 457, + 272, + 469 + ], + "score": 0.91, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "of this image. We assume that the class labels are drawn", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 156, + 480 + ], + "score": 1.0, + "content": "from the set", + "type": "text" + }, + { + "bbox": [ + 157, + 468, + 202, + 480 + ], + "score": 0.93, + "content": "\\{ 1 , \\ldots , C \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 467, + 289, + 480 + ], + "score": 1.0, + "content": ", i.e., we have a set of", + "type": "text" + }, + { + "bbox": [ + 289, + 468, + 318, + 478 + ], + "score": 0.9, + "content": "C \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 467, + 498, + 480 + ], + "score": 1.0, + "content": "possible labels. We assume that images have", + "type": "text" + }, + { + "bbox": [ + 498, + 468, + 504, + 478 + ], + "score": 0.74, + "content": "\\ell", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 371, + 492 + ], + "score": 1.0, + "content": "channels (in experiments we use the RGB format) and are of width", + "type": "text" + }, + { + "bbox": [ + 371, + 479, + 399, + 489 + ], + "score": 0.9, + "content": "w \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 477, + 443, + 492 + ], + "score": 1.0, + "content": "and height", + "type": "text" + }, + { + "bbox": [ + 444, + 479, + 470, + 489 + ], + "score": 0.9, + "content": "h \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 477, + 506, + 492 + ], + "score": 1.0, + "content": ". We say", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 124, + 502 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 489, + 156, + 501 + ], + "score": 0.92, + "content": "( b , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 488, + 310, + 502 + ], + "score": 1.0, + "content": "is a coordinate of an image for channel", + "type": "text" + }, + { + "bbox": [ + 311, + 490, + 316, + 499 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 488, + 367, + 502 + ], + "score": 1.0, + "content": "and location", + "type": "text" + }, + { + "bbox": [ + 367, + 489, + 391, + 501 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 488, + 411, + 502 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 412, + 489, + 444, + 501 + ], + "score": 0.93, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "is a pixel of an", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 160, + 513 + ], + "score": 1.0, + "content": "image where", + "type": "text" + }, + { + "bbox": [ + 160, + 501, + 193, + 512 + ], + "score": 0.9, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 499, + 264, + 513 + ], + "score": 1.0, + "content": "represents all the", + "type": "text" + }, + { + "bbox": [ + 264, + 501, + 270, + 510 + ], + "score": 0.8, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "coordinates corresponding to different channels at location", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 129, + 523 + ], + "score": 0.79, + "content": "( x , { \\bar { y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 510, + 136, + 524 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 137, + 512, + 198, + 523 + ], + "score": 0.73, + "content": "I ( b , x , y ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 510, + 262, + 524 + ], + "score": 1.0, + "content": "is the value of", + "type": "text" + }, + { + "bbox": [ + 262, + 512, + 269, + 521 + ], + "score": 0.79, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 510, + 297, + 524 + ], + "score": 1.0, + "content": "at the", + "type": "text" + }, + { + "bbox": [ + 298, + 511, + 330, + 523 + ], + "score": 0.93, + "content": "( b , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 510, + 438, + 524 + ], + "score": 1.0, + "content": "coordinate, and similarly", + "type": "text" + }, + { + "bbox": [ + 439, + 510, + 504, + 523 + ], + "score": 0.92, + "content": "I ( \\star , x , y ) \\in \\mathbb { R } ^ { \\ell }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 333, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 240, + 535 + ], + "score": 1.0, + "content": "represents the vector of values of", + "type": "text" + }, + { + "bbox": [ + 241, + 523, + 247, + 532 + ], + "score": 0.76, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 522, + 273, + 535 + ], + "score": 1.0, + "content": "at the", + "type": "text" + }, + { + "bbox": [ + 273, + 523, + 306, + 534 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 522, + 333, + 535 + ], + "score": 1.0, + "content": "pixel.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "It is a common practice to normalize the image before passing it to the network. A normalized", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "image has the same dimension as the original image, but differs in the coordinate values. In this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "score": 1.0, + "content": "work we treat the normalization procedure as an external procedure and assume that all images are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "normalized. As we always work with normalized images, in the following, a reference to image means", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "a normalized input image. We denote by LB and UB two constants such that all the coordinates of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 360, + 606 + ], + "score": 1.0, + "content": "all the normalized images fall in the range [LB, UB]. Generally,", + "type": "text" + }, + { + "bbox": [ + 360, + 594, + 393, + 605 + ], + "score": 0.83, + "content": "\\mathrm { L B } < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 594, + 411, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 411, + 594, + 445, + 605 + ], + "score": 0.86, + "content": "\\mathrm { U B > 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 594, + 505, + 606 + ], + "score": 1.0, + "content": ". We denote by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 602, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 159, + 615 + ], + "score": 0.91, + "content": "\\mathbb { I } \\subset \\mathbb { R } ^ { \\ell \\times w \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 602, + 465, + 619 + ], + "score": 1.0, + "content": "the space of all (valid) images which satisfy the following property: for every", + "type": "text" + }, + { + "bbox": [ + 465, + 605, + 488, + 615 + ], + "score": 0.89, + "content": "I \\in \\mathbb { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 602, + 507, + 619 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 367, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 167, + 629 + ], + "score": 1.0, + "content": "all coordinates", + "type": "text" + }, + { + "bbox": [ + 168, + 615, + 363, + 628 + ], + "score": 0.77, + "content": "( \\bar { b , \\boldsymbol { x } } , \\boldsymbol { y } ) \\in [ \\ell ] \\times [ w ] \\times [ \\bar { h ] } , I ( b , \\boldsymbol { x } , \\boldsymbol { y } ) \\in [ \\mathrm { \\bar { L } B , U B } ] .", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 616, + 367, + 629 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "We denote by NN a trained neural network (trained on some set of training images). NN takes an", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 133, + 657 + ], + "score": 1.0, + "content": "image", + "type": "text" + }, + { + "bbox": [ + 134, + 644, + 141, + 654 + ], + "score": 0.65, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 644, + 273, + 657 + ], + "score": 1.0, + "content": "as an input and outputs a vector", + "type": "text" + }, + { + "bbox": [ + 273, + 644, + 367, + 656 + ], + "score": 0.92, + "content": "\\mathbf { N N } ( I ) = ( o _ { 1 } , \\dots , o _ { C } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 644, + 398, + 657 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 398, + 645, + 408, + 655 + ], + "score": 0.85, + "content": "o _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "denotes the probability", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 247, + 668 + ], + "score": 1.0, + "content": "as determined by NN that image", + "type": "text" + }, + { + "bbox": [ + 247, + 655, + 254, + 665 + ], + "score": 0.67, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 655, + 325, + 668 + ], + "score": 1.0, + "content": "belongs to class", + "type": "text" + }, + { + "bbox": [ + 325, + 656, + 331, + 667 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 655, + 386, + 668 + ], + "score": 1.0, + "content": ". We denote", + "type": "text" + }, + { + "bbox": [ + 387, + 655, + 439, + 667 + ], + "score": 0.92, + "content": "\\pi ( \\mathrm { N N } ( I ) , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "a function that", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 263, + 678 + ], + "score": 1.0, + "content": "returns a set of indices that are the top-", + "type": "text" + }, + { + "bbox": [ + 264, + 666, + 270, + 676 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "predictions (ranked by decreasing probability scores with", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 690, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 688, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 119, + 688, + 505, + 703 + ], + "score": 1.0, + "content": "1These kind of attacks are also known as differential attacks motivated by the use of the term in differential", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 251, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 251, + 712 + ], + "score": 1.0, + "content": "cryptanalysis (Biham & Shamir, 1991).", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "2Papernot et al. (2016a) have recently extended this attack beyond deep neural networks to other classes of", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 213, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 213, + 733 + ], + "score": 1.0, + "content": "machine learning techniques.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Our focus in this paper is the setting of black-box attacks, where we assume that an adversary has", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 107 + ], + "score": 1.0, + "content": "only the ability to use the network as an oracle. The adversary can obtain output from supplied inputs,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 102, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 102, + 506, + 119 + ], + "score": 1.0, + "content": "and use the observed input-output relationship to craft adversarial images.1 In the context of deep", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "neural networks, a black-box attack was first proposed by Papernot et al. (2016b) with the motivation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "of constructing an attack on a remotely hosted system.2 Their general idea is to first approximate the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "target network by querying it for output labels, which is used to train a substitute network, which is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "then used to craft adversarial examples for the original network. The success of the attack crucially", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "score": 1.0, + "content": "depends on the transferability property to hold between the original and the substitute network. Our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "black-box attack is more direct, and completely avoids the transferability assumption, making it far", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "more applicable. We also avoid the overhead of gathering data and training a substitute network.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 446, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 446, + 205 + ], + "score": 1.0, + "content": "Additionally, our techniques can be adapted to a stronger notion of misclassification.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 82, + 507, + 205 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "A complementary line of work has focused on building defenses against adversarial attacks. Although", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "designing defenses is beyond scope of this paper, it is possible that adapting the previous suggested", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "defense solutions such as Jacobian-based regularization (Gu & Rigazio, 2015) and distillation (Pa-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "pernot et al., 2016d) can reduce the efficacy of our proposed attacks. Moreover, the recently proposed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 480, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 480, + 266 + ], + "score": 1.0, + "content": "technique of differentially private training (Abadi et al., 2016) can also prove beneficial here.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 209, + 506, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "The study of adversarial instability have led to development of solutions that seeks to improve", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "training to in return increase the robustness and classification performance of the network. In", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "some case, adding adversarial examples to the training (adversarial training) set can act like a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "regularizer (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016). The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "phenomenon of adversarial instability has also been theoretically investigated for certain families", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "of classifiers under various models of (semi) random noise (Fawzi et al., 2015; 2016). However, as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 504, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 504, + 348 + ], + "score": 1.0, + "content": "we discuss later, due to peculiar nature of adversarial images generated by our approaches, a simple", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 470, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 470, + 358 + ], + "score": 1.0, + "content": "adversarial training is only mildly effective in preventing future similar adversarial attacks.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 270, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 363, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "The security of machine learning in settings distinct from deep neural networks is also an area of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "active research with various known attacks under different threat models. We refer the reader to a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 360, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 360, + 398 + ], + "score": 1.0, + "content": "recent survey by McDaniel et al. (2016) and references therein.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 363, + 506, + 398 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 410, + 208, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 209, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 209, + 425 + ], + "score": 1.0, + "content": "3 PRELIMINARIES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 434, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 302, + 447 + ], + "score": 1.0, + "content": "Notation and Normalization. We denote by", + "type": "text" + }, + { + "bbox": [ + 302, + 434, + 315, + 446 + ], + "score": 0.88, + "content": "[ n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 433, + 348, + 447 + ], + "score": 1.0, + "content": "the set", + "type": "text" + }, + { + "bbox": [ + 349, + 434, + 392, + 447 + ], + "score": 0.93, + "content": "\\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 433, + 505, + 447 + ], + "score": 1.0, + "content": ". The dataset of images is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 454, + 459 + ], + "score": 1.0, + "content": "partitioned into train and test (or validation) subsets. An element of a dataset is a pair", + "type": "text" + }, + { + "bbox": [ + 455, + 446, + 489, + 458 + ], + "score": 0.93, + "content": "( I , c ( I ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 146, + 469 + ], + "score": 1.0, + "content": "an image", + "type": "text" + }, + { + "bbox": [ + 146, + 457, + 153, + 466 + ], + "score": 0.71, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 456, + 254, + 469 + ], + "score": 1.0, + "content": "and a ground truth label", + "type": "text" + }, + { + "bbox": [ + 254, + 457, + 272, + 469 + ], + "score": 0.91, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "of this image. We assume that the class labels are drawn", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 504, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 156, + 480 + ], + "score": 1.0, + "content": "from the set", + "type": "text" + }, + { + "bbox": [ + 157, + 468, + 202, + 480 + ], + "score": 0.93, + "content": "\\{ 1 , \\ldots , C \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 467, + 289, + 480 + ], + "score": 1.0, + "content": ", i.e., we have a set of", + "type": "text" + }, + { + "bbox": [ + 289, + 468, + 318, + 478 + ], + "score": 0.9, + "content": "C \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 467, + 498, + 480 + ], + "score": 1.0, + "content": "possible labels. We assume that images have", + "type": "text" + }, + { + "bbox": [ + 498, + 468, + 504, + 478 + ], + "score": 0.74, + "content": "\\ell", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 371, + 492 + ], + "score": 1.0, + "content": "channels (in experiments we use the RGB format) and are of width", + "type": "text" + }, + { + "bbox": [ + 371, + 479, + 399, + 489 + ], + "score": 0.9, + "content": "w \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 477, + 443, + 492 + ], + "score": 1.0, + "content": "and height", + "type": "text" + }, + { + "bbox": [ + 444, + 479, + 470, + 489 + ], + "score": 0.9, + "content": "h \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 477, + 506, + 492 + ], + "score": 1.0, + "content": ". We say", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 124, + 502 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 489, + 156, + 501 + ], + "score": 0.92, + "content": "( b , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 488, + 310, + 502 + ], + "score": 1.0, + "content": "is a coordinate of an image for channel", + "type": "text" + }, + { + "bbox": [ + 311, + 490, + 316, + 499 + ], + "score": 0.74, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 488, + 367, + 502 + ], + "score": 1.0, + "content": "and location", + "type": "text" + }, + { + "bbox": [ + 367, + 489, + 391, + 501 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 488, + 411, + 502 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 412, + 489, + 444, + 501 + ], + "score": 0.93, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "is a pixel of an", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 160, + 513 + ], + "score": 1.0, + "content": "image where", + "type": "text" + }, + { + "bbox": [ + 160, + 501, + 193, + 512 + ], + "score": 0.9, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 499, + 264, + 513 + ], + "score": 1.0, + "content": "represents all the", + "type": "text" + }, + { + "bbox": [ + 264, + 501, + 270, + 510 + ], + "score": 0.8, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "coordinates corresponding to different channels at location", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 129, + 523 + ], + "score": 0.79, + "content": "( x , { \\bar { y } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 510, + 136, + 524 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 137, + 512, + 198, + 523 + ], + "score": 0.73, + "content": "I ( b , x , y ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 510, + 262, + 524 + ], + "score": 1.0, + "content": "is the value of", + "type": "text" + }, + { + "bbox": [ + 262, + 512, + 269, + 521 + ], + "score": 0.79, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 510, + 297, + 524 + ], + "score": 1.0, + "content": "at the", + "type": "text" + }, + { + "bbox": [ + 298, + 511, + 330, + 523 + ], + "score": 0.93, + "content": "( b , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 510, + 438, + 524 + ], + "score": 1.0, + "content": "coordinate, and similarly", + "type": "text" + }, + { + "bbox": [ + 439, + 510, + 504, + 523 + ], + "score": 0.92, + "content": "I ( \\star , x , y ) \\in \\mathbb { R } ^ { \\ell }", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 522, + 333, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 240, + 535 + ], + "score": 1.0, + "content": "represents the vector of values of", + "type": "text" + }, + { + "bbox": [ + 241, + 523, + 247, + 532 + ], + "score": 0.76, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 522, + 273, + 535 + ], + "score": 1.0, + "content": "at the", + "type": "text" + }, + { + "bbox": [ + 273, + 523, + 306, + 534 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 522, + 333, + 535 + ], + "score": 1.0, + "content": "pixel.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 433, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "It is a common practice to normalize the image before passing it to the network. A normalized", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "image has the same dimension as the original image, but differs in the coordinate values. In this", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "score": 1.0, + "content": "work we treat the normalization procedure as an external procedure and assume that all images are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "normalized. As we always work with normalized images, in the following, a reference to image means", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "a normalized input image. We denote by LB and UB two constants such that all the coordinates of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 360, + 606 + ], + "score": 1.0, + "content": "all the normalized images fall in the range [LB, UB]. Generally,", + "type": "text" + }, + { + "bbox": [ + 360, + 594, + 393, + 605 + ], + "score": 0.83, + "content": "\\mathrm { L B } < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 594, + 411, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 411, + 594, + 445, + 605 + ], + "score": 0.86, + "content": "\\mathrm { U B > 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 594, + 505, + 606 + ], + "score": 1.0, + "content": ". We denote by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 602, + 507, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 159, + 615 + ], + "score": 0.91, + "content": "\\mathbb { I } \\subset \\mathbb { R } ^ { \\ell \\times w \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 602, + 465, + 619 + ], + "score": 1.0, + "content": "the space of all (valid) images which satisfy the following property: for every", + "type": "text" + }, + { + "bbox": [ + 465, + 605, + 488, + 615 + ], + "score": 0.89, + "content": "I \\in \\mathbb { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 602, + 507, + 619 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 367, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 167, + 629 + ], + "score": 1.0, + "content": "all coordinates", + "type": "text" + }, + { + "bbox": [ + 168, + 615, + 363, + 628 + ], + "score": 0.77, + "content": "( \\bar { b , \\boldsymbol { x } } , \\boldsymbol { y } ) \\in [ \\ell ] \\times [ w ] \\times [ \\bar { h ] } , I ( b , \\boldsymbol { x } , \\boldsymbol { y } ) \\in [ \\mathrm { \\bar { L } B , U B } ] .", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 616, + 367, + 629 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 539, + 507, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "We denote by NN a trained neural network (trained on some set of training images). NN takes an", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 133, + 657 + ], + "score": 1.0, + "content": "image", + "type": "text" + }, + { + "bbox": [ + 134, + 644, + 141, + 654 + ], + "score": 0.65, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 644, + 273, + 657 + ], + "score": 1.0, + "content": "as an input and outputs a vector", + "type": "text" + }, + { + "bbox": [ + 273, + 644, + 367, + 656 + ], + "score": 0.92, + "content": "\\mathbf { N N } ( I ) = ( o _ { 1 } , \\dots , o _ { C } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 644, + 398, + 657 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 398, + 645, + 408, + 655 + ], + "score": 0.85, + "content": "o _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "denotes the probability", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 247, + 668 + ], + "score": 1.0, + "content": "as determined by NN that image", + "type": "text" + }, + { + "bbox": [ + 247, + 655, + 254, + 665 + ], + "score": 0.67, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 655, + 325, + 668 + ], + "score": 1.0, + "content": "belongs to class", + "type": "text" + }, + { + "bbox": [ + 325, + 656, + 331, + 667 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 655, + 386, + 668 + ], + "score": 1.0, + "content": ". We denote", + "type": "text" + }, + { + "bbox": [ + 387, + 655, + 439, + 667 + ], + "score": 0.92, + "content": "\\pi ( \\mathrm { N N } ( I ) , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "a function that", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 263, + 678 + ], + "score": 1.0, + "content": "returns a set of indices that are the top-", + "type": "text" + }, + { + "bbox": [ + 264, + 666, + 270, + 676 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "predictions (ranked by decreasing probability scores with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 354, + 95 + ], + "score": 1.0, + "content": "ties broken arbitrarily) of the network NN. For example, if", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 354, + 82, + 480, + 94 + ], + "score": 0.9, + "content": "\\mathbf { N N } ( I ) = ( 0 . 2 5 , 0 . 1 , 0 . 2 , 0 . 4 5 )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 480, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", then", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 189, + 105 + ], + "score": 0.9, + "content": "\\pi ( \\operatorname { N N } ( I ) , 1 ) = \\{ 4 \\}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 190, + 92, + 439, + 106 + ], + "score": 1.0, + "content": "(corresponding to the location of the entry 0.45). Similarly,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 440, + 94, + 505, + 106 + ], + "score": 0.91, + "content": "\\pi ( { \\mathrm { N N } } ( I ) , 2 ) =", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 257, + 118 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 132, + 117 + ], + "score": 0.75, + "content": "\\{ 4 , 1 \\}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 132, + 104, + 136, + 118 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 136, + 105, + 235, + 117 + ], + "score": 0.9, + "content": "\\pi ( \\mathbf { N N } ( I ) , 3 ) = \\{ 4 , 1 , 3 \\}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 236, + 104, + 257, + 118 + ], + "score": 1.0, + "content": ", etc.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 632, + 506, + 678 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 354, + 95 + ], + "score": 1.0, + "content": "ties broken arbitrarily) of the network NN. For example, if", + "type": "text" + }, + { + "bbox": [ + 354, + 82, + 480, + 94 + ], + "score": 0.9, + "content": "\\mathbf { N N } ( I ) = ( 0 . 2 5 , 0 . 1 , 0 . 2 , 0 . 4 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 189, + 105 + ], + "score": 0.9, + "content": "\\pi ( \\operatorname { N N } ( I ) , 1 ) = \\{ 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 92, + 439, + 106 + ], + "score": 1.0, + "content": "(corresponding to the location of the entry 0.45). Similarly,", + "type": "text" + }, + { + "bbox": [ + 440, + 94, + 505, + 106 + ], + "score": 0.91, + "content": "\\pi ( { \\mathrm { N N } } ( I ) , 2 ) =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 257, + 118 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 132, + 117 + ], + "score": 0.75, + "content": "\\{ 4 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 104, + 136, + 118 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 136, + 105, + 235, + 117 + ], + "score": 0.9, + "content": "\\pi ( \\mathbf { N N } ( I ) , 3 ) = \\{ 4 , 1 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 104, + 257, + 118 + ], + "score": 1.0, + "content": ", etc.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 125, + 505, + 158 + ], + "lines": [ + { + "bbox": [ + 106, + 125, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 506, + 137 + ], + "score": 1.0, + "content": "Adversarial Goal. Before we define the goal of black-box adversarial attacks, we define misclassi-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "fication for a NN. In this paper, we use a stronger notion of misclassification, which we refer to as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 148, + 228, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 113, + 158 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 148, + 198, + 159 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 199, + 148, + 225, + 158 + ], + "score": 0.91, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 148, + 228, + 159 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 504, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 162, + 181 + ], + "score": 1.0, + "content": "Definition 1 (", + "type": "text" + }, + { + "bbox": [ + 162, + 170, + 168, + 179 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 168, + 337, + 181 + ], + "score": 1.0, + "content": "-misclassification) A neural network NN", + "type": "text" + }, + { + "bbox": [ + 337, + 169, + 344, + 179 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 168, + 437, + 181 + ], + "score": 1.0, + "content": "-misclassifies an image", + "type": "text" + }, + { + "bbox": [ + 437, + 169, + 444, + 179 + ], + "score": 0.57, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "with true label", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 179, + 389, + 193 + ], + "spans": [ + { + "bbox": [ + 107, + 180, + 125, + 192 + ], + "score": 0.9, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 179, + 179, + 193 + ], + "score": 1.0, + "content": "iff the output", + "type": "text" + }, + { + "bbox": [ + 180, + 180, + 209, + 192 + ], + "score": 0.71, + "content": "\\mathrm { N N } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 179, + 304, + 193 + ], + "score": 1.0, + "content": "of the network satisfies", + "type": "text" + }, + { + "bbox": [ + 304, + 180, + 386, + 192 + ], + "score": 0.91, + "content": "c ( I ) \\not \\in \\pi ( { \\mathrm { N N } } ( I ) , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 179, + 389, + 193 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 200, + 505, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 169, + 213 + ], + "score": 1.0, + "content": "In other words,", + "type": "text" + }, + { + "bbox": [ + 169, + 201, + 176, + 211 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 201, + 474, + 213 + ], + "score": 1.0, + "content": "-misclassification means that the network ranks the true label below at least", + "type": "text" + }, + { + "bbox": [ + 475, + 201, + 481, + 211 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 201, + 506, + 213 + ], + "score": 1.0, + "content": "other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 212, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 478, + 223 + ], + "score": 1.0, + "content": "labels. Traditionally the literature on adversarial attacks have only considered the case where", + "type": "text" + }, + { + "bbox": [ + 478, + 212, + 503, + 222 + ], + "score": 0.91, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 212, + 506, + 223 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 259, + 236 + ], + "score": 1.0, + "content": "Note that an adversary that achieves a", + "type": "text" + }, + { + "bbox": [ + 259, + 223, + 266, + 233 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 222, + 350, + 236 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 351, + 223, + 375, + 233 + ], + "score": 0.9, + "content": "k > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "is a stronger adversary than one", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 239, + 247 + ], + "score": 1.0, + "content": "achieving an 1-misclassification (", + "type": "text" + }, + { + "bbox": [ + 239, + 235, + 245, + 244 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 233, + 347, + 247 + ], + "score": 1.0, + "content": "-misclassification implies", + "type": "text" + }, + { + "bbox": [ + 347, + 234, + 356, + 244 + ], + "score": 0.85, + "content": "k ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 233, + 453, + 247 + ], + "score": 1.0, + "content": "-misclassification for all", + "type": "text" + }, + { + "bbox": [ + 453, + 234, + 501, + 245 + ], + "score": 0.91, + "content": "1 \\overset { \\cdot } { \\leq } k ^ { \\prime } \\leq k ,", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 233, + 506, + 247 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 336, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 115, + 258 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 245, + 141, + 255 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 243, + 336, + 258 + ], + "score": 1.0, + "content": ", we simply say that NN misclassifies the image.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 262, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 377, + 275 + ], + "score": 1.0, + "content": "In our setting, an adversary ADV is a function that takes in image", + "type": "text" + }, + { + "bbox": [ + 378, + 262, + 384, + 272 + ], + "score": 0.68, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "as input and whose output is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 166, + 285 + ], + "score": 1.0, + "content": "another image", + "type": "text" + }, + { + "bbox": [ + 167, + 273, + 200, + 285 + ], + "score": 0.49, + "content": "\\mathrm { A D V } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 273, + 352, + 285 + ], + "score": 1.0, + "content": "(with same number of coordinates as", + "type": "text" + }, + { + "bbox": [ + 352, + 273, + 358, + 282 + ], + "score": 0.7, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "). We define an adversarial image as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 303, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 222, + 295 + ], + "score": 1.0, + "content": "one that fools a network into", + "type": "text" + }, + { + "bbox": [ + 223, + 284, + 229, + 293 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 284, + 303, + 295 + ], + "score": 1.0, + "content": "-misclassification.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 361, + 318 + ], + "score": 1.0, + "content": "Definition 2 (Adversarial Image) Given access to an image", + "type": "text" + }, + { + "bbox": [ + 362, + 306, + 368, + 315 + ], + "score": 0.64, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 304, + 439, + 318 + ], + "score": 1.0, + "content": ", we say that an", + "type": "text" + }, + { + "bbox": [ + 439, + 305, + 473, + 316 + ], + "score": 0.67, + "content": "\\mathrm { A D V } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 304, + 495, + 318 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 496, + 305, + 502, + 315 + ], + "score": 0.38, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 293, + 329 + ], + "score": 1.0, + "content": "adversarial image (resp. adversarial image) if", + "type": "text" + }, + { + "bbox": [ + 294, + 315, + 376, + 327 + ], + "score": 0.89, + "content": "c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 315, + 395, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 316, + 505, + 327 + ], + "score": 0.86, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( \\mathbf { A D V } ( I ) ) , k )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 348, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 131, + 340 + ], + "score": 1.0, + "content": "(resp.", + "type": "text" + }, + { + "bbox": [ + 131, + 326, + 213, + 339 + ], + "score": 0.92, + "content": "c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 325, + 232, + 340 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 326, + 342, + 339 + ], + "score": 0.91, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( \\mathbf { A D V } ( I ) ) , 1 ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 325, + 348, + 340 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 358 + ], + "score": 1.0, + "content": "The goal of adversarial attacks is to design this function ADV that succeeds in fooling the network for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "a large set of images. Ideally, we would like to achieve this misclassification3 by adding only some", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "small perturbation (under some metric) to the image. The presence of adversarial images shows that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "score": 1.0, + "content": "there exist small perturbations in input that produce large perturbations at the output of the last layer.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 411 + ], + "score": 1.0, + "content": "Adversarial threat models can be divided into two broad classes.4 The first class of models roughly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "assumes that the adversary has a total knowledge of the network architecture and the parameters", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "resulting from training (or access to the labeled training set). The second class of threat models, as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "considered in this paper, make no assumptions about the adversary having access to the network", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "architecture, network parameters, or the training set. In this case, the adversary has only a black-box", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 431, + 465 + ], + "score": 1.0, + "content": "(oracle) access to the network, in that it can query the network NN on an image", + "type": "text" + }, + { + "bbox": [ + 431, + 453, + 438, + 462 + ], + "score": 0.57, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "and observe the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 135, + 475 + ], + "score": 1.0, + "content": "output", + "type": "text" + }, + { + "bbox": [ + 135, + 463, + 164, + 475 + ], + "score": 0.65, + "content": "\\mathrm { N N } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 462, + 504, + 475 + ], + "score": 1.0, + "content": ". In our experimental section (Section 6), we also consider a slight weakening of this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "black-box model where the adversary has only the ability to use a proxy of the network NN as an", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 136, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 136, + 497 + ], + "score": 1.0, + "content": "oracle.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "A black-box threat model in the context of deep neural networks was first considered by Papernot", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "et al. (2016b). There is however one subtle difference between the threat model considered here", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "and that considered by Papernot et al. (2016b) in what the adversary can access as an output. While", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "the adversary presented in (Papernot et al., 2016b) requires access to the class label assigned by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "network which is the same level of access needed by our simple randomized adversary (presented in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 421, + 571 + ], + "score": 1.0, + "content": "Section 4), our local-search adversary (presented in Section 5) requires access to", + "type": "text" + }, + { + "bbox": [ + 421, + 558, + 441, + 570 + ], + "score": 0.9, + "content": "o _ { c ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 556, + 506, + 571 + ], + "score": 1.0, + "content": "(the probability", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 211, + 582 + ], + "score": 1.0, + "content": "assigned to the true label", + "type": "text" + }, + { + "bbox": [ + 211, + 569, + 229, + 581 + ], + "score": 0.91, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 569, + 331, + 582 + ], + "score": 1.0, + "content": "by the network on input", + "type": "text" + }, + { + "bbox": [ + 332, + 569, + 338, + 579 + ], + "score": 0.63, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 569, + 375, + 582 + ], + "score": 1.0, + "content": ") and the", + "type": "text" + }, + { + "bbox": [ + 375, + 570, + 383, + 579 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "vector (for checking whether", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 113, + 590 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "-misclassification has been achieved). Our adversarial approaches does not require access to the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 224, + 604 + ], + "score": 1.0, + "content": "complete probability vector", + "type": "text" + }, + { + "bbox": [ + 225, + 591, + 259, + 603 + ], + "score": 0.72, + "content": "( \\mathrm { N N } ( I ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 591, + 506, + 604 + ], + "score": 1.0, + "content": ". Also as pointed out earlier, compared to (Papernot et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "2016b), our approach is more direct (needs no transferability assumption), requires no retraining, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 613, + 430, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 210, + 625 + ], + "score": 1.0, + "content": "can be adapted to achieve", + "type": "text" + }, + { + "bbox": [ + 211, + 613, + 218, + 623 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 614, + 430, + 625 + ], + "score": 1.0, + "content": "-misclassification rather than just 1-misclassification.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 106, + 635, + 362, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 363, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 363, + 649 + ], + "score": 1.0, + "content": "4 BLACK-BOX GENERATION: A FIRST ATTEMPT", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 108, + 658, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "In this section, we present a simple black-box adversary that operates by perturbing a single pixel (or", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "a small set of pixels) selected at random. In the next section, we build upon this idea to construct an", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 680, + 374, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 374, + 693 + ], + "score": 1.0, + "content": "adversary that achieves better success by making adaptive choices.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 700, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 447, + 712 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 447, + 712 + ], + "score": 1.0, + "content": "3Note that the misclassification is at test time, once the trained network has been deployed.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "4More fine-grained classification has also been considered in (Papernot et al., 2016c) where adversaries are", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 339, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 339, + 732 + ], + "score": 1.0, + "content": "categorized by the information and capabilities at their disposal.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 125, + 505, + 158 + ], + "lines": [ + { + "bbox": [ + 106, + 125, + 506, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 506, + 137 + ], + "score": 1.0, + "content": "Adversarial Goal. Before we define the goal of black-box adversarial attacks, we define misclassi-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "fication for a NN. In this paper, we use a stronger notion of misclassification, which we refer to as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 148, + 228, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 113, + 158 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 148, + 198, + 159 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 199, + 148, + 225, + 158 + ], + "score": 0.91, + "content": "k \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 148, + 228, + 159 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 125, + 506, + 159 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 504, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 162, + 181 + ], + "score": 1.0, + "content": "Definition 1 (", + "type": "text" + }, + { + "bbox": [ + 162, + 170, + 168, + 179 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 168, + 337, + 181 + ], + "score": 1.0, + "content": "-misclassification) A neural network NN", + "type": "text" + }, + { + "bbox": [ + 337, + 169, + 344, + 179 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 168, + 437, + 181 + ], + "score": 1.0, + "content": "-misclassifies an image", + "type": "text" + }, + { + "bbox": [ + 437, + 169, + 444, + 179 + ], + "score": 0.57, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "with true label", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 179, + 389, + 193 + ], + "spans": [ + { + "bbox": [ + 107, + 180, + 125, + 192 + ], + "score": 0.9, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 179, + 179, + 193 + ], + "score": 1.0, + "content": "iff the output", + "type": "text" + }, + { + "bbox": [ + 180, + 180, + 209, + 192 + ], + "score": 0.71, + "content": "\\mathrm { N N } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 179, + 304, + 193 + ], + "score": 1.0, + "content": "of the network satisfies", + "type": "text" + }, + { + "bbox": [ + 304, + 180, + 386, + 192 + ], + "score": 0.91, + "content": "c ( I ) \\not \\in \\pi ( { \\mathrm { N N } } ( I ) , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 179, + 389, + 193 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 106, + 168, + 506, + 193 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 200, + 505, + 256 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 169, + 213 + ], + "score": 1.0, + "content": "In other words,", + "type": "text" + }, + { + "bbox": [ + 169, + 201, + 176, + 211 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 201, + 474, + 213 + ], + "score": 1.0, + "content": "-misclassification means that the network ranks the true label below at least", + "type": "text" + }, + { + "bbox": [ + 475, + 201, + 481, + 211 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 201, + 506, + 213 + ], + "score": 1.0, + "content": "other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 212, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 478, + 223 + ], + "score": 1.0, + "content": "labels. Traditionally the literature on adversarial attacks have only considered the case where", + "type": "text" + }, + { + "bbox": [ + 478, + 212, + 503, + 222 + ], + "score": 0.91, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 212, + 506, + 223 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 259, + 236 + ], + "score": 1.0, + "content": "Note that an adversary that achieves a", + "type": "text" + }, + { + "bbox": [ + 259, + 223, + 266, + 233 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 222, + 350, + 236 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 351, + 223, + 375, + 233 + ], + "score": 0.9, + "content": "k > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "is a stronger adversary than one", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 239, + 247 + ], + "score": 1.0, + "content": "achieving an 1-misclassification (", + "type": "text" + }, + { + "bbox": [ + 239, + 235, + 245, + 244 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 233, + 347, + 247 + ], + "score": 1.0, + "content": "-misclassification implies", + "type": "text" + }, + { + "bbox": [ + 347, + 234, + 356, + 244 + ], + "score": 0.85, + "content": "k ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 233, + 453, + 247 + ], + "score": 1.0, + "content": "-misclassification for all", + "type": "text" + }, + { + "bbox": [ + 453, + 234, + 501, + 245 + ], + "score": 0.91, + "content": "1 \\overset { \\cdot } { \\leq } k ^ { \\prime } \\leq k ,", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 233, + 506, + 247 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 336, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 115, + 258 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 116, + 245, + 141, + 255 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 243, + 336, + 258 + ], + "score": 1.0, + "content": ", we simply say that NN misclassifies the image.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 201, + 506, + 258 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 262, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 377, + 275 + ], + "score": 1.0, + "content": "In our setting, an adversary ADV is a function that takes in image", + "type": "text" + }, + { + "bbox": [ + 378, + 262, + 384, + 272 + ], + "score": 0.68, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "as input and whose output is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 166, + 285 + ], + "score": 1.0, + "content": "another image", + "type": "text" + }, + { + "bbox": [ + 167, + 273, + 200, + 285 + ], + "score": 0.49, + "content": "\\mathrm { A D V } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 273, + 352, + 285 + ], + "score": 1.0, + "content": "(with same number of coordinates as", + "type": "text" + }, + { + "bbox": [ + 352, + 273, + 358, + 282 + ], + "score": 0.7, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "). We define an adversarial image as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 303, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 222, + 295 + ], + "score": 1.0, + "content": "one that fools a network into", + "type": "text" + }, + { + "bbox": [ + 223, + 284, + 229, + 293 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 284, + 303, + 295 + ], + "score": 1.0, + "content": "-misclassification.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 261, + 505, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 361, + 318 + ], + "score": 1.0, + "content": "Definition 2 (Adversarial Image) Given access to an image", + "type": "text" + }, + { + "bbox": [ + 362, + 306, + 368, + 315 + ], + "score": 0.64, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 304, + 439, + 318 + ], + "score": 1.0, + "content": ", we say that an", + "type": "text" + }, + { + "bbox": [ + 439, + 305, + 473, + 316 + ], + "score": 0.67, + "content": "\\mathrm { A D V } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 304, + 495, + 318 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 496, + 305, + 502, + 315 + ], + "score": 0.38, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 293, + 329 + ], + "score": 1.0, + "content": "adversarial image (resp. adversarial image) if", + "type": "text" + }, + { + "bbox": [ + 294, + 315, + 376, + 327 + ], + "score": 0.89, + "content": "c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , k )", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 315, + 395, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 316, + 505, + 327 + ], + "score": 0.86, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( \\mathbf { A D V } ( I ) ) , k )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 348, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 131, + 340 + ], + "score": 1.0, + "content": "(resp.", + "type": "text" + }, + { + "bbox": [ + 131, + 326, + 213, + 339 + ], + "score": 0.92, + "content": "c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 325, + 232, + 340 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 326, + 342, + 339 + ], + "score": 0.91, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( \\mathbf { A D V } ( I ) ) , 1 ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 325, + 348, + 340 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 304, + 506, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 358 + ], + "score": 1.0, + "content": "The goal of adversarial attacks is to design this function ADV that succeeds in fooling the network for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "a large set of images. Ideally, we would like to achieve this misclassification3 by adding only some", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "small perturbation (under some metric) to the image. The presence of adversarial images shows that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "score": 1.0, + "content": "there exist small perturbations in input that produce large perturbations at the output of the last layer.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 348, + 507, + 394 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 411 + ], + "score": 1.0, + "content": "Adversarial threat models can be divided into two broad classes.4 The first class of models roughly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "assumes that the adversary has a total knowledge of the network architecture and the parameters", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "resulting from training (or access to the labeled training set). The second class of threat models, as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "considered in this paper, make no assumptions about the adversary having access to the network", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "architecture, network parameters, or the training set. In this case, the adversary has only a black-box", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 431, + 465 + ], + "score": 1.0, + "content": "(oracle) access to the network, in that it can query the network NN on an image", + "type": "text" + }, + { + "bbox": [ + 431, + 453, + 438, + 462 + ], + "score": 0.57, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "and observe the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 462, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 135, + 475 + ], + "score": 1.0, + "content": "output", + "type": "text" + }, + { + "bbox": [ + 135, + 463, + 164, + 475 + ], + "score": 0.65, + "content": "\\mathrm { N N } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 462, + 504, + 475 + ], + "score": 1.0, + "content": ". In our experimental section (Section 6), we also consider a slight weakening of this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "black-box model where the adversary has only the ability to use a proxy of the network NN as an", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 485, + 136, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 136, + 497 + ], + "score": 1.0, + "content": "oracle.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 396, + 506, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "A black-box threat model in the context of deep neural networks was first considered by Papernot", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "et al. (2016b). There is however one subtle difference between the threat model considered here", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "and that considered by Papernot et al. (2016b) in what the adversary can access as an output. While", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "the adversary presented in (Papernot et al., 2016b) requires access to the class label assigned by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "network which is the same level of access needed by our simple randomized adversary (presented in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 421, + 571 + ], + "score": 1.0, + "content": "Section 4), our local-search adversary (presented in Section 5) requires access to", + "type": "text" + }, + { + "bbox": [ + 421, + 558, + 441, + 570 + ], + "score": 0.9, + "content": "o _ { c ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 556, + 506, + 571 + ], + "score": 1.0, + "content": "(the probability", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 211, + 582 + ], + "score": 1.0, + "content": "assigned to the true label", + "type": "text" + }, + { + "bbox": [ + 211, + 569, + 229, + 581 + ], + "score": 0.91, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 569, + 331, + 582 + ], + "score": 1.0, + "content": "by the network on input", + "type": "text" + }, + { + "bbox": [ + 332, + 569, + 338, + 579 + ], + "score": 0.63, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 569, + 375, + 582 + ], + "score": 1.0, + "content": ") and the", + "type": "text" + }, + { + "bbox": [ + 375, + 570, + 383, + 579 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "vector (for checking whether", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 113, + 590 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "-misclassification has been achieved). Our adversarial approaches does not require access to the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 224, + 604 + ], + "score": 1.0, + "content": "complete probability vector", + "type": "text" + }, + { + "bbox": [ + 225, + 591, + 259, + 603 + ], + "score": 0.72, + "content": "( \\mathrm { N N } ( I ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 591, + 506, + 604 + ], + "score": 1.0, + "content": ". Also as pointed out earlier, compared to (Papernot et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "2016b), our approach is more direct (needs no transferability assumption), requires no retraining, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 613, + 430, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 210, + 625 + ], + "score": 1.0, + "content": "can be adapted to achieve", + "type": "text" + }, + { + "bbox": [ + 211, + 613, + 218, + 623 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 614, + 430, + 625 + ], + "score": 1.0, + "content": "-misclassification rather than just 1-misclassification.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 502, + 506, + 625 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 635, + 362, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 363, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 363, + 649 + ], + "score": 1.0, + "content": "4 BLACK-BOX GENERATION: A FIRST ATTEMPT", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 108, + 658, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "In this section, we present a simple black-box adversary that operates by perturbing a single pixel (or", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "a small set of pixels) selected at random. In the next section, we build upon this idea to construct an", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 680, + 374, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 374, + 693 + ], + "score": 1.0, + "content": "adversary that achieves better success by making adaptive choices.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 658, + 506, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Power of One Pixel. Starting point of our investigation is to understand the influence of a single", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "pixel in an adversarial setting. Most existing adversarial attacks operate by applying the same", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "perturbation on each individual pixel while minimizing the overall perturbation (Szegedy et al., 2014;", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016), while recent research have yielded attacks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "that perturb only a fraction of the pixels (Papernot et al., 2016c;b; Grosse et al., 2016). However, in all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "score": 1.0, + "content": "these cases, no explicit restriction is placed on the number of pixels that can be perturbed. Therefore,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 146, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 506, + 162 + ], + "score": 1.0, + "content": "it is natural to ask: whether it is possible to force the network to misclassify an image by modifying", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "a single pixel? If so, how strong should this perturbation be? We run several experiments to shed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "light on these questions. For simplicity, in this section, we focus the case of 1-misclassification, even", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 312, + 194 + ], + "score": 1.0, + "content": "though all discussions easily extend to the case of", + "type": "text" + }, + { + "bbox": [ + 312, + 182, + 319, + 191 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 181, + 405, + 194 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 406, + 181, + 431, + 191 + ], + "score": 0.89, + "content": "k > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 181, + 506, + 194 + ], + "score": 1.0, + "content": ". We begin with a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 176, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 176, + 204 + ], + "score": 1.0, + "content": "useful definition.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 432, + 225 + ], + "score": 1.0, + "content": "Definition 3 (Critical Pixel) 5 Given a trained neural network NN and an image", + "type": "text" + }, + { + "bbox": [ + 432, + 212, + 438, + 221 + ], + "score": 0.43, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 209, + 471, + 225 + ], + "score": 1.0, + "content": ", a pixel", + "type": "text" + }, + { + "bbox": [ + 472, + 211, + 505, + 223 + ], + "score": 0.91, + "content": "( \\star , x , y )", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 117, + 235 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 223, + 123, + 232 + ], + "score": 0.43, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "is a critical pixel if a perturbation of this pixel generates an image that is misclassified by the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 232, + 223, + 247 + ], + "score": 1.0, + "content": "network NN. In other words,", + "type": "text" + }, + { + "bbox": [ + 223, + 234, + 256, + 245 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 232, + 335, + 247 + ], + "score": 1.0, + "content": "is a critical pixel in", + "type": "text" + }, + { + "bbox": [ + 335, + 234, + 342, + 243 + ], + "score": 0.33, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 232, + 506, + 247 + ], + "score": 1.0, + "content": "if there exists another neighboring image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 244, + 493, + 257 + ], + "spans": [ + { + "bbox": [ + 107, + 245, + 117, + 257 + ], + "score": 0.87, + "content": "I _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 244, + 192, + 257 + ], + "score": 1.0, + "content": "which differs from", + "type": "text" + }, + { + "bbox": [ + 193, + 245, + 199, + 254 + ], + "score": 0.47, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 244, + 340, + 257 + ], + "score": 1.0, + "content": "only in values at the pixel location", + "type": "text" + }, + { + "bbox": [ + 341, + 245, + 364, + 257 + ], + "score": 0.93, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 244, + 404, + 257 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 405, + 244, + 489, + 257 + ], + "score": 0.92, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( I _ { p } ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 244, + 493, + 257 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 104, + 263, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 154, + 279 + ], + "score": 1.0, + "content": "The image", + "type": "text" + }, + { + "bbox": [ + 154, + 266, + 165, + 277 + ], + "score": 0.87, + "content": "I _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 263, + 506, + 279 + ], + "score": 1.0, + "content": "can be generated in multiple ways, here we consider a class of sign-preserving", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 295, + 289 + ], + "score": 1.0, + "content": "perturbation functions defined as follows. Let", + "type": "text" + }, + { + "bbox": [ + 295, + 276, + 360, + 288 + ], + "score": 0.93, + "content": "\\mathtt { P E R T } ( I , p , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "be a function that takes as input an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 101, + 286, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 101, + 286, + 135, + 306 + ], + "score": 1.0, + "content": "image", + "type": "text" + }, + { + "bbox": [ + 135, + 291, + 141, + 300 + ], + "score": 0.74, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 286, + 250, + 306 + ], + "score": 1.0, + "content": ", a perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 251, + 290, + 280, + 302 + ], + "score": 0.91, + "content": "p \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 286, + 347, + 306 + ], + "score": 1.0, + "content": ", and a location", + "type": "text" + }, + { + "bbox": [ + 347, + 290, + 371, + 302 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 286, + 468, + 306 + ], + "score": 1.0, + "content": ", and outputs an image", + "type": "text" + }, + { + "bbox": [ + 468, + 287, + 505, + 302 + ], + "score": 0.92, + "content": "I _ { p } ^ { ( x , y ) } \\in", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 298, + 192, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 141, + 312 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { \\ell \\times w \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 298, + 192, + 315 + ], + "score": 1.0, + "content": ", defined as:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 316, + 428, + 344 + ], + "lines": [ + { + "bbox": [ + 182, + 316, + 428, + 344 + ], + "spans": [ + { + "bbox": [ + 182, + 316, + 428, + 344 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) } ( b , u , v ) \\stackrel { \\mathrm { \\tiny ~ d e f n } } { = } \\left\\{ { \\begin{array} { l l } { { \\scriptstyle I ( b , u , v ) } } & { { \\mathrm { i f ~ } } x \\neq u { \\mathrm { ~ o r ~ } } y \\neq v } \\\\ { { p \\times \\mathrm { s i g n } } ( I ( b , u , v ) ) } & { { \\mathrm { o t h e r w i s e } } } \\end{array} } \\right.", + "type": "interline_equation", + "image_path": "e270b4e8d34f4bc1b3e96bff2bc2a041fad51c462455c9ffddcaf5c5f1b751f2.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 182, + 316, + 428, + 325.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 182, + 325.3333333333333, + 428, + 334.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 182, + 334.66666666666663, + 428, + 343.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 104, + 345, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 212, + 366 + ], + "score": 1.0, + "content": "In other words, the image", + "type": "text" + }, + { + "bbox": [ + 212, + 348, + 313, + 363 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) } = \\mathtt { P E R T } ( I , p , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 345, + 420, + 366 + ], + "score": 1.0, + "content": "has same values as image", + "type": "text" + }, + { + "bbox": [ + 420, + 352, + 426, + 361 + ], + "score": 0.75, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 345, + 507, + 366 + ], + "score": 1.0, + "content": "at all pixels except", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 360, + 507, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 144, + 380 + ], + "score": 1.0, + "content": "the pixel", + "type": "text" + }, + { + "bbox": [ + 145, + 365, + 177, + 377 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 360, + 278, + 380 + ], + "score": 1.0, + "content": ". The value of the image", + "type": "text" + }, + { + "bbox": [ + 278, + 363, + 302, + 377 + ], + "score": 0.92, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 360, + 336, + 380 + ], + "score": 1.0, + "content": "at pixel", + "type": "text" + }, + { + "bbox": [ + 336, + 365, + 369, + 377 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 360, + 397, + 380 + ], + "score": 1.0, + "content": "is just", + "type": "text" + }, + { + "bbox": [ + 398, + 365, + 478, + 377 + ], + "score": 0.9, + "content": "p \\times \\mathrm { s i g n } ( I ( \\star , x , y ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 360, + 507, + 380 + ], + "score": 1.0, + "content": ". Note", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "that unlike some of the previous adversarial attacks (Szegedy et al., 2014; Goodfellow et al., 2015;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c; Grosse et al., 2016), our threat model does not", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "assume access to the true network gradient factors, hence the construction of the perturbed image has", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 409, + 276, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 276, + 421 + ], + "score": 1.0, + "content": "to be oblivious to the network parameters.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 105, + 426, + 467, + 441 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 465, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 235, + 442 + ], + "score": 1.0, + "content": "In the following, we say a pixel", + "type": "text" + }, + { + "bbox": [ + 235, + 429, + 268, + 441 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 425, + 306, + 442 + ], + "score": 1.0, + "content": "in image", + "type": "text" + }, + { + "bbox": [ + 307, + 429, + 313, + 438 + ], + "score": 0.69, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 425, + 365, + 442 + ], + "score": 1.0, + "content": "is critical iff", + "type": "text" + }, + { + "bbox": [ + 366, + 426, + 465, + 441 + ], + "score": 0.86, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( I _ { p } ^ { ( x , y ) } ) , 1 )", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "Critical Pixels are Common. Our first experiment is to investigate existence of critical pixels in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 481, + 475 + ], + "score": 1.0, + "content": "the considered dataset of images. To do so, we perform a simple procedure that picks a location", + "type": "text" + }, + { + "bbox": [ + 481, + 461, + 505, + 473 + ], + "score": 0.9, + "content": "( x , y )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 101, + 468, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 101, + 468, + 158, + 492 + ], + "score": 1.0, + "content": "in the image", + "type": "text" + }, + { + "bbox": [ + 159, + 475, + 165, + 484 + ], + "score": 0.7, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 468, + 453, + 492 + ], + "score": 1.0, + "content": "and applies the PERT function to this pixel to obtain a perturbed image", + "type": "text" + }, + { + "bbox": [ + 453, + 472, + 477, + 487 + ], + "score": 0.92, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 468, + 506, + 492 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "the perturbed image is run through the trained network, and we check whether it was misclassified or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 496, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 217, + 513 + ], + "score": 1.0, + "content": "not. If the perturbed image", + "type": "text" + }, + { + "bbox": [ + 217, + 496, + 241, + 511 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 496, + 506, + 513 + ], + "score": 1.0, + "content": "is misclassified then we have identified a critical pixel. While we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "can exhaustively repeat this procedure for all pixels in an image, for computational efficiency we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "score": 1.0, + "content": "instead perform it only on a fraction of randomly chosen pixels, and our results somewhat surprisingly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "suggest that in many cases this is sufficient to generate an adversarial image. Algorithm RANDADV", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 412, + 555 + ], + "score": 1.0, + "content": "presents the pseudo-code for this experiment. Algorithm RANDADV, selects", + "type": "text" + }, + { + "bbox": [ + 412, + 543, + 422, + 553 + ], + "score": 0.79, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "random pixels (with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "replacement) and performs checks whether the pixel is critical or not. The algorithm output is an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 383, + 577 + ], + "score": 1.0, + "content": "unbiased estimate for the fraction of critical pixels in the input image", + "type": "text" + }, + { + "bbox": [ + 383, + 565, + 389, + 574 + ], + "score": 0.61, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 564, + 505, + 577 + ], + "score": 1.0, + "content": ". Note that the algorithm can", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "fail in generating an adversarial image (i.e., in finding any critical pixel for an image). The following", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 586, + 313, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 313, + 599 + ], + "score": 1.0, + "content": "definition will be useful for our ensuing discussion.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 608, + 503, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 310, + 620 + ], + "score": 1.0, + "content": "Definition 4 (Good Image) We say that an image", + "type": "text" + }, + { + "bbox": [ + 310, + 608, + 317, + 618 + ], + "score": 0.35, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 606, + 379, + 620 + ], + "score": 1.0, + "content": "with true label", + "type": "text" + }, + { + "bbox": [ + 379, + 608, + 397, + 619 + ], + "score": 0.9, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "is good for a network NN", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 618, + 416, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 117, + 631 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 118, + 618, + 202, + 631 + ], + "score": 0.92, + "content": "c ( I ) \\in \\pi ( N N ( I ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 618, + 276, + 631 + ], + "score": 1.0, + "content": "(i.e., NN predicts", + "type": "text" + }, + { + "bbox": [ + 276, + 619, + 294, + 631 + ], + "score": 0.91, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 618, + 403, + 631 + ], + "score": 1.0, + "content": "as the most likely label for", + "type": "text" + }, + { + "bbox": [ + 403, + 619, + 410, + 628 + ], + "score": 0.51, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 618, + 416, + 631 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "Our first observation is that sometimes even small perturbation to a pixel can be sufficient to obtain an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 651, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 449, + 662 + ], + "score": 1.0, + "content": "adversarial image. Table 2 shows two images and their adversarial counterparts, with", + "type": "text" + }, + { + "bbox": [ + 450, + 651, + 474, + 662 + ], + "score": 0.9, + "content": "p = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 651, + 506, + 662 + ], + "score": 1.0, + "content": ". Often,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "original and adversarial images are indistinguishable to the human eye, but sometimes the critical", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 672, + 208, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 208, + 685 + ], + "score": 1.0, + "content": "pixel is visible (Table 2).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 691, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 690, + 441, + 703 + ], + "score": 1.0, + "content": "5In the definition of critical pixel we have not considered how well the original image", + "type": "text" + }, + { + "bbox": [ + 441, + 692, + 447, + 700 + ], + "score": 0.72, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "is classified by", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 171, + 713 + ], + "score": 1.0, + "content": "NN, i.e., whether", + "type": "text" + }, + { + "bbox": [ + 172, + 701, + 246, + 712 + ], + "score": 0.92, + "content": "c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 700, + 306, + 713 + ], + "score": 1.0, + "content": ". In particular, if", + "type": "text" + }, + { + "bbox": [ + 307, + 701, + 381, + 713 + ], + "score": 0.9, + "content": "c ( I ) \\notin \\pi ( \\mathrm { N N } ( I ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "then by definition all pixels in the", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "image are critical even without any perturbation. In our experiments, we ignore these images and only focus on", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 420, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 134, + 732 + ], + "score": 1.0, + "content": "images", + "type": "text" + }, + { + "bbox": [ + 134, + 722, + 140, + 730 + ], + "score": 0.75, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 722, + 165, + 732 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 165, + 721, + 240, + 732 + ], + "score": 0.92, + "content": "c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) ; \\hat { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 722, + 420, + 732 + ], + "score": 1.0, + "content": ", which we refer to as good images (Definition 4).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Power of One Pixel. Starting point of our investigation is to understand the influence of a single", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "pixel in an adversarial setting. Most existing adversarial attacks operate by applying the same", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "perturbation on each individual pixel while minimizing the overall perturbation (Szegedy et al., 2014;", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016), while recent research have yielded attacks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "that perturb only a fraction of the pixels (Papernot et al., 2016c;b; Grosse et al., 2016). However, in all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 507, + 150 + ], + "score": 1.0, + "content": "these cases, no explicit restriction is placed on the number of pixels that can be perturbed. Therefore,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 146, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 506, + 162 + ], + "score": 1.0, + "content": "it is natural to ask: whether it is possible to force the network to misclassify an image by modifying", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "a single pixel? If so, how strong should this perturbation be? We run several experiments to shed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "light on these questions. For simplicity, in this section, we focus the case of 1-misclassification, even", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 312, + 194 + ], + "score": 1.0, + "content": "though all discussions easily extend to the case of", + "type": "text" + }, + { + "bbox": [ + 312, + 182, + 319, + 191 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 181, + 405, + 194 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 406, + 181, + 431, + 191 + ], + "score": 0.89, + "content": "k > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 181, + 506, + 194 + ], + "score": 1.0, + "content": ". We begin with a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 176, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 176, + 204 + ], + "score": 1.0, + "content": "useful definition.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 81, + 507, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 257 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 432, + 225 + ], + "score": 1.0, + "content": "Definition 3 (Critical Pixel) 5 Given a trained neural network NN and an image", + "type": "text" + }, + { + "bbox": [ + 432, + 212, + 438, + 221 + ], + "score": 0.43, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 209, + 471, + 225 + ], + "score": 1.0, + "content": ", a pixel", + "type": "text" + }, + { + "bbox": [ + 472, + 211, + 505, + 223 + ], + "score": 0.91, + "content": "( \\star , x , y )", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 117, + 235 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 223, + 123, + 232 + ], + "score": 0.43, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "is a critical pixel if a perturbation of this pixel generates an image that is misclassified by the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 232, + 223, + 247 + ], + "score": 1.0, + "content": "network NN. In other words,", + "type": "text" + }, + { + "bbox": [ + 223, + 234, + 256, + 245 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 232, + 335, + 247 + ], + "score": 1.0, + "content": "is a critical pixel in", + "type": "text" + }, + { + "bbox": [ + 335, + 234, + 342, + 243 + ], + "score": 0.33, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 232, + 506, + 247 + ], + "score": 1.0, + "content": "if there exists another neighboring image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 244, + 493, + 257 + ], + "spans": [ + { + "bbox": [ + 107, + 245, + 117, + 257 + ], + "score": 0.87, + "content": "I _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 244, + 192, + 257 + ], + "score": 1.0, + "content": "which differs from", + "type": "text" + }, + { + "bbox": [ + 193, + 245, + 199, + 254 + ], + "score": 0.47, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 244, + 340, + 257 + ], + "score": 1.0, + "content": "only in values at the pixel location", + "type": "text" + }, + { + "bbox": [ + 341, + 245, + 364, + 257 + ], + "score": 0.93, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 244, + 404, + 257 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 405, + 244, + 489, + 257 + ], + "score": 0.92, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( I _ { p } ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 244, + 493, + 257 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 209, + 506, + 257 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 265, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 104, + 263, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 154, + 279 + ], + "score": 1.0, + "content": "The image", + "type": "text" + }, + { + "bbox": [ + 154, + 266, + 165, + 277 + ], + "score": 0.87, + "content": "I _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 263, + 506, + 279 + ], + "score": 1.0, + "content": "can be generated in multiple ways, here we consider a class of sign-preserving", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 295, + 289 + ], + "score": 1.0, + "content": "perturbation functions defined as follows. Let", + "type": "text" + }, + { + "bbox": [ + 295, + 276, + 360, + 288 + ], + "score": 0.93, + "content": "\\mathtt { P E R T } ( I , p , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "be a function that takes as input an", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 101, + 286, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 101, + 286, + 135, + 306 + ], + "score": 1.0, + "content": "image", + "type": "text" + }, + { + "bbox": [ + 135, + 291, + 141, + 300 + ], + "score": 0.74, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 286, + 250, + 306 + ], + "score": 1.0, + "content": ", a perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 251, + 290, + 280, + 302 + ], + "score": 0.91, + "content": "p \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 286, + 347, + 306 + ], + "score": 1.0, + "content": ", and a location", + "type": "text" + }, + { + "bbox": [ + 347, + 290, + 371, + 302 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 286, + 468, + 306 + ], + "score": 1.0, + "content": ", and outputs an image", + "type": "text" + }, + { + "bbox": [ + 468, + 287, + 505, + 302 + ], + "score": 0.92, + "content": "I _ { p } ^ { ( x , y ) } \\in", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 298, + 192, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 141, + 312 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { \\ell \\times w \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 298, + 192, + 315 + ], + "score": 1.0, + "content": ", defined as:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 101, + 263, + 506, + 315 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 316, + 428, + 344 + ], + "lines": [ + { + "bbox": [ + 182, + 316, + 428, + 344 + ], + "spans": [ + { + "bbox": [ + 182, + 316, + 428, + 344 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) } ( b , u , v ) \\stackrel { \\mathrm { \\tiny ~ d e f n } } { = } \\left\\{ { \\begin{array} { l l } { { \\scriptstyle I ( b , u , v ) } } & { { \\mathrm { i f ~ } } x \\neq u { \\mathrm { ~ o r ~ } } y \\neq v } \\\\ { { p \\times \\mathrm { s i g n } } ( I ( b , u , v ) ) } & { { \\mathrm { o t h e r w i s e } } } \\end{array} } \\right.", + "type": "interline_equation", + "image_path": "e270b4e8d34f4bc1b3e96bff2bc2a041fad51c462455c9ffddcaf5c5f1b751f2.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 182, + 316, + 428, + 325.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 182, + 325.3333333333333, + 428, + 334.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 182, + 334.66666666666663, + 428, + 343.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 506, + 420 + ], + "lines": [ + { + "bbox": [ + 104, + 345, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 212, + 366 + ], + "score": 1.0, + "content": "In other words, the image", + "type": "text" + }, + { + "bbox": [ + 212, + 348, + 313, + 363 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) } = \\mathtt { P E R T } ( I , p , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 345, + 420, + 366 + ], + "score": 1.0, + "content": "has same values as image", + "type": "text" + }, + { + "bbox": [ + 420, + 352, + 426, + 361 + ], + "score": 0.75, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 345, + 507, + 366 + ], + "score": 1.0, + "content": "at all pixels except", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 360, + 507, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 144, + 380 + ], + "score": 1.0, + "content": "the pixel", + "type": "text" + }, + { + "bbox": [ + 145, + 365, + 177, + 377 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 360, + 278, + 380 + ], + "score": 1.0, + "content": ". The value of the image", + "type": "text" + }, + { + "bbox": [ + 278, + 363, + 302, + 377 + ], + "score": 0.92, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 360, + 336, + 380 + ], + "score": 1.0, + "content": "at pixel", + "type": "text" + }, + { + "bbox": [ + 336, + 365, + 369, + 377 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 360, + 397, + 380 + ], + "score": 1.0, + "content": "is just", + "type": "text" + }, + { + "bbox": [ + 398, + 365, + 478, + 377 + ], + "score": 0.9, + "content": "p \\times \\mathrm { s i g n } ( I ( \\star , x , y ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 360, + 507, + 380 + ], + "score": 1.0, + "content": ". Note", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "that unlike some of the previous adversarial attacks (Szegedy et al., 2014; Goodfellow et al., 2015;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "Moosavi-Dezfooli et al., 2016; Papernot et al., 2016c; Grosse et al., 2016), our threat model does not", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "assume access to the true network gradient factors, hence the construction of the perturbed image has", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 409, + 276, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 276, + 421 + ], + "score": 1.0, + "content": "to be oblivious to the network parameters.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 345, + 507, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 426, + 467, + 441 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 465, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 235, + 442 + ], + "score": 1.0, + "content": "In the following, we say a pixel", + "type": "text" + }, + { + "bbox": [ + 235, + 429, + 268, + 441 + ], + "score": 0.92, + "content": "( \\star , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 425, + 306, + 442 + ], + "score": 1.0, + "content": "in image", + "type": "text" + }, + { + "bbox": [ + 307, + 429, + 313, + 438 + ], + "score": 0.69, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 425, + 365, + 442 + ], + "score": 1.0, + "content": "is critical iff", + "type": "text" + }, + { + "bbox": [ + 366, + 426, + 465, + 441 + ], + "score": 0.86, + "content": "c ( I ) \\notin \\pi ( \\mathbf { N N } ( I _ { p } ^ { ( x , y ) } ) , 1 )", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 425, + 465, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "Critical Pixels are Common. Our first experiment is to investigate existence of critical pixels in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 481, + 475 + ], + "score": 1.0, + "content": "the considered dataset of images. To do so, we perform a simple procedure that picks a location", + "type": "text" + }, + { + "bbox": [ + 481, + 461, + 505, + 473 + ], + "score": 0.9, + "content": "( x , y )", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 101, + 468, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 101, + 468, + 158, + 492 + ], + "score": 1.0, + "content": "in the image", + "type": "text" + }, + { + "bbox": [ + 159, + 475, + 165, + 484 + ], + "score": 0.7, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 468, + 453, + 492 + ], + "score": 1.0, + "content": "and applies the PERT function to this pixel to obtain a perturbed image", + "type": "text" + }, + { + "bbox": [ + 453, + 472, + 477, + 487 + ], + "score": 0.92, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 468, + 506, + 492 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "the perturbed image is run through the trained network, and we check whether it was misclassified or", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 496, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 217, + 513 + ], + "score": 1.0, + "content": "not. If the perturbed image", + "type": "text" + }, + { + "bbox": [ + 217, + 496, + 241, + 511 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 496, + 506, + 513 + ], + "score": 1.0, + "content": "is misclassified then we have identified a critical pixel. While we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "can exhaustively repeat this procedure for all pixels in an image, for computational efficiency we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 535 + ], + "score": 1.0, + "content": "instead perform it only on a fraction of randomly chosen pixels, and our results somewhat surprisingly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "suggest that in many cases this is sufficient to generate an adversarial image. Algorithm RANDADV", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 412, + 555 + ], + "score": 1.0, + "content": "presents the pseudo-code for this experiment. Algorithm RANDADV, selects", + "type": "text" + }, + { + "bbox": [ + 412, + 543, + 422, + 553 + ], + "score": 0.79, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "random pixels (with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 567 + ], + "score": 1.0, + "content": "replacement) and performs checks whether the pixel is critical or not. The algorithm output is an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 383, + 577 + ], + "score": 1.0, + "content": "unbiased estimate for the fraction of critical pixels in the input image", + "type": "text" + }, + { + "bbox": [ + 383, + 565, + 389, + 574 + ], + "score": 0.61, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 564, + 505, + 577 + ], + "score": 1.0, + "content": ". Note that the algorithm can", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "fail in generating an adversarial image (i.e., in finding any critical pixel for an image). The following", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 586, + 313, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 313, + 599 + ], + "score": 1.0, + "content": "definition will be useful for our ensuing discussion.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35, + "bbox_fs": [ + 101, + 449, + 506, + 599 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 608, + 503, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 310, + 620 + ], + "score": 1.0, + "content": "Definition 4 (Good Image) We say that an image", + "type": "text" + }, + { + "bbox": [ + 310, + 608, + 317, + 618 + ], + "score": 0.35, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 606, + 379, + 620 + ], + "score": 1.0, + "content": "with true label", + "type": "text" + }, + { + "bbox": [ + 379, + 608, + 397, + 619 + ], + "score": 0.9, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "is good for a network NN", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 618, + 416, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 117, + 631 + ], + "score": 1.0, + "content": "iff", + "type": "text" + }, + { + "bbox": [ + 118, + 618, + 202, + 631 + ], + "score": 0.92, + "content": "c ( I ) \\in \\pi ( N N ( I ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 618, + 276, + 631 + ], + "score": 1.0, + "content": "(i.e., NN predicts", + "type": "text" + }, + { + "bbox": [ + 276, + 619, + 294, + 631 + ], + "score": 0.91, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 618, + 403, + 631 + ], + "score": 1.0, + "content": "as the most likely label for", + "type": "text" + }, + { + "bbox": [ + 403, + 619, + 410, + 628 + ], + "score": 0.51, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 618, + 416, + 631 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 606, + 505, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "Our first observation is that sometimes even small perturbation to a pixel can be sufficient to obtain an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 651, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 449, + 662 + ], + "score": 1.0, + "content": "adversarial image. Table 2 shows two images and their adversarial counterparts, with", + "type": "text" + }, + { + "bbox": [ + 450, + 651, + 474, + 662 + ], + "score": 0.9, + "content": "p = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 651, + 506, + 662 + ], + "score": 1.0, + "content": ". Often,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "original and adversarial images are indistinguishable to the human eye, but sometimes the critical", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 672, + 208, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 208, + 685 + ], + "score": 1.0, + "content": "pixel is visible (Table 2).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 639, + 506, + 685 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 80, + 497, + 180 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 80, + 497, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 80, + 497, + 180 + ], + "spans": [ + { + "bbox": [ + 115, + 80, + 497, + 180 + ], + "score": 0.973, + "type": "image", + "image_path": "35f61844a2fc87ca9ef855e0bf44a533ed34704baca6fd233c93e79e87e3e179.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 80, + 497, + 113.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 113.33333333333334, + 497, + 146.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 146.66666666666669, + 497, + 180.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 189, + 506, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 104, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "Table 2: The row contains original images followed by misclassified images where only one pixel", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 399, + 212 + ], + "score": 1.0, + "content": "(pointed using a black arrow) was perturbed with perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 399, + 200, + 424, + 211 + ], + "score": 0.9, + "content": "p = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 200, + 506, + 212 + ], + "score": 1.0, + "content": ". After perturbation,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "in the first case (images (a) and (b)) an automobile gets misclassified as a truck, and in the second", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 339, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 339, + 236 + ], + "score": 1.0, + "content": "case (images (c) and (d)) a cat gets misclassified as a dog.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 451, + 269 + ], + "score": 1.0, + "content": "We also tried to understand the effect of larger perturbation parameter values. We set", + "type": "text" + }, + { + "bbox": [ + 451, + 257, + 460, + 266 + ], + "score": 0.76, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 255, + 506, + 269 + ], + "score": 1.0, + "content": "to half the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "number of pixels in each image. After usual training of the neural network using the training set (see", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "Section 6 for more details about training), we ran Algorithm RANDADV on 1000 randomly drawn", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 301 + ], + "score": 1.0, + "content": "images from the test set of the corresponding dataset. In our experiments, we varied perturbation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 198, + 313 + ], + "score": 1.0, + "content": "parameter in the range", + "type": "text" + }, + { + "bbox": [ + 198, + 300, + 256, + 312 + ], + "score": 0.92, + "content": "\\{ 1 , 5 , 1 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 301, + 506, + 313 + ], + "score": 1.0, + "content": ". Before we consider our results, we note some of the perturba-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "tion values that we use to construct the adversarial image might construct images that are not in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "original image space.6 However, these results are still somewhat surprising, because even though we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 333, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 506, + 346 + ], + "score": 1.0, + "content": "allow large (even out-of-range) perturbation, it is applied to exactly one pixel in the image, and it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 344, + 330, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 330, + 357 + ], + "score": 1.0, + "content": "appears that it suffices to even pick the pixel at random.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "Figures 1 and 2 show results for 4 datasets (more details about the datasets and the networks are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 232, + 385 + ], + "score": 1.0, + "content": "presented in Section 6). On the", + "type": "text" + }, + { + "bbox": [ + 232, + 374, + 239, + 382 + ], + "score": 0.51, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 372, + 403, + 385 + ], + "score": 1.0, + "content": "-axis we show the perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 404, + 374, + 410, + 383 + ], + "score": 0.73, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 372, + 505, + 385 + ], + "score": 1.0, + "content": ". In Figure 1, the y-axis", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 396 + ], + "score": 1.0, + "content": "represents the output of Algorithm RANDADV averaged over good images for the network.7 The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 393, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 469, + 407 + ], + "score": 1.0, + "content": "first observation that we can make is that the critical pixels are common, and in fact, as", + "type": "text" + }, + { + "bbox": [ + 470, + 396, + 477, + 405 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 393, + 505, + 407 + ], + "score": 1.0, + "content": "grows", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 399, + 417 + ], + "score": 1.0, + "content": "the fraction of critical pixels increases. For example, in CIFAR10, with", + "type": "text" + }, + { + "bbox": [ + 399, + 405, + 434, + 416 + ], + "score": 0.94, + "content": "p = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 404, + 468, + 417 + ], + "score": 1.0, + "content": ", almost", + "type": "text" + }, + { + "bbox": [ + 468, + 405, + 488, + 416 + ], + "score": 0.88, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "(on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "average) of the pixels randomly selected are critical. In Figure 2, the y-axis represents the fraction", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "of successful adversarial images generated by Algorithm RANDADV, i.e., fraction of inputs where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 435, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 497, + 452 + ], + "score": 1.0, + "content": "Algorithm RANDADV is successful in finding at least one critical pixel. Again we notice that as", + "type": "text" + }, + { + "bbox": [ + 497, + 440, + 504, + 450 + ], + "score": 0.73, + "content": "p", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 448, + 428, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 428, + 462 + ], + "score": 1.0, + "content": "grows it gets easier for Algorithm RANDADV to construct an adversarial image.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 474, + 233, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 234, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 234, + 488 + ], + "score": 1.0, + "content": "Algorithm 1 RANDADV (NN)", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 108, + 489, + 506, + 641 + ], + "lines": [ + { + "bbox": [ + 110, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 110, + 489, + 173, + 502 + ], + "score": 1.0, + "content": "1: Input: Image", + "type": "text" + }, + { + "bbox": [ + 174, + 491, + 180, + 500 + ], + "score": 0.43, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 489, + 235, + 502 + ], + "score": 1.0, + "content": "with true label", + "type": "text" + }, + { + "bbox": [ + 236, + 490, + 305, + 502 + ], + "score": 0.92, + "content": "c ( I ) \\in \\{ 1 , \\ldots , C \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 489, + 378, + 502 + ], + "score": 1.0, + "content": ", perturbation factor", + "type": "text" + }, + { + "bbox": [ + 378, + 491, + 401, + 501 + ], + "score": 0.89, + "content": "p \\in \\mathbb R", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 489, + 453, + 502 + ], + "score": 1.0, + "content": ", and a budget", + "type": "text" + }, + { + "bbox": [ + 453, + 491, + 480, + 500 + ], + "score": 0.88, + "content": "U \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 489, + 506, + 502 + ], + 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We give the following", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 678, + 507, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 507, + 693 + ], + "score": 1.0, + "content": "explanation for this observation. The majority of pixels in an MNIST image belong to the background,", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 700, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 698, + 408, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 698, + 408, + 714 + ], + "score": 1.0, + "content": "6We fix this shortcoming using a local-search based strategy in the next section.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "7Note by focusing on good images, we make sure that we are only accounting for those cases where", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 311, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 311, + 733 + ], + "score": 1.0, + "content": "perturbation is needed for creating an adversarial image.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 80, + 497, + 180 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 80, + 497, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 80, + 497, + 180 + ], + "spans": [ + { + "bbox": [ + 115, + 80, + 497, + 180 + ], + "score": 0.973, + "type": "image", + "image_path": "35f61844a2fc87ca9ef855e0bf44a533ed34704baca6fd233c93e79e87e3e179.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 80, + 497, + 113.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 113.33333333333334, + 497, + 146.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 146.66666666666669, + 497, + 180.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 189, + 506, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 104, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "Table 2: The row contains original images followed by misclassified images where only one pixel", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 399, + 212 + ], + "score": 1.0, + "content": "(pointed using a black arrow) was perturbed with perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 399, + 200, + 424, + 211 + ], + "score": 0.9, + "content": "p = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 200, + 506, + 212 + ], + "score": 1.0, + "content": ". After perturbation,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "in the first case (images (a) and (b)) an automobile gets misclassified as a truck, and in the second", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 339, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 339, + 236 + ], + "score": 1.0, + "content": "case (images (c) and (d)) a cat gets misclassified as a dog.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 451, + 269 + ], + "score": 1.0, + "content": "We also tried to understand the effect of larger perturbation parameter values. We set", + "type": "text" + }, + { + "bbox": [ + 451, + 257, + 460, + 266 + ], + "score": 0.76, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 255, + 506, + 269 + ], + "score": 1.0, + "content": "to half the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "number of pixels in each image. After usual training of the neural network using the training set (see", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 291 + ], + "score": 1.0, + "content": "Section 6 for more details about training), we ran Algorithm RANDADV on 1000 randomly drawn", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 301 + ], + "score": 1.0, + "content": "images from the test set of the corresponding dataset. In our experiments, we varied perturbation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 198, + 313 + ], + "score": 1.0, + "content": "parameter in the range", + "type": "text" + }, + { + "bbox": [ + 198, + 300, + 256, + 312 + ], + "score": 0.92, + "content": "\\{ 1 , 5 , 1 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 301, + 506, + 313 + ], + "score": 1.0, + "content": ". Before we consider our results, we note some of the perturba-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "tion values that we use to construct the adversarial image might construct images that are not in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 336 + ], + "score": 1.0, + "content": "original image space.6 However, these results are still somewhat surprising, because even though we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 333, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 506, + 346 + ], + "score": 1.0, + "content": "allow large (even out-of-range) perturbation, it is applied to exactly one pixel in the image, and it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 344, + 330, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 330, + 357 + ], + "score": 1.0, + "content": "appears that it suffices to even pick the pixel at random.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 255, + 506, + 357 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "Figures 1 and 2 show results for 4 datasets (more details about the datasets and the networks are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 232, + 385 + ], + "score": 1.0, + "content": "presented in Section 6). On the", + "type": "text" + }, + { + "bbox": [ + 232, + 374, + 239, + 382 + ], + "score": 0.51, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 372, + 403, + 385 + ], + "score": 1.0, + "content": "-axis we show the perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 404, + 374, + 410, + 383 + ], + "score": 0.73, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 372, + 505, + 385 + ], + "score": 1.0, + "content": ". In Figure 1, the y-axis", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 396 + ], + "score": 1.0, + "content": "represents the output of Algorithm RANDADV averaged over good images for the network.7 The", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 393, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 469, + 407 + ], + "score": 1.0, + "content": "first observation that we can make is that the critical pixels are common, and in fact, as", + "type": "text" + }, + { + "bbox": [ + 470, + 396, + 477, + 405 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 393, + 505, + 407 + ], + "score": 1.0, + "content": "grows", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 399, + 417 + ], + "score": 1.0, + "content": "the fraction of critical pixels increases. For example, in CIFAR10, with", + "type": "text" + }, + { + "bbox": [ + 399, + 405, + 434, + 416 + ], + "score": 0.94, + "content": "p = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 404, + 468, + 417 + ], + "score": 1.0, + "content": ", almost", + "type": "text" + }, + { + "bbox": [ + 468, + 405, + 488, + 416 + ], + "score": 0.88, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "(on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "average) of the pixels randomly selected are critical. In Figure 2, the y-axis represents the fraction", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "of successful adversarial images generated by Algorithm RANDADV, i.e., fraction of inputs where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 435, + 504, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 497, + 452 + ], + "score": 1.0, + "content": "Algorithm RANDADV is successful in finding at least one critical pixel. Again we notice that as", + "type": "text" + }, + { + "bbox": [ + 497, + 440, + 504, + 450 + ], + "score": 0.73, + "content": "p", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 448, + 428, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 428, + 462 + ], + "score": 1.0, + "content": "grows it gets easier for Algorithm RANDADV to construct an adversarial image.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 361, + 505, + 462 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 474, + 233, + 486 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 234, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 234, + 488 + ], + "score": 1.0, + "content": "Algorithm 1 RANDADV (NN)", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "index", + "bbox": [ + 108, + 489, + 506, + 641 + ], + "lines": [ + { + "bbox": [ + 110, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 110, + 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"index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 617, + 469, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 469, + 630 + ], + "score": 1.0, + "content": "12: {The algorithm succeeds in generating an adversarial image if it finds at least one critical pixel}", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 627, + 177, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 150, + 639 + ], + "score": 1.0, + "content": "13: return", + "type": "text" + }, + { + "bbox": [ + 152, + 627, + 177, + 640 + ], + "score": 0.57, + "content": "\\frac { \\mathrm { \\bar { c r i t i c a l } } } { U }", + "type": "inline_equation" + } + ], + "index": 39, + "is_list_start_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 489, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "Another observation is that for the MNIST and STL10 datasets, Algorithm RANDADV succeeds in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 681 + ], + "score": 1.0, + "content": "finding fewer critical pixels as compared to SVHN and CIFAR10 datasets. We give the following", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 678, + 507, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 507, + 693 + ], + "score": 1.0, + "content": "explanation for this observation. The majority of pixels in an MNIST image belong to the background,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "hence, these pixels are less likely to be critical. On the other hand, STL10 contains high resolution", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 140, + 361 + ], + "score": 1.0, + "content": "images,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 140, + 349, + 174, + 360 + ], + "score": 0.9, + "content": "9 6 \\times 9 6", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 174, + 349, + 505, + 361 + ], + "score": 1.0, + "content": ", where perhaps a single pixel has less of an impact on the output prediction. The", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 360, + 462, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 462, + 374 + ], + "score": 1.0, + "content": "latter observation motivated us to generalize the notion of a critical pixel to a critical set.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 656, + 507, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 81, + 496, + 164 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 81, + 496, + 164 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 81, + 496, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 496, + 164 + ], + "score": 0.969, + "type": "image", + "image_path": "7e8864ae3078d8172ab7a9829d78956096dcab004436e610d6380e7219120e9d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 81, + 496, + 108.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 108.66666666666667, + 496, + 136.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 136.33333333333334, + 496, + 164.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 173, + 506, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "Figure 1: Output of Algorithm RANDADV (averaged over good images). The results are for two networks: a)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 183, + 451, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 333, + 194 + ], + "score": 1.0, + "content": "Network-in-Network and b) VGG. The perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 333, + 185, + 339, + 194 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 183, + 393, + 194 + ], + "score": 1.0, + "content": "is varied from", + "type": "text" + }, + { + "bbox": [ + 393, + 183, + 447, + 194 + ], + "score": 0.89, + "content": "\\{ 1 , 5 , 1 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 183, + 451, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 105, + 204, + 496, + 288 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 204, + 496, + 288 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 204, + 496, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 496, + 288 + ], + "score": 0.968, + "type": "image", + "image_path": "e6719714440c8cfd2fc901d4cd2fef26915fc30b8d7a9db8a45e0c6471206f23.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 105, + 204, + 496, + 232.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 105, + 232.0, + 496, + 260.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 105, + 260.0, + 496, + 288.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 297, + 504, + 318 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "Figure 2: Fraction of images where Algorithm RANDADV succeeds in finding at least one critical pixel. Again", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 307, + 241, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 241, + 320 + ], + "score": 1.0, + "content": "we only start with only good images.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 104, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "hence, these pixels are less likely to be critical. On the other hand, STL10 contains high resolution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 140, + 361 + ], + "score": 1.0, + "content": "images,", + "type": "text" + }, + { + "bbox": [ + 140, + 349, + 174, + 360 + ], + "score": 0.9, + "content": "9 6 \\times 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 349, + 505, + 361 + ], + "score": 1.0, + "content": ", where perhaps a single pixel has less of an impact on the output prediction. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 360, + 462, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 462, + 374 + ], + "score": 1.0, + "content": "latter observation motivated us to generalize the notion of a critical pixel to a critical set.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "Definition 5 (Critical Set) Given a trained neural network NN and an image I, a critical set of I is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 390, + 507, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 163, + 407 + ], + "score": 1.0, + "content": "a set of pixels", + "type": "text" + }, + { + "bbox": [ + 163, + 392, + 233, + 406 + ], + "score": 0.92, + "content": "\\textstyle \\bigcup _ { ( x , y ) } \\{ ( \\star , x , y ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 390, + 244, + 407 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 244, + 392, + 251, + 402 + ], + "score": 0.57, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 390, + 507, + 407 + ], + "score": 1.0, + "content": "such that a perturbation of these pixels generates an image that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 250, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 250, + 416 + ], + "score": 1.0, + "content": "is misclassified by the network NN.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "The general goal will be to find critical sets of small size in an image. With this notion of critical", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "set, we considered constructing adversarial images on the high-resolution ImageNet1000 dataset.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 101, + 446, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 101, + 446, + 237, + 467 + ], + "score": 1.0, + "content": "We can modify the definition of", + "type": "text" + }, + { + "bbox": [ + 237, + 447, + 261, + 462 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 446, + 506, + 467 + ], + "score": 1.0, + "content": "(from (1)) where instead of a single pixel we perturb all the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "pixels in a set. Similarly, we can devise a simple extension to Algorithm RANDADV to operate with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "a set of pixels and to output an unbiased estimate for the fraction of critical sets of some fixed size", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "(50 in our case) in the input image.8 Note that a set size of 50 pixels is still a tiny fraction of all the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 266, + 506 + ], + "score": 1.0, + "content": "pixels in a standard (center) crop of size", + "type": "text" + }, + { + "bbox": [ + 266, + 494, + 309, + 504 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 493, + 360, + 506 + ], + "score": 1.0, + "content": ", namely just", + "type": "text" + }, + { + "bbox": [ + 361, + 493, + 388, + 505 + ], + "score": 0.88, + "content": "0 . 0 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 493, + 505, + 506 + ], + "score": 1.0, + "content": ". We use a larger perturbation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 147, + 518 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 148, + 506, + 154, + 516 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 504, + 234, + 518 + ], + "score": 1.0, + "content": "than before, and set", + "type": "text" + }, + { + "bbox": [ + 234, + 505, + 249, + 516 + ], + "score": 0.78, + "content": "( U )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "the budget on the number of trials on an image as 5000. Figure 3", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 514, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 506, + 529 + ], + "score": 1.0, + "content": "shows our results. Overall, we note that we can draw similar conclusions as before, i.e., increasing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "the perturbation parameter creates more critical sets making them easier to find and relatively small", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 537, + 344, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 344, + 551 + ], + "score": 1.0, + "content": "perturbations are sufficient to construct adversarial images.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 560, + 383, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 385, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 385, + 574 + ], + "score": 1.0, + "content": "5 BLACK-BOX GENERATION: A GREEDY APPROACH", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "The results from Section 4 show that most images have critical pixels such that modifying these pixels", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "significantly leads to a failure of NN to classify the image correctly. However, one shortcoming", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "of Algorithm RANDADV was that to build adversarial images, we sometimes had to apply a large", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "perturbation to a single pixel (or a small set of pixels). Hence, there might exist a pixel (or a set of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "pixels) in the adversarial image whose coordinate value could lie outside the valid range [LB, UB].", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "To overcome this issue, we need to redesign the search procedure to generate adversarial images", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "that still belong to the original image space I (defined in Section 3). Here a brute-force approach is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 672 + ], + "score": 1.0, + "content": "generally not feasible because of computational reasons, especially in high-resolution images. Hence,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "we need to develop an efficient heuristic procedure to find the right small set of pixels to be perturbed.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "Our solution presented in this section is based on performing a greedy local search over the image", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 694, + 134, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 134, + 705 + ], + "score": 1.0, + "content": "space.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "8Searching over all pixel sets of size 50 pixels is computationally prohibitive, which again motivates the need", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 339, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 339, + 732 + ], + "score": 1.0, + "content": "for a randomized strategy as proposed in Algorithm RANDADV.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 105, + 81, + 496, + 164 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 81, + 496, + 164 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 81, + 496, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 496, + 164 + ], + "score": 0.969, + "type": "image", + "image_path": "7e8864ae3078d8172ab7a9829d78956096dcab004436e610d6380e7219120e9d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 105, + 81, + 496, + 108.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 105, + 108.66666666666667, + 496, + 136.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 105, + 136.33333333333334, + 496, + 164.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 173, + 506, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "Figure 1: Output of Algorithm RANDADV (averaged over good images). The results are for two networks: a)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 183, + 451, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 333, + 194 + ], + "score": 1.0, + "content": "Network-in-Network and b) VGG. The perturbation parameter", + "type": "text" + }, + { + "bbox": [ + 333, + 185, + 339, + 194 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 183, + 393, + 194 + ], + "score": 1.0, + "content": "is varied from", + "type": "text" + }, + { + "bbox": [ + 393, + 183, + 447, + 194 + ], + "score": 0.89, + "content": "\\{ 1 , 5 , 1 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 183, + 451, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 105, + 204, + 496, + 288 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 204, + 496, + 288 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 204, + 496, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 496, + 288 + ], + "score": 0.968, + "type": "image", + "image_path": "e6719714440c8cfd2fc901d4cd2fef26915fc30b8d7a9db8a45e0c6471206f23.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 105, + 204, + 496, + 232.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 105, + 232.0, + 496, + 260.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 105, + 260.0, + 496, + 288.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 297, + 504, + 318 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "Figure 2: Fraction of images where Algorithm RANDADV succeeds in finding at least one critical pixel. Again", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 307, + 241, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 241, + 320 + ], + "score": 1.0, + "content": "we only start with only good images.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 505, + 371 + ], + "lines": [], + "index": 11, + "bbox_fs": [ + 104, + 337, + 505, + 374 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "Definition 5 (Critical Set) Given a trained neural network NN and an image I, a critical set of I is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 390, + 507, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 163, + 407 + ], + "score": 1.0, + "content": "a set of pixels", + "type": "text" + }, + { + "bbox": [ + 163, + 392, + 233, + 406 + ], + "score": 0.92, + "content": "\\textstyle \\bigcup _ { ( x , y ) } \\{ ( \\star , x , y ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 390, + 244, + 407 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 244, + 392, + 251, + 402 + ], + "score": 0.57, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 390, + 507, + 407 + ], + "score": 1.0, + "content": "such that a perturbation of these pixels generates an image that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 250, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 250, + 416 + ], + "score": 1.0, + "content": "is misclassified by the network NN.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 380, + 507, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "The general goal will be to find critical sets of small size in an image. With this notion of critical", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "set, we considered constructing adversarial images on the high-resolution ImageNet1000 dataset.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 101, + 446, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 101, + 446, + 237, + 467 + ], + "score": 1.0, + "content": "We can modify the definition of", + "type": "text" + }, + { + "bbox": [ + 237, + 447, + 261, + 462 + ], + "score": 0.93, + "content": "I _ { p } ^ { ( x , y ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 446, + 506, + 467 + ], + "score": 1.0, + "content": "(from (1)) where instead of a single pixel we perturb all the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "pixels in a set. Similarly, we can devise a simple extension to Algorithm RANDADV to operate with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "a set of pixels and to output an unbiased estimate for the fraction of critical sets of some fixed size", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "(50 in our case) in the input image.8 Note that a set size of 50 pixels is still a tiny fraction of all the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 266, + 506 + ], + "score": 1.0, + "content": "pixels in a standard (center) crop of size", + "type": "text" + }, + { + "bbox": [ + 266, + 494, + 309, + 504 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 493, + 360, + 506 + ], + "score": 1.0, + "content": ", namely just", + "type": "text" + }, + { + "bbox": [ + 361, + 493, + 388, + 505 + ], + "score": 0.88, + "content": "0 . 0 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 493, + 505, + 506 + ], + "score": 1.0, + "content": ". We use a larger perturbation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 147, + 518 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 148, + 506, + 154, + 516 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 504, + 234, + 518 + ], + "score": 1.0, + "content": "than before, and set", + "type": "text" + }, + { + "bbox": [ + 234, + 505, + 249, + 516 + ], + "score": 0.78, + "content": "( U )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "the budget on the number of trials on an image as 5000. Figure 3", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 514, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 506, + 529 + ], + "score": 1.0, + "content": "shows our results. Overall, we note that we can draw similar conclusions as before, i.e., increasing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "the perturbation parameter creates more critical sets making them easier to find and relatively small", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 537, + 344, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 344, + 551 + ], + "score": 1.0, + "content": "perturbations are sufficient to construct adversarial images.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 101, + 425, + 506, + 551 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 560, + 383, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 385, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 385, + 574 + ], + "score": 1.0, + "content": "5 BLACK-BOX GENERATION: A GREEDY APPROACH", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "The results from Section 4 show that most images have critical pixels such that modifying these pixels", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "significantly leads to a failure of NN to classify the image correctly. However, one shortcoming", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "of Algorithm RANDADV was that to build adversarial images, we sometimes had to apply a large", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "perturbation to a single pixel (or a small set of pixels). Hence, there might exist a pixel (or a set of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "pixels) in the adversarial image whose coordinate value could lie outside the valid range [LB, UB].", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "To overcome this issue, we need to redesign the search procedure to generate adversarial images", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "that still belong to the original image space I (defined in Section 3). Here a brute-force approach is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 507, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 672 + ], + "score": 1.0, + "content": "generally not feasible because of computational reasons, especially in high-resolution images. Hence,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "we need to develop an efficient heuristic procedure to find the right small set of pixels to be perturbed.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "Our solution presented in this section is based on performing a greedy local search over the image", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 694, + 134, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 134, + 705 + ], + "score": 1.0, + "content": "space.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 582, + 507, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 133, + 81, + 476, + 214 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 133, + 81, + 476, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 81, + 476, + 214 + ], + "spans": [ + { + "bbox": [ + 133, + 81, + 476, + 214 + ], + "score": 0.972, + "type": "image", + "image_path": "83372eea17b2e701cefc45d156d3d9f66be0989cf1e8cc78c4eecbe71d0ecc36.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 133, + 81, + 476, + 125.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 133, + 125.33333333333334, + 476, + 169.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 133, + 169.66666666666669, + 476, + 214.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 224, + 503, + 255 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "Figure 3: Experiments in Figures 1 and 2 for the high-resolution ImageNet1000 dataset. The results are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "again for good images from a set of 1000 randomly selected images. We use a slightly modified version of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 245, + 301, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 301, + 255 + ], + "score": 1.0, + "content": "Algorithm RANDADV that perturbs a set of 50 pixels.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 506, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 210, + 289 + ], + "score": 1.0, + "content": "We consider the general", + "type": "text" + }, + { + "bbox": [ + 210, + 277, + 217, + 287 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 277, + 506, + 289 + ], + "score": 1.0, + "content": "-misclassification problem (Definition 1) where an adversarial attack", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 287, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 327, + 299 + ], + "score": 1.0, + "content": "ensures that the true label does not appear in the top-", + "type": "text" + }, + { + "bbox": [ + 328, + 288, + 335, + 298 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 287, + 506, + 299 + ], + "score": 1.0, + "content": "predictions of the network. We utilize a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 298, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 312 + ], + "score": 1.0, + "content": "local-search procedure, which is an incomplete search procedure that is widely used for solving", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 308, + 507, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 507, + 324 + ], + "score": 1.0, + "content": "combinatorial problems appearing in diverse domains such as graph clustering, scheduling, logistics,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "and verification (Lenstra, 1997). For a general optimization problem it works as follows. Consider an", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 330, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 181, + 345 + ], + "score": 1.0, + "content": "objective function", + "type": "text" + }, + { + "bbox": [ + 181, + 331, + 248, + 343 + ], + "score": 0.92, + "content": "f ( \\mathbf { z } ) : \\mathbb { R } ^ { n } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 330, + 370, + 345 + ], + "score": 1.0, + "content": "where the goal is to minimize", + "type": "text" + }, + { + "bbox": [ + 370, + 331, + 389, + 343 + ], + "score": 0.91, + "content": "f ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 330, + 506, + 345 + ], + "score": 1.0, + "content": ". The local-search procedure", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 349, + 355 + ], + "score": 1.0, + "content": "works in rounds, where each round consists of two steps. Let", + "type": "text" + }, + { + "bbox": [ + 350, + 344, + 370, + 354 + ], + "score": 0.89, + "content": "\\mathbf { z } _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "be the solution iterate after round", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 128, + 364 + ], + "score": 0.83, + "content": "i - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 353, + 199, + 367 + ], + "score": 1.0, + "content": ". Consider round", + "type": "text" + }, + { + "bbox": [ + 199, + 354, + 204, + 363 + ], + "score": 0.63, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 353, + 408, + 367 + ], + "score": 1.0, + "content": ". The first step is to select a small subset of points", + "type": "text" + }, + { + "bbox": [ + 408, + 353, + 482, + 365 + ], + "score": 0.92, + "content": "Z = \\{ \\hat { \\mathbf { z } } _ { 1 } , \\hdots , \\hat { \\mathbf { z } } _ { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 353, + 506, + 367 + ], + "score": 1.0, + "content": ", a so", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 266, + 378 + ], + "score": 1.0, + "content": "called local neighborhood, and evaluate", + "type": "text" + }, + { + "bbox": [ + 267, + 365, + 291, + 377 + ], + "score": 0.93, + "content": "f ( \\hat { \\mathbf { z } } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 364, + 330, + 378 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 330, + 365, + 360, + 376 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { z } } _ { j } \\in Z", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 364, + 428, + 378 + ], + "score": 1.0, + "content": ". Usually, the set", + "type": "text" + }, + { + "bbox": [ + 428, + 365, + 437, + 374 + ], + "score": 0.77, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "consist of points", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 373, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 200, + 390 + ], + "score": 1.0, + "content": "that are close to current", + "type": "text" + }, + { + "bbox": [ + 200, + 377, + 220, + 387 + ], + "score": 0.89, + "content": "\\mathbf { z } _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 373, + 506, + 390 + ], + "score": 1.0, + "content": "for some measure of distance which is domain specific. 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Hence,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 397, + 468, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 248, + 410 + ], + "score": 0.9, + "content": "\\mathbf { z } _ { i } = g ( f ( \\mathbf { z } _ { i - 1 } ) , f ( \\hat { \\mathbf { z } } _ { 1 } ) , \\hdots , f ( \\hat { \\mathbf { z } } _ { n } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 397, + 279, + 411 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 279, + 399, + 286, + 409 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 397, + 468, + 411 + ], + "score": 1.0, + "content": "is some pre-defined transformation function.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "We adapt this general procedure to search critical sets efficiently as explained below. Our optimization", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "problem will try to minimize the probability that the network determines an perturbed image has the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 449 + ], + "score": 1.0, + "content": "class label of the original image, and by using a local-search procedure we generate perturbed images", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "which differ from the original image in only few pixels. Intuitively, in each round, our local-search", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 458, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 472 + ], + "score": 1.0, + "content": "procedure computes an implicit approximation to the gradient of the current image by understanding", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 469, + 469, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 469, + 482 + ], + "score": 1.0, + "content": "the influence of a few pixels on the output, which is then used to update the current image.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 99, + 491, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 98, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 98, + 490, + 289, + 504 + ], + "score": 1.0, + "content": "(a) First, we need to define the cost function", + "type": "text" + }, + { + "bbox": [ + 290, + 492, + 297, + 503 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 490, + 320, + 504 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 321, + 492, + 327, + 501 + ], + "score": 0.75, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 490, + 452, + 504 + ], + "score": 1.0, + "content": "be the image (with true label", + "type": "text" + }, + { + "bbox": [ + 453, + 491, + 472, + 503 + ], + "score": 0.85, + "content": "c ( I ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 490, + 505, + 504 + ], + "score": 1.0, + "content": ") whose", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 114, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 114, + 504, + 496, + 516 + ], + "score": 1.0, + "content": "adversarial image we want to generate for a target neural network NN. For some input image", + "type": "text" + }, + { + "bbox": [ + 496, + 503, + 502, + 514 + ], + "score": 0.81, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 504, + 506, + 516 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 515, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 115, + 516, + 235, + 529 + ], + "score": 1.0, + "content": "we use the objective function", + "type": "text" + }, + { + "bbox": [ + 236, + 515, + 268, + 530 + ], + "score": 0.94, + "content": "f _ { c ( I ) } ( \\hat { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "which equals the probability assigned by the network NN", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 529, + 356, + 544 + ], + "spans": [ + { + "bbox": [ + 114, + 529, + 197, + 544 + ], + "score": 1.0, + "content": "that the input image", + "type": "text" + }, + { + "bbox": [ + 198, + 529, + 204, + 540 + ], + "score": 0.81, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 529, + 271, + 544 + ], + "score": 1.0, + "content": "belongs to class", + "type": "text" + }, + { + "bbox": [ + 271, + 530, + 289, + 542 + ], + "score": 0.9, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 529, + 356, + 544 + ], + "score": 1.0, + "content": ". More formally,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 547, + 404, + 564 + ], + "lines": [ + { + "bbox": [ + 214, + 547, + 404, + 564 + ], + "spans": [ + { + "bbox": [ + 214, + 547, + 404, + 564 + ], + "score": 0.91, + "content": "f _ { c ( I ) } ( \\hat { I } ) = o _ { c ( I ) } \\mathrm { w h e r e } \\mathrm { N N } ( \\hat { I } ) = ( o _ { 1 } , \\dots , o _ { C } ) ,", + "type": "interline_equation", + "image_path": "d3beeb6ee5e9e695c0fde6d8d5c55aac1602ffb5eb31d7d45e0c13de7a87b8dc.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 214, + 547, + 404, + 564 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 112, + 570, + 503, + 594 + ], + "lines": [ + { + "bbox": [ + 114, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 114, + 569, + 136, + 585 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 137, + 573, + 147, + 583 + ], + "score": 0.85, + "content": "o _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 569, + 394, + 585 + ], + "score": 1.0, + "content": "denoting the probability as determined by NN that image", + "type": "text" + }, + { + "bbox": [ + 394, + 570, + 401, + 581 + ], + "score": 0.83, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 569, + 473, + 585 + ], + "score": 1.0, + "content": "belongs to class", + "type": "text" + }, + { + "bbox": [ + 473, + 572, + 479, + 583 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 569, + 505, + 585 + ], + "score": 1.0, + "content": ". Our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 582, + 334, + 594 + ], + "spans": [ + { + "bbox": [ + 115, + 582, + 334, + 594 + ], + "score": 1.0, + "content": "local-search procedure aims to minimize this function.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 99, + 596, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 97, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 97, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "(b) Second, we consider how to form a neighborhood set of images. As mentioned above, the local-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 607, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 114, + 608, + 281, + 622 + ], + "score": 1.0, + "content": "search procedure operates in rounds. Let", + "type": "text" + }, + { + "bbox": [ + 281, + 607, + 300, + 621 + ], + "score": 0.93, + "content": "\\hat { I } _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 608, + 402, + 622 + ], + "score": 1.0, + "content": "be the image after round", + "type": "text" + }, + { + "bbox": [ + 402, + 609, + 424, + 619 + ], + "score": 0.81, + "content": "i - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 608, + 506, + 622 + ], + "score": 1.0, + "content": ". 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For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 114, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 114, + 667, + 174, + 680 + ], + "score": 1.0, + "content": "the first round", + "type": "text" + }, + { + "bbox": [ + 174, + 667, + 218, + 679 + ], + "score": 0.93, + "content": "( P _ { X } , P _ { Y } ) _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "is randomly generated. At each subsequent round, it is formed based on", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 677, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 113, + 677, + 406, + 692 + ], + "score": 1.0, + "content": "a set of pixel locations which were perturbed in the previous round. 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The results are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "again for good images from a set of 1000 randomly selected images. 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For some input image", + "type": "text" + }, + { + "bbox": [ + 496, + 503, + 502, + 514 + ], + "score": 0.81, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 504, + 506, + 516 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 515, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 115, + 516, + 235, + 529 + ], + "score": 1.0, + "content": "we use the objective function", + "type": "text" + }, + { + "bbox": [ + 236, + 515, + 268, + 530 + ], + "score": 0.94, + "content": "f _ { c ( I ) } ( \\hat { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "which equals the probability assigned by the network NN", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 529, + 356, + 544 + ], + "spans": [ + { + "bbox": [ + 114, + 529, + 197, + 544 + ], + "score": 1.0, + "content": "that the input image", + "type": "text" + }, + { + "bbox": [ + 198, + 529, + 204, + 540 + ], + "score": 0.81, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 529, + 271, + 544 + ], + "score": 1.0, + "content": "belongs to class", + "type": "text" + }, + { + "bbox": [ + 271, + 530, + 289, + 542 + ], + "score": 0.9, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 529, + 356, + 544 + ], + "score": 1.0, + "content": ". More formally,", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 98, + 490, + 506, + 544 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 547, + 404, + 564 + ], + "lines": [ + { + "bbox": [ + 214, + 547, + 404, + 564 + ], + "spans": [ + { + "bbox": [ + 214, + 547, + 404, + 564 + ], + "score": 0.91, + "content": "f _ { c ( I ) } ( \\hat { I } ) = o _ { c ( I ) } \\mathrm { w h e r e } \\mathrm { N N } ( \\hat { I } ) = ( o _ { 1 } , \\dots , o _ { C } ) ,", + "type": "interline_equation", + "image_path": "d3beeb6ee5e9e695c0fde6d8d5c55aac1602ffb5eb31d7d45e0c13de7a87b8dc.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 214, + 547, + 404, + 564 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 112, + 570, + 503, + 594 + ], + "lines": [ + { + "bbox": [ + 114, + 569, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 114, + 569, + 136, + 585 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 137, + 573, + 147, + 583 + ], + "score": 0.85, + "content": "o _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 569, + 394, + 585 + ], + "score": 1.0, + "content": "denoting the probability as determined by NN that image", + "type": "text" + }, + { + "bbox": [ + 394, + 570, + 401, + 581 + ], + "score": 0.83, + "content": "\\hat { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 569, + 473, + 585 + ], + "score": 1.0, + "content": "belongs to class", + "type": "text" + }, + { + "bbox": [ + 473, + 572, + 479, + 583 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 569, + 505, + 585 + ], + "score": 1.0, + "content": ". Our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 582, + 334, + 594 + ], + "spans": [ + { + "bbox": [ + 115, + 582, + 334, + 594 + ], + "score": 1.0, + "content": "local-search procedure aims to minimize this function.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 114, + 569, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 99, + 596, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 97, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 97, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "(b) Second, we consider how to form a neighborhood set of images. As mentioned above, the local-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 114, + 607, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 114, + 608, + 281, + 622 + ], + "score": 1.0, + "content": "search procedure operates in rounds. Let", + "type": "text" + }, + { + "bbox": [ + 281, + 607, + 300, + 621 + ], + "score": 0.93, + "content": "\\hat { I } _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 608, + 402, + 622 + ], + "score": 1.0, + "content": "be the image after round", + "type": "text" + }, + { + "bbox": [ + 402, + 609, + 424, + 619 + ], + "score": 0.81, + "content": "i - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 608, + 506, + 622 + ], + "score": 1.0, + "content": ". Our neighborhood", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 114, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 114, + 621, + 393, + 634 + ], + "score": 1.0, + "content": "will consist of images that are different in one pixel from the image", + "type": "text" + }, + { + "bbox": [ + 393, + 620, + 412, + 633 + ], + "score": 0.92, + "content": "\\hat { I } _ { i - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 621, + 505, + 634 + ], + "score": 1.0, + "content": ". 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The value of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 115, + 452, + 121, + 459 + ], + "score": 0.67, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "is kept small in the range [0, 2]. On the other hand, we do not put any explicit restrictions on the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 115, + 461, + 149, + 473 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 149, + 462, + 156, + 472 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 461, + 235, + 473 + ], + "score": 1.0, + "content": ". The best choice of", + "type": "text" + }, + { + "bbox": [ + 235, + 462, + 242, + 472 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "will be one that facilitates the identification of the “best” pixels to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 114, + 471, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 114, + 471, + 371, + 484 + ], + "score": 1.0, + "content": "perturb in each round. In our experiments, we adjust the value of", + "type": "text" + }, + { + "bbox": [ + 371, + 473, + 378, + 483 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 471, + 507, + 484 + ], + "score": 1.0, + "content": "automatically during the search.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 481, + 326, + 496 + ], + "spans": [ + { + "bbox": [ + 114, + 481, + 326, + 496 + ], + "score": 1.0, + "content": "We defer this discussion to the experimental section.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 504, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 516 + ], + "score": 1.0, + "content": "Algorithm LOCSEARCHADV shows the complete pseudocode of our local-search procedure. At the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "high level, the algorithm takes an image as input, and in each round, finds some pixel locations to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "perturb using the above defined objective function and then applies the above defined transformation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "function to these selected pixels to construct a new (perturbed) image. It terminates if it succeeds", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 234, + 561 + ], + "score": 1.0, + "content": "to push the true label below the", + "type": "text" + }, + { + "bbox": [ + 234, + 549, + 241, + 558 + ], + "score": 0.26, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "th place in the confidence score vector at any round. Otherwise, it", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 289, + 572 + ], + "score": 1.0, + "content": "proceeds to the next round (for a maximum of", + "type": "text" + }, + { + "bbox": [ + 290, + 560, + 299, + 569 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "rounds). 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In round", + "type": "text" + }, + { + "bbox": [ + 263, + 582, + 267, + 591 + ], + "score": 0.64, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 581, + 505, + 594 + ], + "score": 1.0, + "content": ", we query the network at most the number of times as the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 188, + 605 + ], + "score": 1.0, + "content": "number of pixels in", + "type": "text" + }, + { + "bbox": [ + 188, + 592, + 232, + 604 + ], + "score": 0.93, + "content": "( P _ { X } , P _ { Y } ) _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 591, + 381, + 605 + ], + "score": 1.0, + "content": "which after the first round is at most", + "type": "text" + }, + { + "bbox": [ + 382, + 592, + 431, + 603 + ], + "score": 0.92, + "content": "2 d \\times 2 d \\times t", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "(again in practice", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 603, + 391, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 391, + 616 + ], + "score": 1.0, + "content": "this is much less because of the overlaps in the neighborhood squares).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "In Section 6, we demonstrate the efficacy of Algorithm LOCSEARCHADV in constructing adversarial", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "images. 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The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 114, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 114, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "perturbation is performed in a cyclic way (as explained in Algorithm CYCLIC) so that we make", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 114, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 114, + 415, + 244, + 428 + ], + "score": 1.0, + "content": "sure that all coordinate values in", + "type": "text" + }, + { + "bbox": [ + 244, + 414, + 253, + 426 + ], + "score": 0.89, + "content": "\\hat { I } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "are within the valid bounds of LB and UB. Note that at the end", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 425, + 357, + 439 + ], + "spans": [ + { + "bbox": [ + 115, + 426, + 176, + 439 + ], + "score": 1.0, + "content": "of every round", + "type": "text" + }, + { + "bbox": [ + 176, + 426, + 181, + 437 + ], + "score": 0.54, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 426, + 184, + 439 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 185, + 425, + 194, + 438 + ], + "score": 0.84, + "content": "\\hat { I } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 426, + 347, + 439 + ], + "score": 1.0, + "content": "is a valid image from the image space", + "type": "text" + }, + { + "bbox": [ + 348, + 427, + 353, + 437 + ], + "score": 0.47, + "content": "\\mathbb { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 426, + 357, + 439 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 114, + 377, + 506, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 115, + 439, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 124, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 124, + 439, + 276, + 451 + ], + "score": 1.0, + "content": "We want to point out that the function", + "type": "text" + }, + { + "bbox": [ + 276, + 441, + 282, + 450 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 439, + 418, + 451 + ], + "score": 1.0, + "content": "uses two perturbation parameters,", + "type": "text" + }, + { + "bbox": [ + 419, + 441, + 425, + 450 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 439, + 442, + 451 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 443, + 441, + 449, + 448 + ], + "score": 0.74, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 439, + 505, + 451 + ], + "score": 1.0, + "content": ". The value of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 115, + 452, + 121, + 459 + ], + "score": 0.67, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "is kept small in the range [0, 2]. On the other hand, we do not put any explicit restrictions on the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 461, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 115, + 461, + 149, + 473 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 149, + 462, + 156, + 472 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 461, + 235, + 473 + ], + "score": 1.0, + "content": ". The best choice of", + "type": "text" + }, + { + "bbox": [ + 235, + 462, + 242, + 472 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "will be one that facilitates the identification of the “best” pixels to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 114, + 471, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 114, + 471, + 371, + 484 + ], + "score": 1.0, + "content": "perturb in each round. In our experiments, we adjust the value of", + "type": "text" + }, + { + "bbox": [ + 371, + 473, + 378, + 483 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 471, + 507, + 484 + ], + "score": 1.0, + "content": "automatically during the search.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 481, + 326, + 496 + ], + "spans": [ + { + "bbox": [ + 114, + 481, + 326, + 496 + ], + "score": 1.0, + "content": "We defer this discussion to the experimental section.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 114, + 439, + 507, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 504, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 516 + ], + "score": 1.0, + "content": "Algorithm LOCSEARCHADV shows the complete pseudocode of our local-search procedure. At the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "high level, the algorithm takes an image as input, and in each round, finds some pixel locations to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "perturb using the above defined objective function and then applies the above defined transformation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "function to these selected pixels to construct a new (perturbed) image. It terminates if it succeeds", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 234, + 561 + ], + "score": 1.0, + "content": "to push the true label below the", + "type": "text" + }, + { + "bbox": [ + 234, + 549, + 241, + 558 + ], + "score": 0.26, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "th place in the confidence score vector at any round. Otherwise, it", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 289, + 572 + ], + "score": 1.0, + "content": "proceeds to the next round (for a maximum of", + "type": "text" + }, + { + "bbox": [ + 290, + 560, + 299, + 569 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "rounds). Note that the number of pixels in an image", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 325, + 583 + ], + "score": 1.0, + "content": "perturbed by Algorithm LOCSEARCHADV is at most", + "type": "text" + }, + { + "bbox": [ + 326, + 571, + 351, + 581 + ], + "score": 0.91, + "content": "t \\times R", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "and in practice (see Tables 4, 5,and 6", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 262, + 594 + ], + "score": 1.0, + "content": "in Section 6) it is much less. In round", + "type": "text" + }, + { + "bbox": [ + 263, + 582, + 267, + 591 + ], + "score": 0.64, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 581, + 505, + 594 + ], + "score": 1.0, + "content": ", we query the network at most the number of times as the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 591, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 188, + 605 + ], + "score": 1.0, + "content": "number of pixels in", + "type": "text" + }, + { + "bbox": [ + 188, + 592, + 232, + 604 + ], + "score": 0.93, + "content": "( P _ { X } , P _ { Y } ) _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 591, + 381, + 605 + ], + "score": 1.0, + "content": "which after the first round is at most", + "type": "text" + }, + { + "bbox": [ + 382, + 592, + 431, + 603 + ], + "score": 0.92, + "content": "2 d \\times 2 d \\times t", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 591, + 505, + 605 + ], + "score": 1.0, + "content": "(again in practice", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 603, + 391, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 391, + 616 + ], + "score": 1.0, + "content": "this is much less because of the overlaps in the neighborhood squares).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 505, + 506, + 616 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "In Section 6, we demonstrate the efficacy of Algorithm LOCSEARCHADV in constructing adversarial", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "images. We first highlight an interesting connection between the pixels perturbed and their influences", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 641, + 289, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 289, + 656 + ], + "score": 1.0, + "content": "measured by a notion of called saliency map.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 620, + 506, + 656 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "score": 1.0, + "content": "A Relation to Saliency Maps. Simonyan et al. (2014) introduced the notion of saliency map as", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 675, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 687 + ], + "score": 1.0, + "content": "a way to rank pixels of the original images w.r.t. their influence on the output of the network. The", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "score": 1.0, + "content": "intuition is that influential pixels in the saliency map are more likely to be important pixels that", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 696, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 506, + 710 + ], + "score": 1.0, + "content": "represent objects and, for example, can be used for weakly supervised object localization. Formally, let", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 706, + 507, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 150, + 720 + ], + "score": 0.93, + "content": "\\mathrm { N } \\mathrm { \\bar { N } } _ { c ( I ) } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 706, + 321, + 721 + ], + "score": 1.0, + "content": "denote the probability assigned to true class", + "type": "text" + }, + { + "bbox": [ + 321, + 708, + 339, + 720 + ], + "score": 0.92, + "content": "c ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 706, + 450, + 721 + ], + "score": 1.0, + "content": "by the network NN on input", + "type": "text" + }, + { + "bbox": [ + 451, + 707, + 503, + 718 + ], + "score": 0.92, + "content": "I \\in \\mathbb { R } ^ { \\ell \\times \\mathbf { \\bar { w } } \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 706, + 507, + 721 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 717, + 508, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 122, + 735 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 720, + 194, + 734 + ], + "score": 0.93, + "content": "\\dot { W _ { c ( I ) } } \\in R ^ { \\ell \\times w \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 717, + 290, + 735 + ], + "score": 1.0, + "content": "denote the derivative of", + "type": "text" + }, + { + "bbox": [ + 291, + 720, + 321, + 734 + ], + "score": 0.91, + "content": "\\mathrm { N N } _ { c ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 717, + 496, + 735 + ], + "score": 1.0, + "content": "with respect to the input evaluated at image", + "type": "text" + }, + { + "bbox": [ + 497, + 721, + 503, + 730 + ], + "score": 0.75, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 717, + 508, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 104, + 663, + 508, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 104, + 137, + 504, + 252 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 104, + 137, + 504, + 252 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 137, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 504, + 252 + ], + "score": 0.395, + "html": "
Algorithm 2 CYCLIC (r,b,x,y)
Assumptions:Perturbation parameter r ∈ [O,2] and LB≤O≤UB
Output:Perturbed image value at the coordinate (b,x,y) which lies in the range [LB,UB]
if rI(b,x,y)<LB then return rI(b,x,y)+(UB-LB)
else if rI(b,x,y)>UB then
return rI(b,x,y)-(UB-LB)
else
return rI(b,x,y)
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Algorithm 2 CYCLIC (r,b,x,y)
Assumptions:Perturbation parameter r ∈ [O,2] and LB≤O≤UB
Output:Perturbed image value at the coordinate (b,x,y) which lies in the range [LB,UB]
if rI(b,x,y)<LB then return rI(b,x,y)+(UB-LB)
else if rI(b,x,y)>UB then
return rI(b,x,y)-(UB-LB)
else
return rI(b,x,y)
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Pixels with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "higher scores are considered more influential. In subsequent works, this notion has been extended to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "adversarial saliency maps that can be useful in generating adversarial perturbations (Papernot et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 140, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 140, + 141 + ], + "score": 1.0, + "content": "2016c).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 506, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 157 + ], + "score": 1.0, + "content": "Computing the exact saliency scores for an image requires complete access to the network NN, which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 507, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 507, + 168 + ], + "score": 1.0, + "content": "we do not assume. However, a natural hypothesis is that the pixels selected by Algorithm LOC-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 167, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 506, + 179 + ], + "score": 1.0, + "content": "SEARCHADV for perturbation are related to pixels with large saliency scores. We use the Ima-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 191 + ], + "score": 1.0, + "content": "geNet1000 dataset to test this hypothesis. In Figure 3, we present some qualitative results. As can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "seen from the pictures, the pixels perturbed by Algorithm LOCSEARCHADV appear correlated with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 485, + 212 + ], + "score": 1.0, + "content": "pixels with high saliency scores. Quantitatively, we observed that the pixels that occupy top-", + "type": "text" + }, + { + "bbox": [ + 485, + 199, + 505, + 210 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 507, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 316, + 223 + ], + "score": 1.0, + "content": "of the saliency map, on average contain more than", + "type": "text" + }, + { + "bbox": [ + 316, + 210, + 337, + 221 + ], + "score": 0.88, + "content": "23 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 210, + 507, + 223 + ], + "score": 1.0, + "content": "of the pixels chosen by Algorithm LOC-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "score": 1.0, + "content": "SEARCHADV for perturbation (and this overlap only grows when we consider a bigger chunk of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 231, + 507, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 231, + 507, + 246 + ], + "score": 1.0, + "content": "pixels picked by their saliency scores). Note that this is correlation is not though a random occurrence.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 164, + 257 + ], + "score": 1.0, + "content": "For an image", + "type": "text" + }, + { + "bbox": [ + 164, + 244, + 171, + 253 + ], + "score": 0.64, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 242, + 188, + 257 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 188, + 243, + 200, + 254 + ], + "score": 0.87, + "content": "S _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 242, + 311, + 257 + ], + "score": 1.0, + "content": "denote the set of pixels in", + "type": "text" + }, + { + "bbox": [ + 311, + 244, + 318, + 253 + ], + "score": 0.65, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 242, + 422, + 257 + ], + "score": 1.0, + "content": "that rank among the top-", + "type": "text" + }, + { + "bbox": [ + 422, + 243, + 442, + 254 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 242, + 506, + 257 + ], + "score": 1.0, + "content": "in the saliency", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "map. If we pick a random set of around 200 pixels (this is on average number of pixels perturbed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 473, + 277 + ], + "score": 1.0, + "content": "per image by Algorithm LOCSEARCHADV perturbs, see Table 5), we expect only about", + "type": "text" + }, + { + "bbox": [ + 473, + 265, + 493, + 275 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "t o", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 199, + 288 + ], + "score": 1.0, + "content": "them to intersect with", + "type": "text" + }, + { + "bbox": [ + 199, + 276, + 211, + 287 + ], + "score": 0.88, + "content": "S _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 276, + 472, + 288 + ], + "score": 1.0, + "content": "and standard tail bounds show that the probability that at least", + "type": "text" + }, + { + "bbox": [ + 473, + 276, + 492, + 286 + ], + "score": 0.86, + "content": "23 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 507, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 287, + 300 + ], + "score": 1.0, + "content": "the pixels of this random set intersects with", + "type": "text" + }, + { + "bbox": [ + 288, + 287, + 299, + 298 + ], + "score": 0.87, + "content": "S _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 286, + 507, + 300 + ], + "score": 1.0, + "content": "is extremely small.9 Therefore, it appears that Al-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "gorithm LOCSEARCHADV rediscovers part of the high salient score pixels but without explicitly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 208, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 208, + 322 + ], + "score": 1.0, + "content": "computing the gradients.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 332, + 276, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 278, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 278, + 346 + ], + "score": 1.0, + "content": "6 EXPERIMENTAL EVALUATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "We start by describing our experimental setup. We used Caffe and Torch machine learning frameworks", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "to train the networks. All algorithms to generate adversarial images were implemented in Lua within", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 492, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 492, + 390 + ], + "score": 1.0, + "content": "Torch 7. All experiments were performed on a cluster of GPUs using a single GPU for each run.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 397, + 503, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Datasets. We use 5 popular datasets: MNIST (handwritten digits recognition dataset), CIFAR10", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "(objects recognition dataset), SVHN (digits recognition dataset), STL10 (objects recognition dataset),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 420, + 303, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 303, + 432 + ], + "score": 1.0, + "content": "and ImageNet1000 (objects recognition dataset).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 507, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 507, + 455 + ], + "score": 1.0, + "content": "Models. We trained Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 450, + 507, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 507, + 465 + ], + "score": 1.0, + "content": "2014) for MNIST, CIFAR, SVHN, STL10, with minor adjustments for the corresponding image sizes.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "Network-in-Network is a building block of the commonly used GoogLeNet architecture that has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "demonstrated very good performance on medium size datasets, e.g. CIFAR10 (Zagoruyko, 2015).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "score": 1.0, + "content": "VGG is another powerful network that proved to be useful in many applications beyond image", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "classification, like object localization (Ren et al., 2015). We trained each model in two variants: with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "and without batch normalization (Ioffe & Szegedy, 2015). Batch normalization was placed before", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "a ReLU layer in all networks. For the ImageNet1000 dataset, we used pre-trained VGG models", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "from (Chatfield et al., 2014b) (we did not train them from scratch due to limited resources). All", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "Caffe VGG models were converted to Torch models using the loadcaffe package (Zagoruyko, 2016a).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "These models use different normalization procedures which we reproduced for each model based", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "on provided descriptions. Tables 4 and 5 (the second column ERRTOP-1) show the top-1 (base)", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "score": 1.0, + "content": "error for all datasets and models that we considered. The results are comparable with the known", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 583, + 342, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 342, + 595 + ], + "score": 1.0, + "content": "state-of-the-art results on these datasets (Benenson, 2016).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "Related Techniques. There are quite a few approaches for generating adversarial images (as", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "discussed in Section 2). Most of these approaches require access to the network architecture and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 638 + ], + "score": 1.0, + "content": "its parameter values (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016;", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "Papernot et al., 2016c).10 The general idea behind these attacks is based on the evaluating the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "network’s sensitivity to the input components in order to determine a perturbation that achieves", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "the adversarial misclassification goal. Among these approaches, the attack approach (known as", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 679, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 676, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 117, + 676, + 506, + 693 + ], + "score": 1.0, + "content": "9We can also use here a standard hypothesis testing for a proportion. The null-hypothesis is that the", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 446, + 701 + ], + "score": 1.0, + "content": "probability of intersection equals 0.1 as with random Bernoulli trails, and test statistic", + "type": "text" + }, + { + "bbox": [ + 446, + 689, + 505, + 700 + ], + "score": 0.88, + "content": "Z ~ = ~ ( 0 . 2 3 ~ -", + "type": "inline_equation" + } + ] + }, + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 241, + 712 + ], + "score": 0.89, + "content": "0 . 1 ) / \\sqrt { ( 0 . 1 ) ( 1 - 0 . 1 ) / 2 0 0 } = 6 . 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "indicates that the null-hypothesis can be rejected at significance level", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 710, + 128, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 124, + 720 + ], + "score": 0.37, + "content": "0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 710, + 128, + 722 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + }, + { + "bbox": [ + 115, + 718, + 424, + 735 + ], + "spans": [ + { + "bbox": [ + 115, + 718, + 424, + 735 + ], + "score": 1.0, + "content": "10Therefore, not entirely suited for a direct comparison with our black-box approach.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 104, + 78, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 78, + 190, + 98 + ], + "score": 1.0, + "content": "The saliency map of", + "type": "text" + }, + { + "bbox": [ + 191, + 83, + 197, + 92 + ], + "score": 0.75, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 78, + 251, + 98 + ], + "score": 1.0, + "content": "is the matrix", + "type": "text" + }, + { + "bbox": [ + 252, + 82, + 300, + 93 + ], + "score": 0.92, + "content": "M \\in \\mathbb { R } ^ { w \\times h }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 78, + 340, + 98 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 340, + 82, + 474, + 95 + ], + "score": 0.81, + "content": "\\begin{array} { r } { M _ { i , j } = \\operatorname* { m a x } _ { b \\in [ \\ell ] } \\ | W _ { c ( I ) } ( b , x , y ) | } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 78, + 506, + 98 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 107, + 94, + 163, + 107 + ], + "score": 0.92, + "content": "W _ { c ( I ) } ( b , x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 93, + 235, + 108 + ], + "score": 1.0, + "content": "is the element of", + "type": "text" + }, + { + "bbox": [ + 236, + 95, + 260, + 107 + ], + "score": 0.92, + "content": "W _ { c ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 93, + 367, + 108 + ], + "score": 1.0, + "content": "corresponding to channel", + "type": "text" + }, + { + "bbox": [ + 368, + 95, + 374, + 105 + ], + "score": 0.62, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 93, + 428, + 108 + ], + "score": 1.0, + "content": "and location", + "type": "text" + }, + { + "bbox": [ + 428, + 95, + 452, + 106 + ], + "score": 0.9, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 93, + 505, + 108 + ], + "score": 1.0, + "content": ". Pixels with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "higher scores are considered more influential. In subsequent works, this notion has been extended to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "adversarial saliency maps that can be useful in generating adversarial perturbations (Papernot et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 140, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 140, + 141 + ], + "score": 1.0, + "content": "2016c).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 78, + 506, + 141 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 506, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 506, + 157 + ], + "score": 1.0, + "content": "Computing the exact saliency scores for an image requires complete access to the network NN, which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 507, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 507, + 168 + ], + "score": 1.0, + "content": "we do not assume. However, a natural hypothesis is that the pixels selected by Algorithm LOC-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 167, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 506, + 179 + ], + "score": 1.0, + "content": "SEARCHADV for perturbation are related to pixels with large saliency scores. We use the Ima-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 191 + ], + "score": 1.0, + "content": "geNet1000 dataset to test this hypothesis. In Figure 3, we present some qualitative results. As can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "seen from the pictures, the pixels perturbed by Algorithm LOCSEARCHADV appear correlated with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 485, + 212 + ], + "score": 1.0, + "content": "pixels with high saliency scores. Quantitatively, we observed that the pixels that occupy top-", + "type": "text" + }, + { + "bbox": [ + 485, + 199, + 505, + 210 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 210, + 507, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 316, + 223 + ], + "score": 1.0, + "content": "of the saliency map, on average contain more than", + "type": "text" + }, + { + "bbox": [ + 316, + 210, + 337, + 221 + ], + "score": 0.88, + "content": "23 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 210, + 507, + 223 + ], + "score": 1.0, + "content": "of the pixels chosen by Algorithm LOC-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "score": 1.0, + "content": "SEARCHADV for perturbation (and this overlap only grows when we consider a bigger chunk of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 231, + 507, + 246 + ], + "spans": [ + { + "bbox": [ + 104, + 231, + 507, + 246 + ], + "score": 1.0, + "content": "pixels picked by their saliency scores). Note that this is correlation is not though a random occurrence.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 164, + 257 + ], + "score": 1.0, + "content": "For an image", + "type": "text" + }, + { + "bbox": [ + 164, + 244, + 171, + 253 + ], + "score": 0.64, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 242, + 188, + 257 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 188, + 243, + 200, + 254 + ], + "score": 0.87, + "content": "S _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 242, + 311, + 257 + ], + "score": 1.0, + "content": "denote the set of pixels in", + "type": "text" + }, + { + "bbox": [ + 311, + 244, + 318, + 253 + ], + "score": 0.65, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 242, + 422, + 257 + ], + "score": 1.0, + "content": "that rank among the top-", + "type": "text" + }, + { + "bbox": [ + 422, + 243, + 442, + 254 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 242, + 506, + 257 + ], + "score": 1.0, + "content": "in the saliency", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "map. If we pick a random set of around 200 pixels (this is on average number of pixels perturbed", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 473, + 277 + ], + "score": 1.0, + "content": "per image by Algorithm LOCSEARCHADV perturbs, see Table 5), we expect only about", + "type": "text" + }, + { + "bbox": [ + 473, + 265, + 493, + 275 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "t o", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 199, + 288 + ], + "score": 1.0, + "content": "them to intersect with", + "type": "text" + }, + { + "bbox": [ + 199, + 276, + 211, + 287 + ], + "score": 0.88, + "content": "S _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 276, + 472, + 288 + ], + "score": 1.0, + "content": "and standard tail bounds show that the probability that at least", + "type": "text" + }, + { + "bbox": [ + 473, + 276, + 492, + 286 + ], + "score": 0.86, + "content": "23 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 507, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 287, + 300 + ], + "score": 1.0, + "content": "the pixels of this random set intersects with", + "type": "text" + }, + { + "bbox": [ + 288, + 287, + 299, + 298 + ], + "score": 0.87, + "content": "S _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 286, + 507, + 300 + ], + "score": 1.0, + "content": "is extremely small.9 Therefore, it appears that Al-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "gorithm LOCSEARCHADV rediscovers part of the high salient score pixels but without explicitly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 208, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 208, + 322 + ], + "score": 1.0, + "content": "computing the gradients.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 144, + 507, + 322 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 332, + 276, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 278, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 278, + 346 + ], + "score": 1.0, + "content": "6 EXPERIMENTAL EVALUATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "We start by describing our experimental setup. We used Caffe and Torch machine learning frameworks", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "to train the networks. All algorithms to generate adversarial images were implemented in Lua within", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 492, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 492, + 390 + ], + "score": 1.0, + "content": "Torch 7. All experiments were performed on a cluster of GPUs using a single GPU for each run.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 353, + 505, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 397, + 503, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Datasets. We use 5 popular datasets: MNIST (handwritten digits recognition dataset), CIFAR10", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "(objects recognition dataset), SVHN (digits recognition dataset), STL10 (objects recognition dataset),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 420, + 303, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 303, + 432 + ], + "score": 1.0, + "content": "and ImageNet1000 (objects recognition dataset).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 397, + 505, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 507, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 507, + 455 + ], + "score": 1.0, + "content": "Models. We trained Network-in-Network (Lin et al., 2014) and VGG (Simonyan & Zisserman,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 450, + 507, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 507, + 465 + ], + "score": 1.0, + "content": "2014) for MNIST, CIFAR, SVHN, STL10, with minor adjustments for the corresponding image sizes.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "score": 1.0, + "content": "Network-in-Network is a building block of the commonly used GoogLeNet architecture that has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "demonstrated very good performance on medium size datasets, e.g. CIFAR10 (Zagoruyko, 2015).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 498 + ], + "score": 1.0, + "content": "VGG is another powerful network that proved to be useful in many applications beyond image", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "classification, like object localization (Ren et al., 2015). We trained each model in two variants: with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "and without batch normalization (Ioffe & Szegedy, 2015). Batch normalization was placed before", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "a ReLU layer in all networks. For the ImageNet1000 dataset, we used pre-trained VGG models", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "from (Chatfield et al., 2014b) (we did not train them from scratch due to limited resources). All", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "Caffe VGG models were converted to Torch models using the loadcaffe package (Zagoruyko, 2016a).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "These models use different normalization procedures which we reproduced for each model based", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "on provided descriptions. Tables 4 and 5 (the second column ERRTOP-1) show the top-1 (base)", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "score": 1.0, + "content": "error for all datasets and models that we considered. The results are comparable with the known", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 583, + 342, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 342, + 595 + ], + "score": 1.0, + "content": "state-of-the-art results on these datasets (Benenson, 2016).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 438, + 507, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "Related Techniques. There are quite a few approaches for generating adversarial images (as", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "discussed in Section 2). Most of these approaches require access to the network architecture and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 638 + ], + "score": 1.0, + "content": "its parameter values (Szegedy et al., 2014; Goodfellow et al., 2015; Moosavi-Dezfooli et al., 2016;", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "Papernot et al., 2016c).10 The general idea behind these attacks is based on the evaluating the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "network’s sensitivity to the input components in order to determine a perturbation that achieves", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "the adversarial misclassification goal. Among these approaches, the attack approach (known as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "the “fast-gradient sign method”) suggested by Goodfellow et al. (2015) stands out for being able to", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "efficiently generate adversarial images. Here we compare the performance of our local-search based", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 531, + 292, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 292, + 543 + ], + "score": 1.0, + "content": "attack against this fast-gradient sign method.11", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 604, + 506, + 672 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 114, + 81, + 497, + 412 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 81, + 497, + 412 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 81, + 497, + 412 + ], + "spans": [ + { + "bbox": [ + 114, + 81, + 497, + 412 + ], + "score": 0.977, + "type": "image", + "image_path": "6a4d9a530eb9e0cd10fe492164133a18e40afd6afa68fbfcd68a866b16882a02.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 114, + 81, + 497, + 191.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 114, + 191.33333333333331, + 497, + 301.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 301.66666666666663, + 497, + 411.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 421, + 506, + 487 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 420, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 507, + 433 + ], + "score": 1.0, + "content": "Table 3: Results on ImageNet1000 using VGG CNN-S (Caffe) network (Chatfield et al., 2014a).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 430, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 446 + ], + "score": 1.0, + "content": "Columns from left to right: the original image, top 150 pixels chosen according to their saliency", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "scores (in white), the absolute difference between the perturbed image and the true image (the pixels", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 455, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 466 + ], + "score": 1.0, + "content": "that are perturbed appear in white), and the perturbed image. Adversarial misclassification (rows", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "from top to bottom): a ruffed grouse misclassified as a frilled lizard, an artichoke misclassified as a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 477, + 457, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 457, + 487 + ], + "score": 1.0, + "content": "sleeping bag, a bubble misclassified as a fountain, and a hare misclassified as a cheetah.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "the “fast-gradient sign method”) suggested by Goodfellow et al. (2015) stands out for being able to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "efficiently generate adversarial images. Here we compare the performance of our local-search based", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 531, + 292, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 292, + 543 + ], + "score": 1.0, + "content": "attack against this fast-gradient sign method.11", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 560 + ], + "score": 1.0, + "content": "For completeness, we now briefly explain the fast-gradient sign method of Goodfellow et al. (2015).", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 176, + 572 + ], + "score": 1.0, + "content": "Given an image", + "type": "text" + }, + { + "bbox": [ + 176, + 560, + 186, + 570 + ], + "score": 0.86, + "content": "I _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 559, + 223, + 572 + ], + "score": 1.0, + "content": ", a label", + "type": "text" + }, + { + "bbox": [ + 223, + 559, + 290, + 571 + ], + "score": 0.94, + "content": "a \\in \\{ 1 , \\ldots , C \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 559, + 506, + 572 + ], + "score": 1.0, + "content": ", and a network NN, the fast-gradient sign method", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 101, + 568, + 509, + 588 + ], + "spans": [ + { + "bbox": [ + 101, + 568, + 142, + 588 + ], + "score": 1.0, + "content": "perturbs", + "type": "text" + }, + { + "bbox": [ + 142, + 572, + 153, + 583 + ], + "score": 0.85, + "content": "I _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 568, + 289, + 588 + ], + "score": 1.0, + "content": "using the following update rule:", + "type": "text" + }, + { + "bbox": [ + 289, + 571, + 475, + 584 + ], + "score": 0.89, + "content": "I _ { 0 } ^ { \\mathrm { p e r t } } = I _ { 0 } + \\epsilon \\cdot \\mathrm { s i g n } ( \\nabla _ { I = I _ { 0 } } \\mathrm { L o s s } ( \\mathbf { N N } ( I ) , a ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 568, + 509, + 588 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 581, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 225, + 595 + ], + "score": 0.82, + "content": "\\operatorname { s i g n } ( \\nabla _ { I = I _ { 0 } } \\operatorname { L o s s } ( \\mathbf { N N } ( I ) , a ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 581, + 441, + 597 + ], + "score": 1.0, + "content": "is the sign of the network’s cost function gradient (here", + "type": "text" + }, + { + "bbox": [ + 442, + 583, + 505, + 595 + ], + "score": 0.82, + "content": "\\mathrm { L o s s } ( \\mathbf { N N } ( I ) , a )", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 333, + 606 + ], + "score": 1.0, + "content": "denotes the loss function of the network NN given input", + "type": "text" + }, + { + "bbox": [ + 334, + 594, + 340, + 604 + ], + "score": 0.68, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 594, + 380, + 606 + ], + "score": 1.0, + "content": "and class", + "type": "text" + }, + { + "bbox": [ + 380, + 596, + 387, + 604 + ], + "score": 0.71, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 594, + 429, + 606 + ], + "score": 1.0, + "content": "). We vary", + "type": "text" + }, + { + "bbox": [ + 430, + 596, + 436, + 604 + ], + "score": 0.7, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "over all possible", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 618 + ], + "score": 1.0, + "content": "labels in the dataset and choose the best result where this procedure is successful in generating an", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 615, + 504, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 332, + 629 + ], + "score": 1.0, + "content": "adversarial image. Without general guidelines for setting", + "type": "text" + }, + { + "bbox": [ + 332, + 618, + 338, + 626 + ], + "score": 0.51, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 615, + 498, + 629 + ], + "score": 1.0, + "content": ", we experimented with several values of", + "type": "text" + }, + { + "bbox": [ + 498, + 618, + 504, + 626 + ], + "score": 0.63, + "content": "\\epsilon", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 396, + 639 + ], + "score": 1.0, + "content": "starting from 0.07 and increasing this number. We found that the value", + "type": "text" + }, + { + "bbox": [ + 396, + 626, + 435, + 637 + ], + "score": 0.92, + "content": "\\epsilon = 0 . 2 ^ { 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "was the smallest", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "value where the fast-gradient sign method started to yield competitive performance compared to our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 224, + 662 + ], + "score": 1.0, + "content": "algorithm. Smaller values of", + "type": "text" + }, + { + "bbox": [ + 224, + 650, + 230, + 658 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 648, + 455, + 662 + ], + "score": 1.0, + "content": "leads to generation of fewer adversarial images, e.g., at", + "type": "text" + }, + { + "bbox": [ + 455, + 649, + 486, + 659 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 648, + 506, + 662 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 371, + 672 + ], + "score": 1.0, + "content": "percentage of generated adversarial images is reduced by around", + "type": "text" + }, + { + "bbox": [ + 372, + 660, + 391, + 670 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "as compared to the value at", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 140, + 681 + ], + "score": 0.88, + "content": "\\epsilon = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 671, + 463, + 682 + ], + "score": 1.0, + "content": "for the CIFAR10 dataset on the Network-in-Network model. 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Another", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "important point to remember is that unlike the fast-gradient sign method, our approach is based on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "a weaker and more realistic assumption on the adversarial power, making our attacks more widely", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 152, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 152, + 149 + ], + "score": 1.0, + "content": "applicable.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 158, + 506, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 384, + 172 + ], + "score": 1.0, + "content": "Implementing Algorithm LOCSEARCHADV. 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At each round,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 205, + 275 + ], + "score": 1.0, + "content": "we increase the value of", + "type": "text" + }, + { + "bbox": [ + 206, + 263, + 212, + 273 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 261, + 222, + 275 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 222, + 262, + 242, + 273 + ], + "score": 0.87, + "content": "\\bar { o } _ { c ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 261, + 357, + 275 + ], + "score": 1.0, + "content": "is close to one and decrease", + "type": "text" + }, + { + "bbox": [ + 357, + 263, + 364, + 273 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 261, + 374, + 275 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 374, + 262, + 394, + 274 + ], + "score": 0.91, + "content": "\\bar { o } _ { c ( I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 261, + 506, + 275 + ], + "score": 1.0, + "content": "is low, e.g., below 0.3. For", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 219, + 286 + ], + "score": 1.0, + "content": "Algorithm CYCLIC, we set", + "type": "text" + }, + { + "bbox": [ + 219, + 273, + 254, + 285 + ], + "score": 0.9, + "content": "r = 3 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 273, + 505, + 286 + ], + "score": 1.0, + "content": ". To avoid perturbing the most sensitive pixels frequently, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "make sure that if a pixel is perturbed in a round then we exclude it from consideration for the next 30", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 139, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 139, + 307 + ], + "score": 1.0, + "content": "rounds.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "Experimental Observations. For ease of comparison with the fast-gradient sign method (Good-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 216, + 340 + ], + "score": 1.0, + "content": "fellow et al., 2015), we set", + "type": "text" + }, + { + "bbox": [ + 216, + 328, + 241, + 338 + ], + "score": 0.9, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "and focus on achieving 1-misclassification. Tables 4 and 5 show", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "the results of our experiments on the test sets. The first column shows the dataset name. The sec-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 350, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 506, + 362 + ], + "score": 1.0, + "content": "ond column (ERRTOP-1) presents the top-1 misclassification rate on the corresponding test dataset", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "without any perturbation (base error). ERRTOP-1(ADV) is the top-1 misclassification rate where", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "each original image in the test set was replaced with an generated perturbed image (using either our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 382, + 503, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 503, + 394 + ], + "score": 1.0, + "content": "approach or the fast-gradient sign method (Goodfellow et al., 2015) which is denoted as FGSM).13", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 457, + 412 + ], + "score": 1.0, + "content": "In the following, we say an adversarial generation technique ADV, given an input image", + "type": "text" + }, + { + "bbox": [ + 457, + 400, + 464, + 409 + ], + "score": 0.74, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 399, + 505, + 412 + ], + "score": 1.0, + "content": ", succeeds", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 250, + 424 + ], + "score": 1.0, + "content": "in generating an adversarial image", + "type": "text" + }, + { + "bbox": [ + 250, + 410, + 284, + 423 + ], + "score": 0.47, + "content": "\\mathrm { A D V } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 410, + 373, + 424 + ], + "score": 1.0, + "content": "for a network NN iff", + "type": "text" + }, + { + "bbox": [ + 374, + 410, + 456, + 423 + ], + "score": 0.92, + "content": "c ( I ) \\in \\pi ( \\mathrm { N N } ( I ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 410, + 475, + 424 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 476, + 410, + 505, + 423 + ], + "score": 0.9, + "content": "c ( I ) \\notin", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 107, + 421, + 187, + 433 + ], + "score": 0.88, + "content": "\\pi ( \\bar { \\mathrm { N N } } ( \\mathrm { A D V } ( I ) ) , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 420, + 505, + 433 + ], + "score": 1.0, + "content": ". 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More formally, let", + "type": "text" + }, + { + "bbox": [ + 451, + 444, + 460, + 453 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "denote the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 453, + 466, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 153, + 468 + ], + "score": 1.0, + "content": "test set and", + "type": "text" + }, + { + "bbox": [ + 154, + 454, + 202, + 467 + ], + "score": 0.92, + "content": "\\tau _ { \\mathrm { A D V } } \\subseteq \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 453, + 312, + 468 + ], + "score": 1.0, + "content": "denote the set of images in", + "type": "text" + }, + { + "bbox": [ + 312, + 455, + 321, + 464 + ], + "score": 0.83, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 453, + 466, + 468 + ], + "score": 1.0, + "content": "on which ADV is successful. 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The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 549 + ], + "score": 1.0, + "content": "#PTBPIXELS column shows the average percentage of perturbed pixels in the successful adversarial", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "images. Similarly, TIME column shows the average time (in seconds) to generate a successful", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 448, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 448, + 572 + ], + "score": 1.0, + "content": "adversarial image. Finally, the last column indicates the type of network architecture.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 586 + ], + "score": 1.0, + "content": "As is quite evident from these results, Algorithm LOCSEARCHADV is more effective than the fast-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "gradient sign method in generating adversarial images, even without having access to the network", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "architecture and its parameter values. The difference is quite prominent for networks trained with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 621 + ], + "score": 1.0, + "content": "batch normalization as here we noticed that the fast-gradient sign method has difficulties producing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 615, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 506, + 632 + ], + "score": 1.0, + "content": "adversarial images.14 Another advantage with our approach is that it modifies a very tiny fraction", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "of pixels as compared to all the pixels perturbed by the fast-gradient sign method, and also in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "many cases with far less average perturbation. Putting these points together demonstrates that", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 660, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 115, + 658, + 507, + 675 + ], + "spans": [ + { + "bbox": [ + 115, + 658, + 507, + 675 + ], + "score": 1.0, + "content": "13Note that by explicitly constraining the number of pixels that can be perturbed, as we do in our approach,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 232, + 682 + ], + "score": 1.0, + "content": "it might be impossible to get to a", + "type": "text" + }, + { + "bbox": [ + 232, + 671, + 254, + 680 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "misclassification rate on some datasets. Similarly, the fast-gradient", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 680, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 214, + 692 + ], + "score": 1.0, + "content": "sign method fails to achieve a", + "type": "text" + }, + { + "bbox": [ + 215, + 681, + 236, + 690 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 680, + 408, + 692 + ], + "score": 1.0, + "content": "misclassification rate even with larger values of", + "type": "text" + }, + { + "bbox": [ + 408, + 682, + 414, + 690 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 680, + 506, + 692 + ], + "score": 1.0, + "content": "(Moosavi-Dezfooli et al.,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 690, + 132, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 132, + 702 + ], + "score": 1.0, + "content": "2016).", + "type": "text" + } + ] + }, + { + "bbox": [ + 114, + 697, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 114, + 697, + 506, + 716 + ], + "score": 1.0, + "content": "14In general, we observed that models trained with batch normalization are somewhat more resilient to", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "adversarial perturbations probably because of the regularization properties of batch normalization (Ioffe &", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 166, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 166, + 733 + ], + "score": 1.0, + "content": "Szegedy, 2015).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 158, + 506, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 384, + 172 + ], + "score": 1.0, + "content": "Implementing Algorithm LOCSEARCHADV. 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To avoid perturbing the most sensitive pixels frequently, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "make sure that if a pixel is perturbed in a round then we exclude it from consideration for the next 30", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 139, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 139, + 307 + ], + "score": 1.0, + "content": "rounds.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 157, + 507, + 307 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "Experimental Observations. 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The sec-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 350, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 506, + 362 + ], + "score": 1.0, + "content": "ond column (ERRTOP-1) presents the top-1 misclassification rate on the corresponding test dataset", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "without any perturbation (base error). 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Note that the inner summation is measuring the", + "type": "text" + }, + { + "bbox": [ + 335, + 525, + 347, + 535 + ], + "score": 0.86, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 523, + 422, + 537 + ], + "score": 1.0, + "content": "-distance between", + "type": "text" + }, + { + "bbox": [ + 422, + 525, + 429, + 534 + ], + "score": 0.7, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 523, + 447, + 537 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 448, + 524, + 481, + 536 + ], + "score": 0.81, + "content": "\\mathrm { A D V } ( I )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 523, + 505, + 537 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 549 + ], + "score": 1.0, + "content": "#PTBPIXELS column shows the average percentage of perturbed pixels in the successful adversarial", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "images. Similarly, TIME column shows the average time (in seconds) to generate a successful", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 448, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 448, + 572 + ], + "score": 1.0, + "content": "adversarial image. Finally, the last column indicates the type of network architecture.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 510, + 506, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 586 + ], + "score": 1.0, + "content": "As is quite evident from these results, Algorithm LOCSEARCHADV is more effective than the fast-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "gradient sign method in generating adversarial images, even without having access to the network", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "architecture and its parameter values. The difference is quite prominent for networks trained with", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 621 + ], + "score": 1.0, + "content": "batch normalization as here we noticed that the fast-gradient sign method has difficulties producing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 615, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 506, + 632 + ], + "score": 1.0, + "content": "adversarial images.14 Another advantage with our approach is that it modifies a very tiny fraction", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "of pixels as compared to all the pixels perturbed by the fast-gradient sign method, and also in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "many cases with far less average perturbation. Putting these points together demonstrates that", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 575, + 506, + 653 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Algorithm LOCSEARCHADV is successful in generating more adversarial images than the fast-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "gradient sign method, while modifying far fewer pixels and adding less noise per image. On the other", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "side, the fast-gradient sign method takes lesser time in the generation process and generally seems to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 411, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 411, + 128 + ], + "score": 1.0, + "content": "produce higher confidence scores for the adversarial (misclassified) images.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "score": 1.0, + "content": "Table 5 shows the results for several variants of VGG network trained on the ImageNet1000 dataset.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "These networks do not have batch normalization layers (Chatfield et al., 2014b; Zagoruyko, 2016a).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 135, + 167 + ], + "score": 1.0, + "content": "We set", + "type": "text" + }, + { + "bbox": [ + 135, + 155, + 159, + 164 + ], + "score": 0.9, + "content": "\\epsilon = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "for the fast-gradient sign method as a different pre-processing technique was used for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "score": 1.0, + "content": "this network (we converted these networks from pre-trained Caffe models). Results are similar to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "that observed on the smaller datasets. In most cases, our proposed local-search based approach is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 451, + 200 + ], + "score": 1.0, + "content": "more successful in generating adversarial images while on average perturbing less than", + "type": "text" + }, + { + "bbox": [ + 452, + 187, + 479, + 198 + ], + "score": 0.86, + "content": "0 . 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 198, + 136, + 210 + ], + "spans": [ + { + "bbox": [ + 104, + 198, + 136, + 210 + ], + "score": 1.0, + "content": "pixels.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 229, + 506, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 177, + 241 + ], + "score": 1.0, + "content": "Case of Larger", + "type": "text" + }, + { + "bbox": [ + 177, + 230, + 184, + 239 + ], + "score": 0.45, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 229, + 323, + 241 + ], + "score": 1.0, + "content": "’s. We now consider achieving", + "type": "text" + }, + { + "bbox": [ + 323, + 230, + 330, + 239 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 229, + 420, + 241 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 420, + 230, + 451, + 240 + ], + "score": 0.9, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 229, + 506, + 241 + ], + "score": 1.0, + "content": "using Algo-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 506, + 252 + ], + "score": 1.0, + "content": "rithm LOCSEARCHADV. In Table 6, we present the results as we change the goal from 1-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "misclassification to 4-misclassification on the CIFAR10 dataset. We use the same parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 496, + 275 + ], + "score": 1.0, + "content": "as before for Algorithm LOCSEARCHADV. As one would expect, as we increase the value of", + "type": "text" + }, + { + "bbox": [ + 496, + 263, + 502, + 272 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 262, + 506, + 275 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "score": 1.0, + "content": "the effectiveness of the attack decreases, perturbation and time needed increases. But overall our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 479, + 297 + ], + "score": 1.0, + "content": "local-search procedure is still able to generate a large fraction of adversarial images at even", + "type": "text" + }, + { + "bbox": [ + 479, + 284, + 505, + 294 + ], + "score": 0.89, + "content": "k = 4", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "with a small perturbation and computation time, meaning that these images will fool even a system", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 306, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 506, + 318 + ], + "score": 1.0, + "content": "that is evaluated on a top-4 classification criteria. We are not aware of a straightforward extension of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 317, + 451, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 371, + 329 + ], + "score": 1.0, + "content": "the fast-gradient sign method (Goodfellow et al., 2015) to achieve", + "type": "text" + }, + { + "bbox": [ + 371, + 317, + 378, + 327 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 317, + 451, + 329 + ], + "score": 1.0, + "content": "-misclassification.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "Even Weaker Adversarial Models. We also consider a weaker model where the adversary does", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "not even have a black-box (oracle) access to the network (NN) of interest, and has to rely on a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 368, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 384 + ], + "score": 1.0, + "content": "black-box access to somewhat of a “similar” (proxy) network as NN. For example, the adversary", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "score": 1.0, + "content": "might want to evade a spam filter A, but might have to develop adversarial images by utilizing the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 392, + 379, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 379, + 404 + ], + "score": 1.0, + "content": "output of a spam filter B, which might share properties similar to A.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 423 + ], + "score": 1.0, + "content": "We trained several modifications of Network-in-Network model for the CIFAR10 dataset, varying", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "the initial value of the learning rate, the size of filters, and the number of layers in the network. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 196, + 443 + ], + "score": 1.0, + "content": "observed that between", + "type": "text" + }, + { + "bbox": [ + 196, + 431, + 216, + 441 + ], + "score": 0.86, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 430, + 227, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 227, + 431, + 246, + 441 + ], + "score": 0.87, + "content": "43 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "of adversarial images generated by Algorithm LOCSEARCHADV", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 446, + 454 + ], + "score": 1.0, + "content": "using the original network were also adversarial for these modified networks (at", + "type": "text" + }, + { + "bbox": [ + 446, + 442, + 476, + 452 + ], + "score": 0.88, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "). The", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "transferability of adversarial images that we observe here has also been observed with other attacks", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "too (Szegedy et al., 2014; Goodfellow et al., 2015; Papernot et al., 2016b;a) and demonstrates the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 475, + 263, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 263, + 487 + ], + "score": 1.0, + "content": "wider applicability of all these attacks.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 507, + 195, + 520 + ], + "lines": [ + { + "bbox": [ + 104, + 505, + 198, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 198, + 524 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "We investigate the inherent vulnerabilities in modern CNNs to practical black-box adversarial attacks.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "We present approaches that can efficiently locate a small set of pixels, without using any gradient", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "information, which when perturbed lead to misclassification by a deep neural network. Our extensive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "experimental results, somewhat surprisingly, demonstrates the effectiveness of our simple approaches", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 577, + 250, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 250, + 590 + ], + "score": 1.0, + "content": "in generating adversarial examples.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Defenses against these attacks is an interesting research direction. However, we note that here that by", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "limiting the perturbation to some pixels (being localized) the adversarial images generated by our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "local-search based approach do not represent the distribution of the original data. This means for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "these adversarial images, the use of adversarial training (or fine-tuning), a technique of training (or", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "fine-tuning) networks on adversarial images to build more robust classifiers, is not very effective. In", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "fact, even with adversarial training we noticed that the networks ability to resist new local-search", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 398, + 673 + ], + "score": 1.0, + "content": "based adversarial attack improves only marginally (on average between", + "type": "text" + }, + { + "bbox": [ + 398, + 660, + 421, + 671 + ], + "score": 0.84, + "content": "1 - 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "). On the other hand,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "we suspect that one possible counter-measure to these localized adversarial attacks could be based on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "performing a careful analysis of the oracle queries to thwart the attempts to generate an adversarial", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 692, + 136, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 136, + 708 + ], + "score": 1.0, + "content": "image.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Finally, we believe that our local-search approach can also be used for attacks against other machine", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 483, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 483, + 733 + ], + "score": 1.0, + "content": "learning systems and can serve as an useful tool in measuring the robustness of these systems.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Algorithm LOCSEARCHADV is successful in generating more adversarial images than the fast-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "gradient sign method, while modifying far fewer pixels and adding less noise per image. On the other", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "side, the fast-gradient sign method takes lesser time in the generation process and generally seems to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 411, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 411, + 128 + ], + "score": 1.0, + "content": "produce higher confidence scores for the adversarial (misclassified) images.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "score": 1.0, + "content": "Table 5 shows the results for several variants of VGG network trained on the ImageNet1000 dataset.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "These networks do not have batch normalization layers (Chatfield et al., 2014b; Zagoruyko, 2016a).", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 135, + 167 + ], + "score": 1.0, + "content": "We set", + "type": "text" + }, + { + "bbox": [ + 135, + 155, + 159, + 164 + ], + "score": 0.9, + "content": "\\epsilon = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "for the fast-gradient sign method as a different pre-processing technique was used for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "score": 1.0, + "content": "this network (we converted these networks from pre-trained Caffe models). Results are similar to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "that observed on the smaller datasets. In most cases, our proposed local-search based approach is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 104, + 187, + 451, + 200 + ], + "score": 1.0, + "content": "more successful in generating adversarial images while on average perturbing less than", + "type": "text" + }, + { + "bbox": [ + 452, + 187, + 479, + 198 + ], + "score": 0.86, + "content": "0 . 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 198, + 136, + 210 + ], + "spans": [ + { + "bbox": [ + 104, + 198, + 136, + 210 + ], + "score": 1.0, + "content": "pixels.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 131, + 507, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 229, + 506, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 177, + 241 + ], + "score": 1.0, + "content": "Case of Larger", + "type": "text" + }, + { + "bbox": [ + 177, + 230, + 184, + 239 + ], + "score": 0.45, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 229, + 323, + 241 + ], + "score": 1.0, + "content": "’s. We now consider achieving", + "type": "text" + }, + { + "bbox": [ + 323, + 230, + 330, + 239 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 229, + 420, + 241 + ], + "score": 1.0, + "content": "-misclassification for", + "type": "text" + }, + { + "bbox": [ + 420, + 230, + 451, + 240 + ], + "score": 0.9, + "content": "k \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 229, + 506, + 241 + ], + "score": 1.0, + "content": "using Algo-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 506, + 252 + ], + "score": 1.0, + "content": "rithm LOCSEARCHADV. In Table 6, we present the results as we change the goal from 1-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "misclassification to 4-misclassification on the CIFAR10 dataset. We use the same parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 496, + 275 + ], + "score": 1.0, + "content": "as before for Algorithm LOCSEARCHADV. As one would expect, as we increase the value of", + "type": "text" + }, + { + "bbox": [ + 496, + 263, + 502, + 272 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 262, + 506, + 275 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "score": 1.0, + "content": "the effectiveness of the attack decreases, perturbation and time needed increases. But overall our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 479, + 297 + ], + "score": 1.0, + "content": "local-search procedure is still able to generate a large fraction of adversarial images at even", + "type": "text" + }, + { + "bbox": [ + 479, + 284, + 505, + 294 + ], + "score": 0.89, + "content": "k = 4", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "with a small perturbation and computation time, meaning that these images will fool even a system", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 306, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 506, + 318 + ], + "score": 1.0, + "content": "that is evaluated on a top-4 classification criteria. We are not aware of a straightforward extension of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 317, + 451, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 371, + 329 + ], + "score": 1.0, + "content": "the fast-gradient sign method (Goodfellow et al., 2015) to achieve", + "type": "text" + }, + { + "bbox": [ + 371, + 317, + 378, + 327 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 317, + 451, + 329 + ], + "score": 1.0, + "content": "-misclassification.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 229, + 506, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 504, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "Even Weaker Adversarial Models. We also consider a weaker model where the adversary does", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "not even have a black-box (oracle) access to the network (NN) of interest, and has to rely on a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 368, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 384 + ], + "score": 1.0, + "content": "black-box access to somewhat of a “similar” (proxy) network as NN. For example, the adversary", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "score": 1.0, + "content": "might want to evade a spam filter A, but might have to develop adversarial images by utilizing the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 392, + 379, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 379, + 404 + ], + "score": 1.0, + "content": "output of a spam filter B, which might share properties similar to A.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 348, + 506, + 404 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 408, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 423 + ], + "score": 1.0, + "content": "We trained several modifications of Network-in-Network model for the CIFAR10 dataset, varying", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "the initial value of the learning rate, the size of filters, and the number of layers in the network. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 196, + 443 + ], + "score": 1.0, + "content": "observed that between", + "type": "text" + }, + { + "bbox": [ + 196, + 431, + 216, + 441 + ], + "score": 0.86, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 430, + 227, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 227, + 431, + 246, + 441 + ], + "score": 0.87, + "content": "43 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "of adversarial images generated by Algorithm LOCSEARCHADV", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 446, + 454 + ], + "score": 1.0, + "content": "using the original network were also adversarial for these modified networks (at", + "type": "text" + }, + { + "bbox": [ + 446, + 442, + 476, + 452 + ], + "score": 0.88, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "). The", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "transferability of adversarial images that we observe here has also been observed with other attacks", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "too (Szegedy et al., 2014; Goodfellow et al., 2015; Papernot et al., 2016b;a) and demonstrates the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 475, + 263, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 263, + 487 + ], + "score": 1.0, + "content": "wider applicability of all these attacks.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 406, + 505, + 487 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 507, + 195, + 520 + ], + "lines": [ + { + "bbox": [ + 104, + 505, + 198, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 505, + 198, + 524 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "We investigate the inherent vulnerabilities in modern CNNs to practical black-box adversarial attacks.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "We present approaches that can efficiently locate a small set of pixels, without using any gradient", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "information, which when perturbed lead to misclassification by a deep neural network. Our extensive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "experimental results, somewhat surprisingly, demonstrates the effectiveness of our simple approaches", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 577, + 250, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 250, + 590 + ], + "score": 1.0, + "content": "in generating adversarial examples.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 533, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Defenses against these attacks is an interesting research direction. However, we note that here that by", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "limiting the perturbation to some pixels (being localized) the adversarial images generated by our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "local-search based approach do not represent the distribution of the original data. This means for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "these adversarial images, the use of adversarial training (or fine-tuning), a technique of training (or", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "fine-tuning) networks on adversarial images to build more robust classifiers, is not very effective. In", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "fact, even with adversarial training we noticed that the networks ability to resist new local-search", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 398, + 673 + ], + "score": 1.0, + "content": "based adversarial attack improves only marginally (on average between", + "type": "text" + }, + { + "bbox": [ + 398, + 660, + 421, + 671 + ], + "score": 0.84, + "content": "1 - 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "). On the other hand,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "we suspect that one possible counter-measure to these localized adversarial attacks could be based on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "performing a careful analysis of the oracle queries to thwart the attempts to generate an adversarial", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 692, + 136, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 136, + 708 + ], + "score": 1.0, + "content": "image.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 594, + 506, + 708 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Finally, we believe that our local-search approach can also be used for attacks against other machine", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 483, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 483, + 733 + ], + "score": 1.0, + "content": "learning systems and can serve as an useful tool in measuring the robustness of these systems.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 101, + 531, + 377 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 101, + 531, + 377 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 101, + 531, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 531, + 377 + ], + "score": 0.984, + "html": "
DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)TechniqueNetwork
NNs trained with batch normalization
CIFAR1011.6597.630.47 0.550.04 0.203.750.68LOCSEARCHADV (Ours)NinN
CIFAR1070.69
11.6297.510.74100.000.01FGSM(Goodfellow et al., 2015)NinN
CIFAR100.043.160.78LOCSEARCHADV (Ours)VGG
CIFAR1011.621111FGSM(Goodfellow et al.,2015)VGG
STL1058.170.420.021.207.15LOCSEARCHADV (Ours)NinN
STL1029.8154.850.530.20100.000.03FGSM(Goodfellow et al.,2015)NinN
STL1065.760.470.021.1113.90LOCSEARCHADV (Ours)VGG
STL1026.5026.50二1 0.05二二FGSM(Goodfellow et al., 2015)VGG
SVHN SVHN9.7197.06 48.620.47 0.490.204.51 100.001.02 0.02LOCSEARCHADV (Ours) FGSM(Goodfellow et al.,2015)NinN NinN
SVHN81.100.660.07 15.432.15LOCSEARCHADV (Ours)VGG VGG
4.77
SVHN4.771 0.54 0.201 2.241 0.64FGSM(Goodfellow et al.,2015)NinN
MNIST91.42
MNIST0.331.650.580.20100.000.02LOCSEARCHADV (Ours) FGSM (Goodfellow et al., 2015)
93.480.630.21 2.200.64LOCSEARCHADV(Ours)NinN VGG
MNIST MNIST0.440.44111 NNs trained without batch normalization1FGSM(Goodfellow et al., 2015)VGG
97.8916.540.720.043.240.58LOCSEARCHADV (Ours)NinN
CIFAR10 CIFAR1093.670.930.20100.000.02FGSM(Goodfellow et al., 2015)NinN
CIFAR1019.7997.980.770.042.990.72LOCSEARCHADV (Ours)VGG
90.930.900.20100.000.04FGSM(Goodfellow et al.,015)VGG
CIFAR1035.470.021.176.42LOCSEARCHADV (Ours)
STL1052.650.56 0.940.200.04NinN
STL1043.9187.160.520.01100.00FGSM(Goodfellow et al.,2015)NinN
STL1059.380.930.201.09 100.0019.65 0.10LOCSEARCHADV (Ours) FGSM(Goodfellow et al., 2015)VGG VGG
STL10SVHN 6.1573.9791.36 92.310.680.05
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DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%）TIME (in sec)TechniqueNetwork
ImageNet100058.2793.590.290.290.4312.72LOCSEARCHADV (Ours)VGG CNN-S (Caffe)
ImageNet100085.510.491.00100.004.74FGSM(Goodfellow et al.,2015)VGG CNN-S (Caffe)
ImageNet100058.9691.360.280.290.4010.01LOCSEARCHADV (Ours)VGG CNN-M (Caffe)
ImageNet100087.850.481.00100.004.36FGSM(Goodfellow etal.,015)VGG CNN-M (Caffe)
ImageNet100058.8092.820.290.300.4111.09LOCSEARCHADV (Ours)VGG CNN-M 2048 (Caffe)
ImageNet100088.430.521.00100.004.42FGSM(Goodfellow et al.,2015)VGG CNN-M 2048 (Caffe)
ImageNet100046.4072.070.300.540.5573.64LOCSEARCHADV (Ours)VGG ILSVRC 19 (Caffe)
ImageNet100085.050.521.00100.0023.94FGSM(Goodfellow et al., 2015)VGG ILSVRC 19 (Caffe)
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DatasetkERRTOP-kERRTOP-k(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)Network
CIFAR1016.5497.890.720.043.240.58NinN
CIFAR106.8876.650.880.075.501.02NinN
CIFAR1012343.5859.020.900.087.091.85NinN
CIFAR101.8448.890.900.097.632.12NinN
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DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)TechniqueNetwork
NNs trained with batch normalization
CIFAR1011.6597.630.47 0.550.04 0.203.750.68LOCSEARCHADV (Ours)NinN
CIFAR1070.69
11.6297.510.74100.000.01FGSM(Goodfellow et al., 2015)NinN
CIFAR100.043.160.78LOCSEARCHADV (Ours)VGG
CIFAR1011.621111FGSM(Goodfellow et al.,2015)VGG
STL1058.170.420.021.207.15LOCSEARCHADV (Ours)NinN
STL1029.8154.850.530.20100.000.03FGSM(Goodfellow et al.,2015)NinN
STL1065.760.470.021.1113.90LOCSEARCHADV (Ours)VGG
STL1026.5026.50二1 0.05二二FGSM(Goodfellow et al., 2015)VGG
SVHN SVHN9.7197.06 48.620.47 0.490.204.51 100.001.02 0.02LOCSEARCHADV (Ours) FGSM(Goodfellow et al.,2015)NinN NinN
SVHN81.100.660.07 15.432.15LOCSEARCHADV (Ours)VGG VGG
4.77
SVHN4.771 0.54 0.201 2.241 0.64FGSM(Goodfellow et al.,2015)NinN
MNIST91.42
MNIST0.331.650.580.20100.000.02LOCSEARCHADV (Ours) FGSM (Goodfellow et al., 2015)
93.480.630.21 2.200.64LOCSEARCHADV(Ours)NinN VGG
MNIST MNIST0.440.44111 NNs trained without batch normalization1FGSM(Goodfellow et al., 2015)VGG
97.8916.540.720.043.240.58LOCSEARCHADV (Ours)NinN
CIFAR10 CIFAR1093.670.930.20100.000.02FGSM(Goodfellow et al., 2015)NinN
CIFAR1019.7997.980.770.042.990.72LOCSEARCHADV (Ours)VGG
90.930.900.20100.000.04FGSM(Goodfellow et al.,015)VGG
CIFAR1035.470.021.176.42LOCSEARCHADV (Ours)
STL1052.650.56 0.940.200.04NinN
STL1043.9187.160.520.01100.00FGSM(Goodfellow et al.,2015)NinN
STL1059.380.930.201.09 100.0019.65 0.10LOCSEARCHADV (Ours) FGSM(Goodfellow et al., 2015)VGG VGG
STL10SVHN 6.1573.9791.36 92.310.680.05
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DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%）TIME (in sec)TechniqueNetwork
ImageNet100058.2793.590.290.290.4312.72LOCSEARCHADV (Ours)VGG CNN-S (Caffe)
ImageNet100085.510.491.00100.004.74FGSM(Goodfellow et al.,2015)VGG CNN-S (Caffe)
ImageNet100058.9691.360.280.290.4010.01LOCSEARCHADV (Ours)VGG CNN-M (Caffe)
ImageNet100087.850.481.00100.004.36FGSM(Goodfellow etal.,015)VGG CNN-M (Caffe)
ImageNet100058.8092.820.290.300.4111.09LOCSEARCHADV (Ours)VGG CNN-M 2048 (Caffe)
ImageNet100088.430.521.00100.004.42FGSM(Goodfellow et al.,2015)VGG CNN-M 2048 (Caffe)
ImageNet100046.4072.070.300.540.5573.64LOCSEARCHADV (Ours)VGG ILSVRC 19 (Caffe)
ImageNet100085.050.521.00100.0023.94FGSM(Goodfellow et al., 2015)VGG ILSVRC 19 (Caffe)
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DatasetkERRTOP-kERRTOP-k(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)Network
CIFAR1016.5497.890.720.043.240.58NinN
CIFAR106.8876.650.880.075.501.02NinN
CIFAR1012343.5859.020.900.087.091.85NinN
CIFAR101.8448.890.900.097.632.12NinN
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Algorithm 2 CYCLIC (r,b,x,y)
Assumptions:Perturbation parameter r ∈ [O,2] and LB≤O≤UB
Output:Perturbed image value at the coordinate (b,x,y) which lies in the range [LB,UB]
if rI(b,x,y)<LB then return rI(b,x,y)+(UB-LB)
else if rI(b,x,y)>UB then
return rI(b,x,y)-(UB-LB)
else
return rI(b,x,y)
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DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)TechniqueNetwork
NNs trained with batch normalization
CIFAR1011.6597.630.47 0.550.04 0.203.750.68LOCSEARCHADV (Ours)NinN
CIFAR1070.69
11.6297.510.74100.000.01FGSM(Goodfellow et al., 2015)NinN
CIFAR100.043.160.78LOCSEARCHADV (Ours)VGG
CIFAR1011.621111FGSM(Goodfellow et al.,2015)VGG
STL1058.170.420.021.207.15LOCSEARCHADV (Ours)NinN
STL1029.8154.850.530.20100.000.03FGSM(Goodfellow et al.,2015)NinN
STL1065.760.470.021.1113.90LOCSEARCHADV (Ours)VGG
STL1026.5026.50二1 0.05二二FGSM(Goodfellow et al., 2015)VGG
SVHN SVHN9.7197.06 48.620.47 0.490.204.51 100.001.02 0.02LOCSEARCHADV (Ours) FGSM(Goodfellow et al.,2015)NinN NinN
SVHN81.100.660.07 15.432.15LOCSEARCHADV (Ours)VGG VGG
4.77
SVHN4.771 0.54 0.201 2.241 0.64FGSM(Goodfellow et al.,2015)NinN
MNIST91.42
MNIST0.331.650.580.20100.000.02LOCSEARCHADV (Ours) FGSM (Goodfellow et al., 2015)
93.480.630.21 2.200.64LOCSEARCHADV(Ours)NinN VGG
MNIST MNIST0.440.44111 NNs trained without batch normalization1FGSM(Goodfellow et al., 2015)VGG
97.8916.540.720.043.240.58LOCSEARCHADV (Ours)NinN
CIFAR10 CIFAR1093.670.930.20100.000.02FGSM(Goodfellow et al., 2015)NinN
CIFAR1019.7997.980.770.042.990.72LOCSEARCHADV (Ours)VGG
90.930.900.20100.000.04FGSM(Goodfellow et al.,015)VGG
CIFAR1035.470.021.176.42LOCSEARCHADV (Ours)
STL1052.650.56 0.940.200.04NinN
STL1043.9187.160.520.01100.00FGSM(Goodfellow et al.,2015)NinN
STL1059.380.930.201.09 100.0019.65 0.10LOCSEARCHADV (Ours) FGSM(Goodfellow et al., 2015)VGG VGG
STL10SVHN 6.1573.9791.36 92.310.680.05
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DatasetkERRTOP-kERRTOP-k(ADV)CONFPTB#PTBPIXELS (%)TIME (in sec)Network
CIFAR1016.5497.890.720.043.240.58NinN
CIFAR106.8876.650.880.075.501.02NinN
CIFAR1012343.5859.020.900.087.091.85NinN
CIFAR101.8448.890.900.097.632.12NinN
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DatasetERRTOP-1ERRTOP-1(ADV)CONFPTB#PTBPIXELS (%）TIME (in sec)TechniqueNetwork
ImageNet100058.2793.590.290.290.4312.72LOCSEARCHADV (Ours)VGG CNN-S (Caffe)
ImageNet100085.510.491.00100.004.74FGSM(Goodfellow et al.,2015)VGG CNN-S (Caffe)
ImageNet100058.9691.360.280.290.4010.01LOCSEARCHADV (Ours)VGG CNN-M (Caffe)
ImageNet100087.850.481.00100.004.36FGSM(Goodfellow etal.,015)VGG CNN-M (Caffe)
ImageNet100058.8092.820.290.300.4111.09LOCSEARCHADV (Ours)VGG CNN-M 2048 (Caffe)
ImageNet100088.430.521.00100.004.42FGSM(Goodfellow et al.,2015)VGG CNN-M 2048 (Caffe)
ImageNet100046.4072.070.300.540.5573.64LOCSEARCHADV (Ours)VGG ILSVRC 19 (Caffe)
ImageNet100085.050.521.00100.0023.94FGSM(Goodfellow et al., 2015)VGG ILSVRC 19 (Caffe)
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AI Research, NAVER Corp., Seongnam, Korea kekepa15@snu.ac.kr, blue378@snu.ac.kr, jaesung.huh@navercorp.com, adrian.kim@navercorp.com, jungwoo.ha@navercorp.com, kglee@snu.ac.kr + +# ABSTRACT + +Most deep learning-based models for speech enhancement have mainly focused on estimating the magnitude of spectrogram while reusing the phase from noisy speech for reconstruction. This is due to the difficulty of estimating the phase of clean speech. To improve speech enhancement performance, we tackle the phase estimation problem in three ways. First, we propose Deep Complex U-Net, an advanced U-Net structured model incorporating well-defined complex-valued building blocks to deal with complex-valued spectrograms. Second, we propose a polar coordinate-wise complex-valued masking method to reflect the distribution of complex ideal ratio masks. Third, we define a novel loss function, weighted source-to-distortion ratio (wSDR) loss, which is designed to directly correlate with a quantitative evaluation measure. Our model was evaluated on a mixture of the Voice Bank corpus and DEMAND database, which has been widely used by many deep learning models for speech enhancement. Ablation experiments were conducted on the mixed dataset showing that all three proposed approaches are empirically valid. Experimental results show that the proposed method achieves state-of-the-art performance in all metrics, outperforming previous approaches by a large margin1. + +# 1 INTRODUCTION + +Speech enhancement is one of the most important and challenging tasks in speech applications where the goal is to separate clean speech from noise when noisy speech is given as an input. As a fundamental component for speech-related systems, the applications of speech enhancement vary from speech recognition front-end modules to hearing aid systems for the hearing-impaired (Erdogan et al., 2015; Weninger et al., 2015; Wang, 2017). + +Due to recent advances in deep learning, the speech enhancement task has been able to reach high levels in performance through significant improvements. When using audio signals with deep learning models, it has been a common practice to transform a time-domain waveform to a time-frequency (TF) representation (i.e. spectrograms) via short-time-Fourier-transform (STFT). Spectrograms are represented as complex matrices, which are normally decomposed into magnitude and phase components to be used in real-valued networks. In tasks involving audio signal reconstruction, such as speech enhancement, it is ideal to perform correct estimation of both components. Unfortunately, complex-valued phase has been often neglected due to the difficulty of its estimation. This has led to the situation where most approaches focus only on the estimation of a magnitude spectrogram while reusing noisy phase information (Huang et al., 2014; Xu et al., 2015; Grais et al., 2016; Nugraha et al., 2016; Takahashi et al., 2018b). However, reusing phase from noisy speech has clear limitations, particularly under extremely noisy conditions, in other words, when signal-to-noise ratio (SNR) is low. This can be easily verified by simply using the magnitude spectrogram of clean speech with the phase spectrogram of noisy speech to reconstruct clean speech, as illustrated in Figure 1. We can clearly see that the difference between clean and estimated speech signals gets larger as the input SNR gets lower. + +![](images/44281215127d595e91a725f5d3e2398f9d143edac8059550ca1138d5b01daf99.jpg) +Figure 1: Effects of reusing phase of noisy speech under different SNR conditions. + +A popular approach to speech enhancement is to optimize a mask which produces a spectrogram of clean speech when applied to noisy input audio. One of the first mask-based attempts to perform the task by incorporating phase information was the proposal of the phase-sensitive mask (PSM) (Erdogan et al., 2015). Since the performance of PSM was limited because of reusing noisy phase, later studies proposed using complex-valued ratio mask (cRM) to directly optimize on complex values (Williamson et al., 2016; Ephrat et al., 2018). We found this direction promising for phase estimation because it has been shown that a complex ideal ratio mask (cIRM) is guaranteed to give the best oracle performance out of other ideal masks such as ideal binary masks, ideal ratio masks, or PSMs (Wang et al., 2016). Moreover, this approach jointly estimates magnitude and phase, removing the need of separate models. To estimate a complex-valued mask, a natural desire would be to use an architecture which can handle complex-domain operations. Recent work gives a solution to this by providing deep learning building blocks adapted to complex arithmetic (Trabelsi et al., 2018). + +In this paper, we build upon previous studies to design a new complex-valued masking framework, based on a proposed variant of U-Net (Ronneberger et al., 2015), named Deep Complex U-Net (DCUnet). In our proposed framework, DCUnet is trained to estimate a complex ratio mask represented in polar coordinates with prior knowledge observable from ideal complex-valued masks. With the complex-valued estimation of clean speech, we can use inverse short-time-Fourier-transform (ISTFT) to convert a spectrogram into a time-domain waveform. Taking this as an advantage, we introduce a novel loss function which directly optimizes source-to-distortion ratio (SDR) (Vincent et al., 2006), a quantitative evaluation measure widely used in many source separation tasks. + +Our contributions can be summarized as follows: + +1. We propose a new neural architecture, Deep Complex U-Net, which combines the advantages of both deep complex networks and U-Net, yielding state-of-the-art performance. +2. While pointing out limitations of current masking strategies, we design a new complexvalued masking method based on polar coordinates. +3. We propose a new loss function weighted-SDR loss, which directly optimizes a well known quantitative evaluation measure. + +# 2 RELATED WORKS + +Phase estimation for audio signal reconstruction has been a recent major interest within the audio source separation community because of its importance and difficulty. While iterative methods such as the Griffin-Lim algorithm and its variants (Griffin & Lim, 1984; Perraudin et al., 2013) aimed to address this problem, neural network-based approaches are recently attracting attention as noniterative alternatives. + +One major approach is to use an end-to-end model that takes audio as raw waveform inputs without using any explicit time-frequency (TF) representation computed via STFT (Pascual et al., 2017; Rethage et al., 2018; Stoller et al., 2018; Germain et al., 2018). Since raw waveforms inherently contain phase information, it is expected to achieve phase estimation naturally. Another method is to estimate magnitude and phase using two separate neural network modules which serially estimate magnitude and phase (Afouras et al., 2018; Takahashi et al., 2018a). In this framework, the phase estimation module uses noisy phase with predicted magnitude to estimate phase of clean speech. There is also a recent study which proposed to use additional layers with trainable discrete values for phase estimation (Roux et al., 2018). + +A more straightforward method would be to jointly estimate magnitude and phase by using a continuous complex-valued ratio mask (cRM). Previous studies tried this joint estimation approach bounding the range of the cRM (Williamson et al., 2016; Ephrat et al., 2018). Despite the advantages of the cRM approach, previously proposed methods had limitations with regard to the loss function and the range of the mask which we will be returning with more details in Section 3 along with our proposed methods to alleviate these issues. As a natural extension to the works above, some studies have also undergone to examine whether complex-valued networks are useful when dealing with intrinsically complex-valued data. In the series of two works, complex-valued networks were shown to help singing voice separation performance with both fully connected neural networks and recurrent neural networks (Lee et al., 2017a;b). However, the approaches were limited as it ended up only switching the real-valued network into a complex-valued counterpart and leaving the other deep learning building blocks such as weight initialization and normalization technique in a realvalued manner. Also, the works do not show whether the phase was actually well estimated either quantitatively or qualitatively, only ending up showing that there was a performance gain. + +# 3 PHASE-AWARE SPEECH ENHANCEMENT + +In this section we will provide details on our approach, starting with our proposed model Deep Complex U-Net, followed by the masking framework based on the model. Finally, we will introduce a new loss function to optimize our model, which takes a critical role for proper phase estimation. + +Before getting into details, here are some notations used throughout the paper. The input mixture signal $\bar { x ( n ) } \overset { \cdot } { = } y ( n ) + z ( n ) \in \mathbb { R }$ is assumed to be a linear sum of the clean speech signal $y ( n ) \in \mathbb { R }$ and noise $z ( n ) \in \mathbb { R }$ , where estimated speech is denoted as $\hat { y } ( n ) \in \mathbb { R }$ . Each of the corresponding time-frequency $( t , f )$ representations computed by STFT is denoted as $X _ { t , f } \in \mathbb { C } , Y _ { t , f } \in \bar { \mathbb { C } } , Z _ { t , f } \in$ $\mathbb { C }$ , and $\hat { Y } _ { t , f } \in \mathbb { C }$ . The ground truth mask cIRM is denoted as $M _ { t , f } \in \mathbb { C }$ and the estimated cRM is denoted as $\hat { M } _ { t , f } \in \mathbb { C }$ , where $M _ { t , f } = Y _ { t , f } / X _ { t , f }$ . + +# 3.1 DEEP COMPLEX U-NET + +![](images/97b1a0b158f74dedaa4e3df7048d9ac98c9b57d80f09e60aa2417cc5b10ed965.jpg) +Figure 2: Illustration of speech enhancement framework with DCUnet. + +The U-Net structure is a well known architecture composed as a convolutional autoencoder with skip-connections, originally proposed for medical imaging in computer vision community (Ronneberger et al., 2015). Furthermore, the use of real-valued U-Net has been shown to be also effective in many recent audio source separation tasks such as music source separation (Jansson et al., 2017; Stoller et al., 2018; Takahashi et al., 2018b), and speech enhancement (Pascual et al., 2017). Deep Complex U-Net (DCUnet) is an extended U-Net, refined specifically to explicitly handle complex domain operations. In this section, we will describe how U-Net is modified using the complex building blocks originally proposed by (Trabelsi et al., 2018). + +Complex-valued Building Blocks. Given a complex-valued convolutional filter $W = A + i B$ with real-valued matrices $\pmb { A }$ and $\textbf { { B } }$ , the complex convolution operation on complex vector $\pmb { h } = \pmb { x } + i \pmb { y }$ with $W$ is done by $W * h = ( A * x - B * y ) + i ( B * x + A * y )$ . In practice, complex convolutions can be implemented as two different real-valued convolution operations with shared real-valued convolution filters. Details are illustrated in Appendix A. + +![](images/9fa5f650774f98c9967d1e4710a3fc31bbdda2f0d1c1716e387a73a5f234097c.jpg) +Figure 3: Illustration of three cRM methods and their corresponding output range in complex space. The Neural Network (NN) can be either DCUnet $\left( \mathbb { C } \right)$ or a corresponding real-valued U-Net $( \mathbb { R } )$ with the same parameter size. Three different output ranges of cRM methods (1, 2 and 3) in (b) are shown in (1, 2 and 3) in (a). 1. Unbounded mask: a masking method without bounded outputs. 2. Bounded (sigmoid-sigmoid) mask: The proposed method by Ephrat et al. 3. Bounded (tanh) mask: Our proposed masking method. The detailed model specification is described in Appendix B. + +Activation functions like ReLU were also adapted to the complex domain. In previous work, CReLU, an activation function which applies ReLU on both real and imaginary values, was shown to produce the best results out of many suggestions. Details on batch normalization and weight initialization for complex networks can be found in Trabelsi et al. (2018). + +Modifying U-Net. The proposed Deep Complex U-Net is a refined U-Net architecture applied in STFT-domain. Modifications done to the original U-Net are as follows. Convolutional layers of UNet are all replaced to complex convolutional layers, initialized to meet the Glorot’s criteria (Glorot & Bengio, 2010). Here, the convolution kernels are set to be independent to each other by initializing the weight tensors as unitary matrices for better generalization and fast learning (Cogswell et al., 2015). Complex batch normalization is implemented on every convolutional layer except the last layer of the network. In the encoding stage, max pooling operations are replaced with strided complex convolutional layers to prevent spatial information loss. In the decoding stage, strided complex deconvolutional operations are used to restore the size of input. For the activation function, we modified the previously suggested CReLU into leaky CReLU, where we simply replace ReLU into leaky ReLU (Maas et al., 2013), making training more stable. Note that all experiments performed in Section 4 are done with these modifications. + +# 3.2 COMPLEX-VALUED MASKING ON POLAR COORDINATES + +As our proposed model can handle complex values, we aim to estimate cRM for speech enhancement. Although it is possible to directly estimate the spectrogram of a clean source signal, it has been shown that better performance can be achieved by applying a weighting mask to the mixture spectrogram (Wang et al., 2014). One thing to note is that real-valued ratio masks (RM) only change the scale of the magnitude without changing phase, resulting in irreducible errors as illustrated in Appendix D. On the other hand, cRM also perform a rotation on the polar coordinates, allowing to correct phase errors. In other words, the estimated speech spectrogram $\hat { Y } _ { t , f }$ is computed by multiplying the estimated mask $\hat { M } _ { t , f }$ on the input spectrogram $X _ { t , f }$ as follows: + +$$ +\hat { Y } _ { t , f } = \hat { M } _ { t , f } \cdot X _ { t , f } = \Big | \hat { M } _ { t , f } \Big | \cdot \big | X _ { t , f } \big | \cdot e ^ { i ( \theta _ { \hat { M } _ { t , f } } + \theta _ { X _ { t , f } } ) } +$$ + +In this state, the real and imaginary values of the estimated cRM is unbounded. Although estimating an unbounded mask makes the problem well-posed (see Appendix $\mathrm { D }$ for more information), we can imagine the difficulty of optimizing from an infinite search space compared to a bounded one. + +Therefore, a few techniques have been tried to bound the range of cRM. For example, Williamson et al. tried to directly optimize a complex mask into a cIRM compressed to a heuristic bound (Williamson et al., 2016). However, this method was limited since it was only able to succeed in training the model by computing the error between cIRM and the predicted cRM which often leads to a degradation of performance (Wang et al., 2014; Yu et al., 2017). More recently, Ephrat et al. proposed a rectangular coordinate-wise cRM made with sigmoid compressions onto each of the real and imaginary parts of the output of the model (Ephrat et al., 2018). After then MSE between clean source $Y$ and estimated source $\hat { Y }$ was computed in STFT-domain to train the model. However, the proposed masking method has two main problems regarding phase estimation. First, it suffers from the inherent problem of not being able to reflect the distribution of cIRM as shown in Figure 3 and Appendix E. Second, this approach results in a cRM with a restricted rotation range of $0 ^ { \circ }$ to $9 0 °$ (only clock-wise), which makes it hard to correct noisy phase. + +To alleviate these problems, we propose a polar coordinate-wise cRM method that imposes nonlinearity only on the magnitude part. More specifically, we use a hyperbolic tangent non-linearity to bound the range of magnitude part of the cRM be $[ 0 , 1 )$ which makes the mask bounded in an unit-circle in complex space. The corresponding phase mask is naturally obtained by dividing the output of the model with the magnitude of it. More formally, let $g ( \cdot )$ be our neural network and the output of it be $O _ { t , f } = g ( X _ { t , f } )$ . The proposed complex-valued mask $\hat { M } _ { t , f }$ is estimated as follows: + +$$ +\hat { M } _ { t , f } = \Bigr | \hat { M } _ { t , f } \Bigr | \cdot e ^ { i \theta _ { \hat { M } _ { t , f } } } = \hat { M } _ { t , f } ^ { m a g } \cdot \hat { M } _ { t , f } ^ { p h a s e } +$$ + +$$ +\hat { M } _ { t , f } ^ { m a g } = \left\{ { t a n h } ( | O _ { t , f } | ) \begin{array} { l } { { ( { \mathrm { b o u n d e d ~ c o n d . } } ) } } \\ { { { | O _ { t , f } | } } } \end{array} , \right. \hat { M } _ { t , f } ^ { p h a s e } = O _ { t , f } / { | O _ { t , f } | } +$$ + +A summarized illustration of cRM methods is depicted in Figure 3. + +# 3.3 WEIGHTED-SDR LOSS + +A popular loss function for audio source separation is mean squared error (MSE) between clean source $Y$ and estimated source $\hat { Y }$ on the STFT-domain. However, it has been reported that optimizing the model with MSE in complex STFT-domain fails in phase estimation due to the randomness in phase structure (Williamson et al., 2016). As an alternative, it is possible to use a loss function defined in the time-domain instead, as raw waveforms contain inherent phase information. While MSE on waveforms can be an easy solution, we can expect it to be more effective if the loss function is directly correlated with well-known evaluation measures defined in the time-domain. + +Here, we propose an improved loss function weighted-SDR loss by building upon a previous work which attempts to optimize a standard quality measure, source-to-distortion ratio (SDR) (Venkataramani et al., 2017). The original loss function $l o s s _ { V e n }$ suggested by Venkataramani et al. is formulated upon the observation from Equation 4, where $y$ is the clean source signal and $\hat { y }$ is the estimated source signal. In practice, the negative reciprocal is optimized as in Equation 5. + +$$ +\begin{array} { l } { \displaystyle \underset { \hat { y } } { \mathrm { m a x } } \ S D R ( y , \hat { y } ) : = \operatorname* { m a x } _ { \hat { y } } \frac { < y , \hat { y } > ^ { 2 } } { | | y | | ^ { 2 } | | \hat { y } | | ^ { 2 } - < y , \hat { y } > ^ { 2 } } \propto \operatorname* { m i n } _ { \hat { y } } \frac { | | \hat { y } | | ^ { 2 } } { < y , \hat { y } > ^ { 2 } } } \\ { l o s s _ { V e n } ( y , \hat { y } ) : = - \frac { < y , \hat { y } > ^ { 2 } } { | | \hat { y } | | ^ { 2 } } } \end{array} +$$ + +Although using Equation 5 works as a loss function, there are a few critical flaws in the design. First, the lower bound becomes $- \| y \| ^ { 2 }$ , which depends on the value of $y$ causing fluctuation in the loss values when training. Second, when the target $y$ is empty (i.e., $y = 0$ ) the loss becomes zero, preventing the model to learn from noisy-only data due to zero gradients. Finally, the loss function is not scale sensitive, meaning that the loss value is the same for $\hat { y }$ and $c \hat { y }$ , where $c \in \mathbb { R }$ . + +To resolve these issues, we redesigned the loss function by giving several modifications to Equation 5. First, we made the lower bound of the loss function independent to the source $y$ by restoring back the term $\left\| y \right\| ^ { 2 }$ and applying square root as in Equation 6. This makes the loss function bounded within the range [-1, 1] and also be more phase sensitive, as inverted phase gets penalized as well. + +$$ +\mathit { l o s s } _ { \mathit { S D R } } ( y , \hat { y } ) : = - \sqrt { - \frac { \mathit { l o s s } _ { \mathit { V e n } } } { \left\| y \right\| ^ { 2 } } } = - \frac { < y , \hat { y } > } { \left\| y \right\| \left\| \hat { y } \right\| } +$$ + +Expecting to be complementary to source prediction and to propagate errors for noise-only samples, we also added a noise prediction term $l o s s _ { S D R } ( z , \hat { z } )$ . To properly balance the contributions of each loss term and solve the scale insensitivity problem, we weighted each term proportional to the energy of each signal. The final form of the suggested weighted-SDR loss is as follows: + +$$ +l o s s _ { w S D R } ( x , y , \hat { y } ) : = \alpha l o s s _ { S D R } ( y , \hat { y } ) + ( 1 - \alpha ) l o s s _ { S D R } ( z , \hat { z } ) +$$ + +where, $\hat { z } = x - \hat { y }$ is estimated noise and $\alpha = | | y | | ^ { 2 } / ( | | y | | ^ { 2 } + | | z | | ^ { 2 } )$ is the energy ratio between clean speech $y$ and noise $z$ . Note that although weighted SDR loss is a time-domain loss function, it can be backpropagated through our framework. Specifically, STFT and ISTFT operations are implemented as 1-D convolution and deconvolution layers consisting of fixed filters initialized with the discrete Fourier transform matrix. The detailed properties of the proposed loss function are in Appendix C. + +# 4 EXPERIMENTS + +# 4.1 EXPERIMENT SETUP + +Dataset. For all experiments, we used the same experimental setups as previous works in order to perform direct performance comparison (Pascual et al., 2017; Rethage et al., 2018; Soni et al., 2018; Germain et al., 2018). Noise and clean speech recordings were provided from the Diverse Environments Multichannel Acoustic Noise Database (DEMAND) (Thiemann et al., 2013) and the Voice Bank corpus (Veaux et al., 2013), respectively, each recorded with sampling rate of $4 8 \mathrm { k H z }$ . Mixed audio inputs used for training were composed by mixing the two datasets with four signalto-noise ratio (SNR) settings (15, 10, 5, and 0 (dB)), using 10 types of noise (2 synthetic $+ \ 8$ from DEMAND) and 28 speakers from the Voice Bank corpus, creating 40 conditional patterns for each speech sample. The test set inputs were made with four SNR settings different from the training set (17.5, 12.5, 7.5, and 2.5 (dB)), using the remaining 5 noise types from DEMAND and 2 speakers from the Voice Bank corpus. Note that the speaker and noise classes were uniquely selected for the training and test sets. + +Pre-processing. The original raw waveforms were first downsampled from 48kHz to 16kHz. For the actual model input, complex-valued spectrograms were obtained from the downsampled waveforms via STFT with a 64ms sized Hann window and 16ms hop length. + +Implementation. All experiments were implemented and fine-tuned with NAVER Smart Machine Learning (NSML) platform (Sung et al., 2017; Kim et al., 2018). + +# 4.2 COMPARISON RESULTS + +In this subsection, we compare overall speech enhancement performance of our method with previously proposed algorithms. As a baseline approach, Wiener filtering (Wiener) with a priori noise SNR estimation was used, along with recent deep-learning based models which are briefly described as the following: SEGAN: a time-domain U-Net model optimized with generative adversarial networks. Wavenet: a time-domain non-causal dilated wavenet-based network. MMSE-GAN: a timefrequency masking-based method with modified adversarial training method. Deep Feature Loss: a time-domain dilated convolution network trained with feature loss from a classifier network. + +Table 1: Quantitative evaluation results of other algorithms and proposed methods (DCUnet-20, Large-DCUnet-20). Higher score means better performance where bold text indicates highest score per evaluation measure. CSIG: Mean opinion score (MOS) predictor of signal distortion CBAK: MOS predictor of background-noise intrusiveness COVL: MOS predictor of overall signal quality PESQ: Perceptual evaluation of speech quality SSNR: Segmental SNR. All evaluation measures were computed by using open source implementation.2 + +
CSIGCBAKCOVLPESQSSNR
Wiener (Scalart et al., 1996)3.232.682.672.225.07
SEGAN (Pascual et al., 2017)3.482.942.802.167.73
Wavenet (Rethage et al., 2018)3.623.232.9811
MMSE-GAN (Soni et al., 2018)3.803.123.142.53-
Deep Feature Loss (Germain et al., 2018)3.863.333.2211
DCUnet-20 (ours)4.244.003.693.1315.95
Large-DCUnet-20 (ours)4.344.103.813.2416.85
+ +For comparison, we used the configuration of using a 20-layer Deep Complex U-Net (DCUnet20) to estimate a tanh bounded cRM, optimized with weighted-SDR loss. As a showcase for the potential of our approach, we also show results from a larger DCUnet-20 (Large-DCUnet-20) which has more channels in each layer. Both architectures are specified in detail in Appendix B. Results show that our proposed method outperforms the previous state-of-the-art methods with respect to all metrics by a large margin. Additionally, we can also see that larger models yield better performance. We see the reason to this significant improvement coming from the phase estimation quality of our method, which we plan to investigate in later sections. + +# 4.3 ABLATION STUDIES + +Table 2: Table of quantitative evaluation results with corresponding mask and loss function in three different model configurations (DCU-10, DCU-16 and DCU-20). The bold font indicates the best loss function when fixing the masking method. The underline indicates the best masking method when fixing the loss function. + +
CSIGCBAKCOVLPESQSSNR
DCU-10|Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR||Spc Wav wSDR|SpcWav wSDR
UBD|3.51 3.713.72 |3.50 3.523.56|3.13 3.153.19|2.78 2.61 2.67[11.86 13.1813.37
BDSS2.94 3.193.21 3.43 3.353.392.81 2.852.892.79 2.61 2.6711.61 12.30 12.30
BDT3.30 3.743.723.52 3.57 3.603.02 3.193.222.79 2.66 2.7212.30 13.5413.60
DCU-16| |SpcWav wSDR||Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD3.97 4.004.07 13.66 3.723.753.46 3.433.482.96 2.85 2.9312.75 14.2514.44
BDSS3.63 3.573.72 3.59 3.513.563.28 3.153.272.98 2.82 2.8712.05 12.8912.89
BDT3.97 4.034.10 3.70 3.743.773.49 3.453.523.01 2.88 2.9312.96 14.3914.44
DCU-20| |Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD|4.02 4.154.25 3.88 3.934.01|3.57 3.593.70[3.11 3.01 3.1314.81 16.0316.36
BDSS3.77 3.743.93 3.68 3.653.723.40 3.333.473.06 2.98 3.0512.80 13.6613.68
BDT4.02 4.184.24 3.87 3.954.003.58 3.643.693.13 3.08 3.1314.59 15.85 15.95
+ +Masking strategy and loss functions. In this experiment, we show the evaluation results on how the various masking strategies and loss functions affect the performance of speech enhancement. Three masking strategies (Unbounded (UBD); Ephrat et al.: Bounded (sig-sig) (BDSS); Bounded (tanh) (BDT)) are compared to see the effect of each different masking method. Then, to compare our proposed loss function (weighted-SDR) with MSE, we compared two MSE in STFT-domain and time-domain (Spectrogram-MSE (Spc); Wave-MSE (Wav); and proposed: weighted-SDR (wSDR)). + +Table 2 shows the jointly combined results on varied masking strategies and loss functions, where three models (DCU-10 (1.4M), DCU-16 (2.3M), and DCU-20 (3.5M)) are investigated to see how architectural differences in the model affect quantitative results. In terms of masking strategy, the proposed BDT mask mostly yields better results than UBD mask in DCU-10 and DCU-16, implying the importance of limiting optimization space with prior knowledge. However, in the case of DCU-20, UBD mask was able to frequently surpass the performance of BDT mask. Intuitively, this indicates that when the number of parameter gets large enough, the model is able to fit the distribution of data well even when the optimization space is not bounded. In terms of the loss function, almost every result shows that optimizing with wSDR loss gives the best result. However, we found out that Spc loss often provides better PESQ results than wSDR loss for DCU-10 and DCU-16 except DCU-20 case where Spc and wSDR gave similar PESQ results. + +Table 3: Table of quantitative evaluation results from three different settings (cRMCn: Complexvalued output/Complex-valued network, cRMRn: Complex-valued output/Real-valued network, and $\mathbf { R M } \mathbb { R } \mathbf { n }$ : Real-valued output/Real-valued network) to show the appropriateness of using complexvalued networks for speech enhancement. Bold font indicates the best results. + +
20-layer16-layer10-layer
cRMCncRMRnRMRncRMCncRMRnRMRncRMCncRMRnRMRn
CSIG4.244.214.064.104.063.883.743.713.71
CBAK4.003.923.403.773.753.333.603.593.23
COVL3.693.653.403.523.493.263.223.203.01
PESQ3.133.072.742.932.912.662.722.722.51
SSNR15.9515.549.9014.4414.259.6613.6013.499.43
+ +Validation on complex-valued network and mask. In order to show that complex neural networks are effective, we compare evaluation results of DCUnet $( \mathbb { C } \mathrm { n } )$ and its corresponding real-valued UNet setting with the same parameter size $( \mathbb { R } \mathbf { n } )$ . For the real-valued network, we tested two settings $\mathbf { c R M } \mathbb { R } \mathbf { n }$ and $\mathbf { R M } \mathbb { R } \mathbf { n }$ to show the effectiveness of phase estimation. The first setting takes a complexvalued spectrogram as an input, estimating a complex ratio mask (cRM) with a tanh bound. The second setting takes a magnitude spectrogram as an input, estimating a magnitude ratio mask (RM) with a sigmoid bound. All models were trained with weighted-SDR loss, where the ground truth phase was given while training $\mathbf { R M } \mathbb { R } \mathbf { n }$ . Additionally, all models were trained on different number of parameters (20-layer (3.5M), 16-layer (2.3M), and 10-layer (1.4M)) to show that the results are consistent regardless of model capacity. Detailed network architectures for each model are illustrated in Appendix B. + +In Table 3, evaluation results show that our approach cRMCn makes better results than conventional method $\mathbf { R M } \mathbb { R } \mathbf { n }$ for all cases, showing the effectiveness of phase correction. Also, cRMCn gives better results than $\mathbf { c R M } \mathbb { R } \mathbf { n }$ , which indicates that using complex-valued networks consistently improve the performance of the network. Note that these results are consistent through every evaluation measure and model size. + +# 4.4 QUALITATIVE EVALUATION + +We performed qualitative evaluations by obtaining preference scores between the proposed DCUnet (Large-DCUnet-20) and baseline methods. 15 utterance samples with different noise levels were selected from the test set and used for subjective listening tests. For each noisy sample, all possible six pairs of denoised audio samples from four different algorithms were presented to the participants in a random order, resulting in 90 pairwise comparisons to be made by each subject. For each comparison, participants were presented with three audio samples - original noisy speech and two denoised speech samples by two randomly selected algorithms - and instructed to choose either a preferred sample (score 1) or “can’t decide” (score 0.5). A total of 30 subjects participated in the listening test, and the results are presented in Table 4 and in Table 7. + +Table 4: Preference scores of DCUnet compared to other baseline models. The scores were obtained by calculating the relative frequency the subjects prefer one method to the other method. Hard/Medium/Easy denote $2 . 5 / 7 . 5 / 1 7 . 5$ SNR conditions in dB, respectively. All statistics had significance of $\mathrm { p } { < } 0 . 0 0 1$ . Complete pairwise preference scores are presented in Appendix H. + +
HardMediumEasyTotal
DCUnet > Deep Feature Loss0.900.820.690.82
DCUnet > Wavenet0.980.950.750.93
DCUnet > SEGAN0.990.930.710.92
+ +![](images/cb0f9851f16949be8934410c702ff74a5beae189f043cb1f8fd000d8e2394515.jpg) +Figure 4: Scatter plots of estimated cRMs with 9 different mask and loss function configurations for a randomly picked noisy speech signal. Each scatter plot shows the distribution of complex values from an estimated cRM. The leftmost plot is from the cIRM for the given input. We can observe that most real-values are distributed around 0 and 1, while being relatively sparse in between. The configuration that fits the most to this distribution pattern is observed in the red dotted box which is achieved by the combination of our proposed methods (Bounded (tanh) and weighted-SDR). + +Table 4 shows that DCUnet clearly outperforms the other methods in terms of preference scores in every SNR condition. These differences are statistically significant as confirmed by pairwise one-tailed t-tests. Furthermore, the difference becomes more obvious as the input SNR condition gets worse, which supports our motivation that accurate phase estimation is even more important under harsh noisy conditions. This is further confirmed by in-depth quantitative analysis of the phase distance as described in Section 5 and Table 5. + +# 5 IN-DEPTH ANALYSIS ON PHASE + +In this section, we aim to provide constructive insights on phase estimation by analyzing how and why our proposed method is effective. We first visualized estimated complex masks with scatter plots in Figure 4 for each masking method and loss function configuration from Table 2. The plots clearly show that the Bounded (sig-sig) mask by Ephrat et al. fails to reflect the distribution of the complex ideal ratio mask (cIRM), whereas the proposed Bounded (tanh) or Unbounded mask tries to follow the ground truth. One interesting finding is that, while time-domain loss functions such as weighted-SDR or Wave-MSE can capture the distribution of the ground truth, Spectrogram-MSE fails to capture the distribution of the cIRM. More specifically, when we use Spectrogram-MSE, the imaginary values of the predicted cRMs are almost always zero, indicating that it ends up only scalin(e.g., $\bar { ( } X _ { t , f } ^ { r e a l } + i X _ { t , f } ^ { i m a g } ) ( a + \bar { i } \cdot \bar { 0 } ) = a ( X _ { t , f } ^ { r e a l } + i X _ { t , f } ^ { i m a g } ) )$ he phase of noisy speech with rotations. + +![](images/2fee5faa352a331c4b09236534209a0c3b7a0321fef337a23dd190b8e99fd896.jpg) +Figure 5: Illustration of four wave-plot segments of estimated speech with reference clean speech. (a) shows the wave-plot of synthesized speech with clean magnitude speech and noisy phase (CMNP). Three other cases show the wave-plots of estimated speech with different loss functions (b) Spectrogram-MSE, (c) Wave-MSE, and (d) weighted-SDR. + +In order to demonstrate this effect in an alternate perspective, we also plotted estimated waveforms for each loss function in Figure 5. As one can notice from Figure 5 (c) $\&$ (d), estimated speech from models optimized with time-domain loss functions are well-aligned with clean speech. However, Figure 5 (a) $\&$ (b) align poorly to the ground truth while showing almost identical estimations. Note that Figure 5 (a) is the best possible result without estimating the phase of clean speech. This again confirms that optimizing the model with Spectrogram-MSE makes it difficult to learn phase correction, meaning that the model ends up reusing noisy phase just like conventional approaches. + +To explicitly support these observations, we would need a quantitative measure for phase estimation. Here, we define the phase distance between target spectrogram $( A )$ and estimated spectrogram $( B )$ as the weighted average of angle between corresponding complex TF bins, where each bin is weighted by the magnitude of target speech $\big \langle \big _ { { A t } , f \big | } \big \rangle$ to emphasize the relative importance of each TF bin. Phase distance is formulated as the following: + +$$ +P h a s e D i s t ( A , B ) = \sum _ { t , f } \frac { \left| A _ { t , f } \right| } { \sum _ { t ^ { \prime } , f ^ { \prime } } \left| A _ { t ^ { \prime } , f ^ { \prime } } \right| } \angle ( A _ { t , f } , B _ { t , f } ) +$$ + +where, $\angle ( A _ { t , f } , B _ { t , f } )$ represents the angle between $A _ { t , f }$ and $B _ { t , f }$ , having a range of [0,180]. + +The phase distance between clean and noisy speech (PhaseDist $( \mathbf { C } , \mathbf { N } ) ,$ ) and the phase distance between clean and estimated speech (PhaseDist $( \mathbf { C } , \mathbf { E } ) )$ are presented in Table 5. The results show that the best phase improvement (Phase Improvement $=$ PhaseDist(C, N) − PhaseDist(C, E)) is obtained with wSDR loss under every SNR condition. Also Spc loss gives the worst results, again reinforcing our observation. Analysis between the phase improvement and performance improvement is further discussed in Appendix G. + +# 6 CONCLUSION + +In this paper, we proposed Deep Complex U-Net which combines two models to deal with complexvalued spectrograms for speech enhancement. In doing so, we designed a new complex-valued masking method optimized with a novel loss function, weighted-SDR loss. Through ablation studies, we showed that the proposed approaches are effective for more precise phase estimation, resulting in state-of-the-art performance for speech enhancement. Furthermore, we conducted both quantitative and qualitative studies and demonstrated that the proposed method is consistently superior to the previously proposed algorithms. + +Table 5: Phase distance and phase improvement under four different SNR conditions. The Spc, Wav, and wSDR column show results from models trained with three different objectives SpectrogramMSE, Wave-MSE, and weighted-SDR, respectively. + +
SNR (dB)PhaseDist(C, N)PhaseDist(C,E)Phase Improvement
SpcWavwSDRSpcWavwSDR
2.514.521°10.512°9.288°7.807°4.009°5.233°6.714°
7.510.066°7.548°6.919°5.658°2.518°3.147°4.408°
12.57.197°5.455°5.105°4.215°1.742°2.092°2.982°
17.54.853°3.949°3.872°3.151°0.905°0.981°1.702°
+ +In the near future, we plan to apply our system to various separation tasks such as speaker separation or music source separation. Another important direction is to extend the proposed model to deal with multichannel audio since accurate estimation of phase is even more critical in multichannel environments (Wang et al., 2018). Apart from separation, our approach can be generalized to various audio-related tasks such as dereverberation, bandwidth extension or phase estimation networks for text-to-speech systems. Taking advantage of sequence modeling, it may also be interesting to find further extensions with complex-valued LSTMs (Arjovsky et al., 2016; Wisdom et al., 2016). + +# ACKNOWLEDGEMENTS + +This work was partly supported by Next-Generation Information Computing Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (MSIT; Grant No. NRF-2017M3C4A7078548), partly supported by National Research Foundation of Korea (NRF) funded by the Korea government (MSIT; Grant No. NRF2017R1E1A1A01076284), and partly supported by the Creative Industrial Technology Development Program (10053249) funded by the Ministry of Trade, Industry and Energy (MOTIE, Korea). + +# REFERENCES + +Triantafyllos Afouras, Joon Son Chung, and Andrew Zisserman. The conversation: Deep audiovisual speech enhancement. Proc. Interspeech 2018, pp. 3244–3248, 2018. + +Martin Arjovsky, Amar Shah, and Yoshua Bengio. Unitary evolution recurrent neural networks. 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IEEE Transactions on signal processing, 40(9):2281–2289, 1992. + +Dong Yu, Morten Kolbæk, Zheng-Hua Tan, and Jesper Jensen. Permutation invariant training of deep models for speaker-independent multi-talker speech separation. In Acoustics, Speech and Signal Processing (ICASSP), pp. 241–245. IEEE, 2017. + +# APPENDIX + +A REAL-VALUED CONVOLUTION & COMPLEX-VALUED CONVOLUTION + +In this section, we address the difference between the real-valued convolution and the complexvalued convolution. Given a complex-valued convolution filter $W = A + i B$ with real-valued matrices $\pmb { A }$ and $\textbf { { B } }$ , the complex-valued convolution can be interpreted as two different real-valued convolution operations with shared parameters, as illustrated in Figure 6 (b). For a fixed number of #Cha $\AA \ i n e l \ p r o d u c t = \# I n p u t \ c h a n n e l ( M ) \times \# O u t p u t \ c h a n n e l ( N )$ , the number of parameters of the complex-valued convolution becomes double of that of a real-valued convolution. Considering this fact, we built the pair of a real-valued network and a complex-valued network with the same number of parameters by reducing #Channel product of complex-valued convolution by half for a fair comparison. The detail of models reflecting this configuration is explained in Appendix B. + +![](images/b2e6920edf71e4738bc877916852adb08f1fa7335b1489df3a69799b1c85cfa3.jpg) +Figure 6: Illustration of (a) real-valued convolution and (b) complex-valued convolution. + +# B MODEL ARCHITECTURE + +In this section, we describe three different model architectures (DCUnet-20 (#params: 3.5M), DCUnet-16 ( $\#$ params: 2.3M), and DCUnet-10 (#params: 1.4M)) each in complex-valued network setting and real-valued network setting in Figure 7, 8, 9. Both complex-valued network $\left( \mathbb { C } \right)$ and realvalued network $( \mathbb { R } )$ have the same size of convolution filters with different number of channels to set the parameter equally. The largest model, Large-DCUnet-20, in Table 1 is also described in Figure 10. Every convolution operation is followed by batch normalization and an activation function as described in Figure 11. For the complex-valued network, the complex-valued version of batch normalization and activation function was used following Deep Complex Networks (Trabelsi et al., 2018). Note that in the very last layer of every model the batch normalization and leaky ReLU activation was not used and non-linearity function for mask was applied instead. The real-valued network configuration was not considered in the case of largest model. + +# C PROPERTIES OF WEIGHTED-SDR LOSS + +In this section, we summarize the properties of the proposed weighted-SDR loss. First, we show that the range of weighted-SDR loss is bounded and explain the conditions under which the minimum value is obtained. Next, we explain the gradients in the case of noise-only input. + +![](images/18fe8b0fe350f7c9f1230e291686d7e6d54e6af58fafb00b3cda3cbcd60f0cf1.jpg) +Figure 7: DCUnet-20 (20-layer): a model with 20 convolutional layers. + +![](images/e27862aec27194bf33be5f24d446a32a9a598117a513c735591da7ffe7d431cd.jpg) +Figure 8: DCUnet-16 (16-layer): a model with 16 convolutional layers. + +![](images/464629eaea84d85782602aef46ea169ce3af4774b1ceaf9ef32fc149e20347f6.jpg) +Figure 9: DCUnet-10 (10-layer): a model with 10 convolutional layers. + +![](images/90ee435567d6811d8a1a1579a4e87668013bb42b6677693c90963420730b17f6.jpg) +Figure 10: Large-DCUnet-20: a model with 20 convolutional layers with more number of channels per layer. + +![](images/bd47132f14f389a06e1bf72c2858b80ef697cccaa9ea67ac29e7b6ab32779ace.jpg) +Figure 11: Description of encoder and decoder block. $F _ { f }$ and $F _ { t }$ denote the convolution filter size along the frequency and time axis, respectively. $S _ { f }$ and $\ddot { S _ { t } }$ denote the stride size of convolution filter along the frequency and time axis, respectively. $O _ { \mathbb { C } }$ and $O _ { \mathbb { R } }$ denote the different number of channels in complex-valued network setting and real-valued network setting, respectively. The number of√ channels of $O _ { \mathbb { R } }$ is set to be roughly $\sqrt { 2 }$ times the number of channels of $O _ { \mathbb { C } }$ so that the number of trainable parameters of real-valued network and complex-valued network becomes approximately the same. + +Let $x$ denotes noisy speech with $T$ time step, $y$ denotes target source and $\hat { y }$ denotes estimated source. Then, $l o s s _ { w S D R } ( x , y , \hat { y } )$ is defined as follows: + +$$ +\mathit { l o s s } _ { \mathit { w S D R } } ( x , y , \hat { y } ) = - \alpha \frac { < y , \hat { y } > } { \| y \| \| \hat { y } \| } - ( 1 - \alpha ) \frac { < x - y , x - \hat { y } > } { \| x - y \| \| x - \hat { y } \| } +$$ + +where, $\alpha$ is the energy ratio between target source and noise, i.e., $\left\| y \right\| ^ { 2 } / ( \left\| y \right\| ^ { 2 } + \left\| x - y \right\| ^ { 2 } )$ + +Proposition 1. $l o s s _ { w S D R } ( x , y , \hat { y } )$ is bounded on $\left. l - I , I \right.$ . Moreover, for fixed $y \ne 0$ and $x - y \neq 0$ , the minimum value - $\mathbf { \nabla } . \mathit { l }$ can only be attained when ${ \hat { y } } = y$ , if $x \neq c y$ for $\forall c \in \mathbb { R }$ . + +Proof. Cauchy-Schwarz inequality states that for $a \in \mathbb { R } ^ { T }$ and $b \in \mathbb { R } ^ { T }$ , $- \| a \| \| b \| \leq < a , b > \leq$ $\| a \| \| b \|$ . By this inequality, [-1,1] becomes the range of $l o s s _ { w S D R }$ . To attain the minimum value, the equality condition of the Cauchy-Schwarz inequality must be satisfied. This equality condition is equivalent to $b = 0$ or $a = t b$ , for $\exists t \in \mathbb { R }$ . Applying the equality condition with the assumption $( y \neq 0 , x - y \neq 0 )$ to Equation 9 leads to $\hat { y } = t _ { 1 } y$ and $x - { \hat { y } } = t _ { 2 } ( x - y )$ , f or $\exists t _ { 1 } \in \mathbb { R }$ and $\exists t _ { 2 } \in \mathbb { R }$ . By adding these two equations, we can get $( 1 - t _ { 1 } ) x = ( t _ { 1 } - t _ { 2 } ) y$ . By the assumption $x \neq c y$ , which is generally satisfied for large $T$ , we can conclude $t _ { 1 } = 1 , t _ { 1 } = t _ { 2 }$ must be satisfied when the minimum value is attained. □ + +The following property of the weighted-SDR loss shows that the network can also learn from noiseonly training data. In experiments, we add small number $\epsilon$ to denominators of Equation 9. Thus for the case of $y = 0$ , Equation 9 becomes + +$$ +\mathit { l o s s } _ { w S D R } ( x , 0 , \hat { y } ) = - \frac { < x , x - \hat { y } > } { \| x \| \| x - \hat { y } \| + \epsilon } +$$ + +Proposition 2. When we parameterize ${ \hat { y } } = g _ { \boldsymbol { \theta } } ( { \boldsymbol { x } } )$ , the $\begin{array} { r } { l o s s _ { w S D R } ( x , y , g _ { \theta } ( x ) ) } \end{array}$ has a non-zero gradient with respect to $\theta$ even if the target source y is empty. + +Proof. We can calculate partial derivatives as follows: + +$$ +\begin{array} { r l } { { \frac { \partial l o s s _ { w S D R } ( x , y , g _ { \theta } ( x ) ) } { \partial \theta } \Big | _ { ( x = x _ { 0 } , y = 0 , \theta = \theta _ { 0 } ) } = \frac { \partial l o s s _ { w S D R } ( x _ { 0 } , 0 , g _ { \theta } ( x _ { 0 } ) ) } { \partial \theta } \Big | _ { \theta = \theta _ { 0 } } } } \\ & { = < \frac { x _ { 0 } } { \| x _ { 0 } \| } , \frac { \partial } { \partial \theta } \{ \frac { x _ { 0 } - g _ { \theta } ( x _ { 0 } ) } { \| x _ { 0 } - g _ { \theta } ( x _ { 0 } ) \| + \epsilon / \| x _ { 0 } \| } \} \Big | _ { \theta = \theta _ { 0 } } > } \end{array} +$$ + +Thus, the non-zero gradients with respect to $\theta$ can be back-propagated. + +# D IRREDUCIBLE ERRORS + +In this section, we illustrate two possible irreducible errors. Figure 12 (a) shows the irreducible phase error due to lack of phase estimation. Figure 12 (b) shows the irreducible error induced when bounding the range of mask. Not bounding the range of the mask makes the problem well-posed but it may suffer from the wide range of optimization search space because of the lack of prior knowledge on the distribution of cIRM. + +![](images/fa32b64835c251e121e7b604cd359d7881d74ae27955ab25b6ba847ad36fa868.jpg) +Figure 12: Two cases of possible irreducible error types in a TF bin in spectrogram. (a) The case where phase is not estimated and phase of mixture is reused. Even if the magnitude estimation is assumed to be perfect, there is still an irreducible phase error between the true source and the estimation. (b) The case where the magnitude of an estimated mask is bounded to 1. This again induces an irreducible error since there can be sources which have a higher magnitude than the magnitude of a mixture. + +# E SCATTER PLOTS OF CIRM + +The scatter plots of cIRM from training set is shown in Figure 13. We show four different scatter plots according to their SNR values of mixture (0, 5, 10, and 15 (dB)). Each scattered point of cIRM, $M _ { t , f }$ , is defined as follows: + +$$ +M _ { t , f } = \frac { \left| Y _ { t , f } \right| } { \left| X _ { t , f } \right| } e ^ { i ( \theta _ { Y _ { t , f } } - \theta _ { X _ { t , f } } ) } +$$ + +The scattered points near origin indicate the TF bins where the value of $\left| Y _ { t , f } \right|$ is significantly small compared to $\left| X _ { t , f } \right|$ . Therefore, those TF bins can be interpreted as the bins dominated with noise rather than source. On the other hand, the scattered points near (1,0) indicates the TF bins where the value of $\left| Y _ { t , f } \right|$ is almost the same as $\left| X _ { t , f } \right|$ . In this case, those TF bins can be interpreted as the bins dominated with source rather than noise. Therefore, as SNR becomes higher, the amount of TF bins dominated with clean source becomes larger compared to the lower SNR cases, and consequently the portion of real part close to 1 becomes larger as in Figure 13. + +# F VISUALIZATION OF ESTIMATED PHASE + +In this section, we show a supplementary visualization of phase of estimated speech. Although the raw phase information itself does not show a distinctive pattern, the hidden structure can be revealed with group delay, which is the negative derivative of the phase along frequency axis (Yegnanarayana & Murthy, 1992). With this technique, the phase information can be explicitly shown as in Figure 14. Figure 14 (d) shows the group delay of clean speech and the corresponding magnitude is shown in Figure 14 (a). The two representations shows that the group delay of phase has a similar structure to that of magnitude spectrogram. The estimated phase by our model is shown in in Figure 14 (c). While the group delay of noisy speech (Figure 14 (b)) does not show a distinctive harmonic pattern, our estimation show the harmonic pattern similar to the group delay of clean speech, as shown in the yellow boxes in Figure 14 (c) and (d). + +![](images/ea2a3771e7eb6841552ebc9bd7d5caa6d96f1359539e5a009709fbebfb836a12.jpg) +Figure 13: Four different scatter plots of cIRM according to the four different SNR values of input mixture in training set, (a) SNR: 0 (dB) (b) SNR: 5 (dB) (c) SNR: 10 (dB) (d) SNR: 15 (dB). + +![](images/7acbbbf14bf8e233f5bf1c11287177383896acb25b1c6becfca763dae0a51b22.jpg) +Figure 14: Illustration of phase group delay. The group delay of the estimated phase from our model (c) shows the similar pattern to that of clean speech (d). For better visualization, we only show the TF bins where the magnitude of clean speech spectrogram exceeds certain threshold 0.01. + +# G IMPORTANCE OF PHASE ESTIMATION + +In this section, to show the limitation of conventional approach (without phase estimation), we emphasize that the phase estimation is important, especially under low SNR condition (harsh condition). + +We first make an assumption that the estimation of phase information becomes more important when the given mixture has low SNR. Our reasoning behind this assumption is that if the SNR of a given mixture is low, the irreducible phase error is likely to be greater, hence a more room for improvement with phase estimation as illustrated in Figure 15. This can also be verified in Table 5 columns PhaseDist(C, N) and Phase Improvement where the values of both columns increase as SNR becomes higher. + +![](images/12b9d9801754fe5cc7bd817b128fda482a6f9e543085ebc1dfc0d1cc6c2c528f.jpg) +Figure 15: (a) The case where SNR of given mixture is high. In this case the source is likely to be dominant in the mixture. Therefore it is relatively easier to estimate ground truth source with better precision even when the phase is not estimated. (b) The case where SNR of given mixture is low. In this case the source is not dominant in the mixture. Therefore, the irreducible phase error is likely to be higher in low SNR conditions than higher SNR conditions. Under this circumstance, we assume the lack of phase estimation will result in a particularly bad system performance. + +To empirically show the importance of phase estimation, we show correlation between phase improvement and performance difference between the conventional method (without phase estimation) and our proposed method (with phase estimation) in Table 6. The performance difference was calculated by simply subtracting the evaluation results of conventional method from the evaluation results of our method with phase estimation. For fair comparison, both conventional method $( \mathbf { R M R } \mathbf { n } )$ and proposed method (cRMCn) were set to have the same number of parameters. Also, both models were trained with weighted-SDR loss. The results show that when the SNR is low, both the phase improvement and the performance difference are relatively higher than the results from higher SNR conditions. Furthermore, almost all results show an incremental increase of phase improvement and performance difference as the SNR decreases, which agrees on our assumption. Therefore we believe that phase estimation is important especially in harsh noisy conditions (low SNR conditions). + +Table 6: The performance difference between conventional method (without phase estimation) and our method (with phase estimation). The performance difference is presented with four different SNR values of mixture in test set. + +
SNR (dB)Phase ImprovementPerformance Difference
PESQCSIGCBAKCOVLSSNR
2.56.714°0.060.440.140.215.32
7.54.408°0.030.420.110.195.21
12.52.982°0.050.390.110.184.96
17.51.702°0.050.300.090.133.93
+ +# H COMPLETE QUALITATIVE RESULTS + +Table 7: Pairwise preference scores of four models including DCUnet. The scores are obtained by calculating the relative frequency the subjects prefer one method to the other method. Hard/Medium/Easy denote 2.5/7.5/17.5 SNR conditions in dB, respectively. Significance for each statistic is also described (n.s.: not significant, $\ast \colon \mathsf { p } { < } 0 . 0 5$ , $^ { * * }$ : $\mathrm { p { < } 0 . 0 1 }$ , $^ { \ast \ast \ast }$ : $\mathrm { p { < } 0 . 0 0 1 }$ ). + +
HardMedium EasyTotal
DCUnet > Deep Feature Loss0.90 (***）0.82 (***)0.69 (***)0.82 (***)
DCUnet > Wavenet0.98 (***)0.95 (***)0.75 (***)0.93 (***)
DCUnet > SEGAN0.99 (***)0.93 (***)0.71 (***)0.92 (***)
Deep Feature Loss > Wavenet0.67 (***)0.83 (***)0.48 (n.s.)0.74 (***)
Deep Feature Loss > SEGAN0.72 (***)0.81 (***)0.40 (**)0.73 (***)
SEGAN > Wavenet0.53 (n.s.)0.56(**)
0.58 (**)0.50 (n.s.)
\ No newline at end of file diff --git a/parse/train/SkeRTsAcYm/SkeRTsAcYm_content_list.json b/parse/train/SkeRTsAcYm/SkeRTsAcYm_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6f9a05829bc33618b0d6a67813e92f92d0aa0716 --- /dev/null +++ b/parse/train/SkeRTsAcYm/SkeRTsAcYm_content_list.json @@ -0,0 +1,2045 @@ +[ + { + "type": "text", + "text": "PHASE-AWARE SPEECH ENHANCEMENT WITH DEEP COMPLEX U-NET ", + "text_level": 1, + "bbox": [ + 176, + 99, + 725, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hyeong-Seok Choi1,3, Jang-Hyun $\\mathbf { K i m ^ { 2 , 3 } }$ , Jaesung $\\mathbf { H u h ^ { 3 } }$ , Adrian $\\mathbf { K i m ^ { 3 } }$ Jung-Woo $\\mathbf { H } \\mathbf { a } ^ { 3 }$ , Kyogu Lee1 ", + "bbox": [ + 183, + 169, + 681, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Department of Transdisciplinary Studies, Seoul National University, Seoul, Korea 2Department of Mathematical Sciences, Seoul National University, Seoul, Korea 3Clova AI Research, NAVER Corp., Seongnam, Korea kekepa15@snu.ac.kr, blue378@snu.ac.kr, jaesung.huh@navercorp.com, adrian.kim@navercorp.com, jungwoo.ha@navercorp.com, kglee@snu.ac.kr ", + "bbox": [ + 184, + 202, + 818, + 270 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 306, + 544, + 321 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Most deep learning-based models for speech enhancement have mainly focused on estimating the magnitude of spectrogram while reusing the phase from noisy speech for reconstruction. This is due to the difficulty of estimating the phase of clean speech. To improve speech enhancement performance, we tackle the phase estimation problem in three ways. First, we propose Deep Complex U-Net, an advanced U-Net structured model incorporating well-defined complex-valued building blocks to deal with complex-valued spectrograms. Second, we propose a polar coordinate-wise complex-valued masking method to reflect the distribution of complex ideal ratio masks. Third, we define a novel loss function, weighted source-to-distortion ratio (wSDR) loss, which is designed to directly correlate with a quantitative evaluation measure. Our model was evaluated on a mixture of the Voice Bank corpus and DEMAND database, which has been widely used by many deep learning models for speech enhancement. Ablation experiments were conducted on the mixed dataset showing that all three proposed approaches are empirically valid. Experimental results show that the proposed method achieves state-of-the-art performance in all metrics, outperforming previous approaches by a large margin1. ", + "bbox": [ + 233, + 338, + 764, + 574 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 601, + 336, + 616 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Speech enhancement is one of the most important and challenging tasks in speech applications where the goal is to separate clean speech from noise when noisy speech is given as an input. As a fundamental component for speech-related systems, the applications of speech enhancement vary from speech recognition front-end modules to hearing aid systems for the hearing-impaired (Erdogan et al., 2015; Weninger et al., 2015; Wang, 2017). ", + "bbox": [ + 174, + 632, + 823, + 702 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Due to recent advances in deep learning, the speech enhancement task has been able to reach high levels in performance through significant improvements. When using audio signals with deep learning models, it has been a common practice to transform a time-domain waveform to a time-frequency (TF) representation (i.e. spectrograms) via short-time-Fourier-transform (STFT). Spectrograms are represented as complex matrices, which are normally decomposed into magnitude and phase components to be used in real-valued networks. In tasks involving audio signal reconstruction, such as speech enhancement, it is ideal to perform correct estimation of both components. Unfortunately, complex-valued phase has been often neglected due to the difficulty of its estimation. This has led to the situation where most approaches focus only on the estimation of a magnitude spectrogram while reusing noisy phase information (Huang et al., 2014; Xu et al., 2015; Grais et al., 2016; Nugraha et al., 2016; Takahashi et al., 2018b). However, reusing phase from noisy speech has clear limitations, particularly under extremely noisy conditions, in other words, when signal-to-noise ratio (SNR) is low. This can be easily verified by simply using the magnitude spectrogram of clean speech with the phase spectrogram of noisy speech to reconstruct clean speech, as illustrated in Figure 1. We can clearly see that the difference between clean and estimated speech signals gets larger as the input SNR gets lower. ", + "bbox": [ + 174, + 709, + 825, + 890 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 821, + 146 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/44281215127d595e91a725f5d3e2398f9d143edac8059550ca1138d5b01daf99.jpg", + "image_caption": [ + "Figure 1: Effects of reusing phase of noisy speech under different SNR conditions. " + ], + "image_footnote": [], + "bbox": [ + 205, + 161, + 797, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A popular approach to speech enhancement is to optimize a mask which produces a spectrogram of clean speech when applied to noisy input audio. One of the first mask-based attempts to perform the task by incorporating phase information was the proposal of the phase-sensitive mask (PSM) (Erdogan et al., 2015). Since the performance of PSM was limited because of reusing noisy phase, later studies proposed using complex-valued ratio mask (cRM) to directly optimize on complex values (Williamson et al., 2016; Ephrat et al., 2018). We found this direction promising for phase estimation because it has been shown that a complex ideal ratio mask (cIRM) is guaranteed to give the best oracle performance out of other ideal masks such as ideal binary masks, ideal ratio masks, or PSMs (Wang et al., 2016). Moreover, this approach jointly estimates magnitude and phase, removing the need of separate models. To estimate a complex-valued mask, a natural desire would be to use an architecture which can handle complex-domain operations. Recent work gives a solution to this by providing deep learning building blocks adapted to complex arithmetic (Trabelsi et al., 2018). ", + "bbox": [ + 173, + 295, + 825, + 462 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we build upon previous studies to design a new complex-valued masking framework, based on a proposed variant of U-Net (Ronneberger et al., 2015), named Deep Complex U-Net (DCUnet). In our proposed framework, DCUnet is trained to estimate a complex ratio mask represented in polar coordinates with prior knowledge observable from ideal complex-valued masks. With the complex-valued estimation of clean speech, we can use inverse short-time-Fourier-transform (ISTFT) to convert a spectrogram into a time-domain waveform. Taking this as an advantage, we introduce a novel loss function which directly optimizes source-to-distortion ratio (SDR) (Vincent et al., 2006), a quantitative evaluation measure widely used in many source separation tasks. ", + "bbox": [ + 174, + 469, + 825, + 580 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions can be summarized as follows: ", + "bbox": [ + 174, + 587, + 495, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We propose a new neural architecture, Deep Complex U-Net, which combines the advantages of both deep complex networks and U-Net, yielding state-of-the-art performance. \n2. While pointing out limitations of current masking strategies, we design a new complexvalued masking method based on polar coordinates. \n3. We propose a new loss function weighted-SDR loss, which directly optimizes a well known quantitative evaluation measure. ", + "bbox": [ + 212, + 613, + 825, + 708 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 174, + 728, + 354, + 744 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Phase estimation for audio signal reconstruction has been a recent major interest within the audio source separation community because of its importance and difficulty. While iterative methods such as the Griffin-Lim algorithm and its variants (Griffin & Lim, 1984; Perraudin et al., 2013) aimed to address this problem, neural network-based approaches are recently attracting attention as noniterative alternatives. ", + "bbox": [ + 174, + 760, + 825, + 829 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "One major approach is to use an end-to-end model that takes audio as raw waveform inputs without using any explicit time-frequency (TF) representation computed via STFT (Pascual et al., 2017; Rethage et al., 2018; Stoller et al., 2018; Germain et al., 2018). Since raw waveforms inherently contain phase information, it is expected to achieve phase estimation naturally. Another method is to estimate magnitude and phase using two separate neural network modules which serially estimate magnitude and phase (Afouras et al., 2018; Takahashi et al., 2018a). In this framework, the phase estimation module uses noisy phase with predicted magnitude to estimate phase of clean speech. There is also a recent study which proposed to use additional layers with trainable discrete values for phase estimation (Roux et al., 2018). ", + "bbox": [ + 174, + 835, + 825, + 920 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A more straightforward method would be to jointly estimate magnitude and phase by using a continuous complex-valued ratio mask (cRM). Previous studies tried this joint estimation approach bounding the range of the cRM (Williamson et al., 2016; Ephrat et al., 2018). Despite the advantages of the cRM approach, previously proposed methods had limitations with regard to the loss function and the range of the mask which we will be returning with more details in Section 3 along with our proposed methods to alleviate these issues. As a natural extension to the works above, some studies have also undergone to examine whether complex-valued networks are useful when dealing with intrinsically complex-valued data. In the series of two works, complex-valued networks were shown to help singing voice separation performance with both fully connected neural networks and recurrent neural networks (Lee et al., 2017a;b). However, the approaches were limited as it ended up only switching the real-valued network into a complex-valued counterpart and leaving the other deep learning building blocks such as weight initialization and normalization technique in a realvalued manner. Also, the works do not show whether the phase was actually well estimated either quantitatively or qualitatively, only ending up showing that there was a performance gain. ", + "bbox": [ + 174, + 154, + 825, + 348 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PHASE-AWARE SPEECH ENHANCEMENT ", + "text_level": 1, + "bbox": [ + 174, + 368, + 532, + 382 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section we will provide details on our approach, starting with our proposed model Deep Complex U-Net, followed by the masking framework based on the model. Finally, we will introduce a new loss function to optimize our model, which takes a critical role for proper phase estimation. ", + "bbox": [ + 174, + 398, + 825, + 440 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Before getting into details, here are some notations used throughout the paper. The input mixture signal $\\bar { x ( n ) } \\overset { \\cdot } { = } y ( n ) + z ( n ) \\in \\mathbb { R }$ is assumed to be a linear sum of the clean speech signal $y ( n ) \\in \\mathbb { R }$ and noise $z ( n ) \\in \\mathbb { R }$ , where estimated speech is denoted as $\\hat { y } ( n ) \\in \\mathbb { R }$ . Each of the corresponding time-frequency $( t , f )$ representations computed by STFT is denoted as $X _ { t , f } \\in \\mathbb { C } , Y _ { t , f } \\in \\bar { \\mathbb { C } } , Z _ { t , f } \\in$ $\\mathbb { C }$ , and $\\hat { Y } _ { t , f } \\in \\mathbb { C }$ . The ground truth mask cIRM is denoted as $M _ { t , f } \\in \\mathbb { C }$ and the estimated cRM is denoted as $\\hat { M } _ { t , f } \\in \\mathbb { C }$ , where $M _ { t , f } = Y _ { t , f } / X _ { t , f }$ . ", + "bbox": [ + 174, + 448, + 825, + 540 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 DEEP COMPLEX U-NET ", + "text_level": 1, + "bbox": [ + 174, + 554, + 380, + 569 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/97b1a0b158f74dedaa4e3df7048d9ac98c9b57d80f09e60aa2417cc5b10ed965.jpg", + "image_caption": [ + "Figure 2: Illustration of speech enhancement framework with DCUnet. " + ], + "image_footnote": [], + "bbox": [ + 191, + 584, + 808, + 661 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The U-Net structure is a well known architecture composed as a convolutional autoencoder with skip-connections, originally proposed for medical imaging in computer vision community (Ronneberger et al., 2015). Furthermore, the use of real-valued U-Net has been shown to be also effective in many recent audio source separation tasks such as music source separation (Jansson et al., 2017; Stoller et al., 2018; Takahashi et al., 2018b), and speech enhancement (Pascual et al., 2017). Deep Complex U-Net (DCUnet) is an extended U-Net, refined specifically to explicitly handle complex domain operations. In this section, we will describe how U-Net is modified using the complex building blocks originally proposed by (Trabelsi et al., 2018). ", + "bbox": [ + 173, + 710, + 825, + 823 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Complex-valued Building Blocks. Given a complex-valued convolutional filter $W = A + i B$ with real-valued matrices $\\pmb { A }$ and $\\textbf { { B } }$ , the complex convolution operation on complex vector $\\pmb { h } = \\pmb { x } + i \\pmb { y }$ with $W$ is done by $W * h = ( A * x - B * y ) + i ( B * x + A * y )$ . In practice, complex convolutions can be implemented as two different real-valued convolution operations with shared real-valued convolution filters. Details are illustrated in Appendix A. ", + "bbox": [ + 174, + 829, + 825, + 898 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/9fa5f650774f98c9967d1e4710a3fc31bbdda2f0d1c1716e387a73a5f234097c.jpg", + "image_caption": [ + "Figure 3: Illustration of three cRM methods and their corresponding output range in complex space. The Neural Network (NN) can be either DCUnet $\\left( \\mathbb { C } \\right)$ or a corresponding real-valued U-Net $( \\mathbb { R } )$ with the same parameter size. Three different output ranges of cRM methods (1, 2 and 3) in (b) are shown in (1, 2 and 3) in (a). 1. Unbounded mask: a masking method without bounded outputs. 2. Bounded (sigmoid-sigmoid) mask: The proposed method by Ephrat et al. 3. Bounded (tanh) mask: Our proposed masking method. The detailed model specification is described in Appendix B. " + ], + "image_footnote": [], + "bbox": [ + 184, + 99, + 818, + 373 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Activation functions like ReLU were also adapted to the complex domain. In previous work, CReLU, an activation function which applies ReLU on both real and imaginary values, was shown to produce the best results out of many suggestions. Details on batch normalization and weight initialization for complex networks can be found in Trabelsi et al. (2018). ", + "bbox": [ + 174, + 502, + 825, + 558 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Modifying U-Net. The proposed Deep Complex U-Net is a refined U-Net architecture applied in STFT-domain. Modifications done to the original U-Net are as follows. Convolutional layers of UNet are all replaced to complex convolutional layers, initialized to meet the Glorot’s criteria (Glorot & Bengio, 2010). Here, the convolution kernels are set to be independent to each other by initializing the weight tensors as unitary matrices for better generalization and fast learning (Cogswell et al., 2015). Complex batch normalization is implemented on every convolutional layer except the last layer of the network. In the encoding stage, max pooling operations are replaced with strided complex convolutional layers to prevent spatial information loss. In the decoding stage, strided complex deconvolutional operations are used to restore the size of input. For the activation function, we modified the previously suggested CReLU into leaky CReLU, where we simply replace ReLU into leaky ReLU (Maas et al., 2013), making training more stable. Note that all experiments performed in Section 4 are done with these modifications. ", + "bbox": [ + 174, + 564, + 825, + 731 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 COMPLEX-VALUED MASKING ON POLAR COORDINATES", + "text_level": 1, + "bbox": [ + 174, + 748, + 602, + 762 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As our proposed model can handle complex values, we aim to estimate cRM for speech enhancement. Although it is possible to directly estimate the spectrogram of a clean source signal, it has been shown that better performance can be achieved by applying a weighting mask to the mixture spectrogram (Wang et al., 2014). One thing to note is that real-valued ratio masks (RM) only change the scale of the magnitude without changing phase, resulting in irreducible errors as illustrated in Appendix D. On the other hand, cRM also perform a rotation on the polar coordinates, allowing to correct phase errors. In other words, the estimated speech spectrogram $\\hat { Y } _ { t , f }$ is computed by multiplying the estimated mask $\\hat { M } _ { t , f }$ on the input spectrogram $X _ { t , f }$ as follows: ", + "bbox": [ + 174, + 775, + 825, + 892 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e0e866a8259332e59e6703d58ddaf55f803864086b94b8f29bfb250427227ebc.jpg", + "text": "$$\n\\hat { Y } _ { t , f } = \\hat { M } _ { t , f } \\cdot X _ { t , f } = \\Big | \\hat { M } _ { t , f } \\Big | \\cdot \\big | X _ { t , f } \\big | \\cdot e ^ { i ( \\theta _ { \\hat { M } _ { t , f } } + \\theta _ { X _ { t , f } } ) }\n$$", + "text_format": "latex", + "bbox": [ + 323, + 113, + 673, + 140 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this state, the real and imaginary values of the estimated cRM is unbounded. Although estimating an unbounded mask makes the problem well-posed (see Appendix $\\mathrm { D }$ for more information), we can imagine the difficulty of optimizing from an infinite search space compared to a bounded one. ", + "bbox": [ + 174, + 148, + 825, + 191 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Therefore, a few techniques have been tried to bound the range of cRM. For example, Williamson et al. tried to directly optimize a complex mask into a cIRM compressed to a heuristic bound (Williamson et al., 2016). However, this method was limited since it was only able to succeed in training the model by computing the error between cIRM and the predicted cRM which often leads to a degradation of performance (Wang et al., 2014; Yu et al., 2017). More recently, Ephrat et al. proposed a rectangular coordinate-wise cRM made with sigmoid compressions onto each of the real and imaginary parts of the output of the model (Ephrat et al., 2018). After then MSE between clean source $Y$ and estimated source $\\hat { Y }$ was computed in STFT-domain to train the model. However, the proposed masking method has two main problems regarding phase estimation. First, it suffers from the inherent problem of not being able to reflect the distribution of cIRM as shown in Figure 3 and Appendix E. Second, this approach results in a cRM with a restricted rotation range of $0 ^ { \\circ }$ to $9 0 °$ (only clock-wise), which makes it hard to correct noisy phase. ", + "bbox": [ + 173, + 196, + 825, + 367 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To alleviate these problems, we propose a polar coordinate-wise cRM method that imposes nonlinearity only on the magnitude part. More specifically, we use a hyperbolic tangent non-linearity to bound the range of magnitude part of the cRM be $[ 0 , 1 )$ which makes the mask bounded in an unit-circle in complex space. The corresponding phase mask is naturally obtained by dividing the output of the model with the magnitude of it. More formally, let $g ( \\cdot )$ be our neural network and the output of it be $O _ { t , f } = g ( X _ { t , f } )$ . The proposed complex-valued mask $\\hat { M } _ { t , f }$ is estimated as follows: ", + "bbox": [ + 173, + 372, + 825, + 460 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1de16af45a61a9850b8419567017c3fca8ec85fc866a29c6dba5b40de1e80d06.jpg", + "text": "$$\n\\hat { M } _ { t , f } = \\Bigr | \\hat { M } _ { t , f } \\Bigr | \\cdot e ^ { i \\theta _ { \\hat { M } _ { t , f } } } = \\hat { M } _ { t , f } ^ { m a g } \\cdot \\hat { M } _ { t , f } ^ { p h a s e }\n$$", + "text_format": "latex", + "bbox": [ + 354, + 464, + 643, + 492 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/91bfadfa9696a33b39862f59902162827610439851b19e20e9ee2230cb2f0c89.jpg", + "text": "$$\n\\hat { M } _ { t , f } ^ { m a g } = \\left\\{ { t a n h } ( | O _ { t , f } | ) \\begin{array} { l } { { ( { \\mathrm { b o u n d e d ~ c o n d . } } ) } } \\\\ { { { | O _ { t , f } | } } } \\end{array} , \\right. \\hat { M } _ { t , f } ^ { p h a s e } = O _ { t , f } / { | O _ { t , f } | }\n$$", + "text_format": "latex", + "bbox": [ + 251, + 501, + 746, + 545 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A summarized illustration of cRM methods is depicted in Figure 3. ", + "bbox": [ + 173, + 554, + 614, + 570 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 WEIGHTED-SDR LOSS", + "text_level": 1, + "bbox": [ + 176, + 585, + 372, + 601 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A popular loss function for audio source separation is mean squared error (MSE) between clean source $Y$ and estimated source $\\hat { Y }$ on the STFT-domain. However, it has been reported that optimizing the model with MSE in complex STFT-domain fails in phase estimation due to the randomness in phase structure (Williamson et al., 2016). As an alternative, it is possible to use a loss function defined in the time-domain instead, as raw waveforms contain inherent phase information. While MSE on waveforms can be an easy solution, we can expect it to be more effective if the loss function is directly correlated with well-known evaluation measures defined in the time-domain. ", + "bbox": [ + 173, + 612, + 825, + 712 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here, we propose an improved loss function weighted-SDR loss by building upon a previous work which attempts to optimize a standard quality measure, source-to-distortion ratio (SDR) (Venkataramani et al., 2017). The original loss function $l o s s _ { V e n }$ suggested by Venkataramani et al. is formulated upon the observation from Equation 4, where $y$ is the clean source signal and $\\hat { y }$ is the estimated source signal. In practice, the negative reciprocal is optimized as in Equation 5. ", + "bbox": [ + 174, + 718, + 825, + 789 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2559934564a994f57fa1bec86f7bcc34e59b93547039804f003197f883ea7b15.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\underset { \\hat { y } } { \\mathrm { m a x } } \\ S D R ( y , \\hat { y } ) : = \\operatorname* { m a x } _ { \\hat { y } } \\frac { < y , \\hat { y } > ^ { 2 } } { | | y | | ^ { 2 } | | \\hat { y } | | ^ { 2 } - < y , \\hat { y } > ^ { 2 } } \\propto \\operatorname* { m i n } _ { \\hat { y } } \\frac { | | \\hat { y } | | ^ { 2 } } { < y , \\hat { y } > ^ { 2 } } } \\\\ { l o s s _ { V e n } ( y , \\hat { y } ) : = - \\frac { < y , \\hat { y } > ^ { 2 } } { | | \\hat { y } | | ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 811, + 725, + 886 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Although using Equation 5 works as a loss function, there are a few critical flaws in the design. First, the lower bound becomes $- \\| y \\| ^ { 2 }$ , which depends on the value of $y$ causing fluctuation in the loss values when training. Second, when the target $y$ is empty (i.e., $y = 0$ ) the loss becomes zero, preventing the model to learn from noisy-only data due to zero gradients. Finally, the loss function is not scale sensitive, meaning that the loss value is the same for $\\hat { y }$ and $c \\hat { y }$ , where $c \\in \\mathbb { R }$ . ", + "bbox": [ + 173, + 892, + 825, + 925 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To resolve these issues, we redesigned the loss function by giving several modifications to Equation 5. First, we made the lower bound of the loss function independent to the source $y$ by restoring back the term $\\left\\| y \\right\\| ^ { 2 }$ and applying square root as in Equation 6. This makes the loss function bounded within the range [-1, 1] and also be more phase sensitive, as inverted phase gets penalized as well. ", + "bbox": [ + 174, + 152, + 825, + 212 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5398494a6a602e0124002b55f96e52bdc1b95abc8c2b58f9229aefd1eb355ea5.jpg", + "text": "$$\n\\mathit { l o s s } _ { \\mathit { S D R } } ( y , \\hat { y } ) : = - \\sqrt { - \\frac { \\mathit { l o s s } _ { \\mathit { V e n } } } { \\left\\| y \\right\\| ^ { 2 } } } = - \\frac { < y , \\hat { y } > } { \\left\\| y \\right\\| \\left\\| \\hat { y } \\right\\| }\n$$", + "text_format": "latex", + "bbox": [ + 338, + 218, + 660, + 260 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Expecting to be complementary to source prediction and to propagate errors for noise-only samples, we also added a noise prediction term $l o s s _ { S D R } ( z , \\hat { z } )$ . To properly balance the contributions of each loss term and solve the scale insensitivity problem, we weighted each term proportional to the energy of each signal. The final form of the suggested weighted-SDR loss is as follows: ", + "bbox": [ + 174, + 266, + 825, + 321 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c823f51cb1dd7e0a902278eb0f9fd88e0a81a4612fcc3dae96decc4d67abe56d.jpg", + "text": "$$\nl o s s _ { w S D R } ( x , y , \\hat { y } ) : = \\alpha l o s s _ { S D R } ( y , \\hat { y } ) + ( 1 - \\alpha ) l o s s _ { S D R } ( z , \\hat { z } )\n$$", + "text_format": "latex", + "bbox": [ + 284, + 329, + 714, + 347 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where, $\\hat { z } = x - \\hat { y }$ is estimated noise and $\\alpha = | | y | | ^ { 2 } / ( | | y | | ^ { 2 } + | | z | | ^ { 2 } )$ is the energy ratio between clean speech $y$ and noise $z$ . Note that although weighted SDR loss is a time-domain loss function, it can be backpropagated through our framework. Specifically, STFT and ISTFT operations are implemented as 1-D convolution and deconvolution layers consisting of fixed filters initialized with the discrete Fourier transform matrix. The detailed properties of the proposed loss function are in Appendix C. ", + "bbox": [ + 174, + 352, + 825, + 424 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 444, + 326, + 460 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 EXPERIMENT SETUP ", + "text_level": 1, + "bbox": [ + 176, + 476, + 357, + 489 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Dataset. For all experiments, we used the same experimental setups as previous works in order to perform direct performance comparison (Pascual et al., 2017; Rethage et al., 2018; Soni et al., 2018; Germain et al., 2018). Noise and clean speech recordings were provided from the Diverse Environments Multichannel Acoustic Noise Database (DEMAND) (Thiemann et al., 2013) and the Voice Bank corpus (Veaux et al., 2013), respectively, each recorded with sampling rate of $4 8 \\mathrm { k H z }$ . Mixed audio inputs used for training were composed by mixing the two datasets with four signalto-noise ratio (SNR) settings (15, 10, 5, and 0 (dB)), using 10 types of noise (2 synthetic $+ \\ 8$ from DEMAND) and 28 speakers from the Voice Bank corpus, creating 40 conditional patterns for each speech sample. The test set inputs were made with four SNR settings different from the training set (17.5, 12.5, 7.5, and 2.5 (dB)), using the remaining 5 noise types from DEMAND and 2 speakers from the Voice Bank corpus. Note that the speaker and noise classes were uniquely selected for the training and test sets. ", + "bbox": [ + 173, + 501, + 825, + 667 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Pre-processing. The original raw waveforms were first downsampled from 48kHz to 16kHz. For the actual model input, complex-valued spectrograms were obtained from the downsampled waveforms via STFT with a 64ms sized Hann window and 16ms hop length. ", + "bbox": [ + 174, + 675, + 825, + 718 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Implementation. All experiments were implemented and fine-tuned with NAVER Smart Machine Learning (NSML) platform (Sung et al., 2017; Kim et al., 2018). ", + "bbox": [ + 173, + 724, + 821, + 752 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 COMPARISON RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 770, + 377, + 784 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this subsection, we compare overall speech enhancement performance of our method with previously proposed algorithms. As a baseline approach, Wiener filtering (Wiener) with a priori noise SNR estimation was used, along with recent deep-learning based models which are briefly described as the following: SEGAN: a time-domain U-Net model optimized with generative adversarial networks. Wavenet: a time-domain non-causal dilated wavenet-based network. MMSE-GAN: a timefrequency masking-based method with modified adversarial training method. Deep Feature Loss: a time-domain dilated convolution network trained with feature loss from a classifier network. ", + "bbox": [ + 173, + 795, + 825, + 893 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/b5aaf5e7d10cde49b1ad7a5468086a135d36b1fa4f1e84dbf703db0caec97890.jpg", + "table_caption": [ + "Table 1: Quantitative evaluation results of other algorithms and proposed methods (DCUnet-20, Large-DCUnet-20). Higher score means better performance where bold text indicates highest score per evaluation measure. CSIG: Mean opinion score (MOS) predictor of signal distortion CBAK: MOS predictor of background-noise intrusiveness COVL: MOS predictor of overall signal quality PESQ: Perceptual evaluation of speech quality SSNR: Segmental SNR. All evaluation measures were computed by using open source implementation.2 " + ], + "table_footnote": [], + "table_body": "
CSIGCBAKCOVLPESQSSNR
Wiener (Scalart et al., 1996)3.232.682.672.225.07
SEGAN (Pascual et al., 2017)3.482.942.802.167.73
Wavenet (Rethage et al., 2018)3.623.232.9811
MMSE-GAN (Soni et al., 2018)3.803.123.142.53-
Deep Feature Loss (Germain et al., 2018)3.863.333.2211
DCUnet-20 (ours)4.244.003.693.1315.95
Large-DCUnet-20 (ours)4.344.103.813.2416.85
", + "bbox": [ + 194, + 202, + 805, + 330 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For comparison, we used the configuration of using a 20-layer Deep Complex U-Net (DCUnet20) to estimate a tanh bounded cRM, optimized with weighted-SDR loss. As a showcase for the potential of our approach, we also show results from a larger DCUnet-20 (Large-DCUnet-20) which has more channels in each layer. Both architectures are specified in detail in Appendix B. Results show that our proposed method outperforms the previous state-of-the-art methods with respect to all metrics by a large margin. Additionally, we can also see that larger models yield better performance. We see the reason to this significant improvement coming from the phase estimation quality of our method, which we plan to investigate in later sections. ", + "bbox": [ + 173, + 361, + 825, + 474 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 ABLATION STUDIES ", + "text_level": 1, + "bbox": [ + 176, + 491, + 352, + 505 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/02806868a0fe890929ffbd9c509f1645d058bb8406bf26c42280a781ff92f1d8.jpg", + "table_caption": [ + "Table 2: Table of quantitative evaluation results with corresponding mask and loss function in three different model configurations (DCU-10, DCU-16 and DCU-20). The bold font indicates the best loss function when fixing the masking method. The underline indicates the best masking method when fixing the loss function. " + ], + "table_footnote": [], + "table_body": "
CSIGCBAKCOVLPESQSSNR
DCU-10|Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR||Spc Wav wSDR|SpcWav wSDR
UBD|3.51 3.713.72 |3.50 3.523.56|3.13 3.153.19|2.78 2.61 2.67[11.86 13.1813.37
BDSS2.94 3.193.21 3.43 3.353.392.81 2.852.892.79 2.61 2.6711.61 12.30 12.30
BDT3.30 3.743.723.52 3.57 3.603.02 3.193.222.79 2.66 2.7212.30 13.5413.60
DCU-16| |SpcWav wSDR||Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD3.97 4.004.07 13.66 3.723.753.46 3.433.482.96 2.85 2.9312.75 14.2514.44
BDSS3.63 3.573.72 3.59 3.513.563.28 3.153.272.98 2.82 2.8712.05 12.8912.89
BDT3.97 4.034.10 3.70 3.743.773.49 3.453.523.01 2.88 2.9312.96 14.3914.44
DCU-20| |Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD|4.02 4.154.25 3.88 3.934.01|3.57 3.593.70[3.11 3.01 3.1314.81 16.0316.36
BDSS3.77 3.743.93 3.68 3.653.723.40 3.333.473.06 2.98 3.0512.80 13.6613.68
BDT4.02 4.184.24 3.87 3.954.003.58 3.643.693.13 3.08 3.1314.59 15.85 15.95
", + "bbox": [ + 173, + 595, + 838, + 824 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Masking strategy and loss functions. In this experiment, we show the evaluation results on how the various masking strategies and loss functions affect the performance of speech enhancement. Three masking strategies (Unbounded (UBD); Ephrat et al.: Bounded (sig-sig) (BDSS); Bounded (tanh) (BDT)) are compared to see the effect of each different masking method. Then, to compare our proposed loss function (weighted-SDR) with MSE, we compared two MSE in STFT-domain and time-domain (Spectrogram-MSE (Spc); Wave-MSE (Wav); and proposed: weighted-SDR (wSDR)). ", + "bbox": [ + 173, + 847, + 825, + 904 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2 shows the jointly combined results on varied masking strategies and loss functions, where three models (DCU-10 (1.4M), DCU-16 (2.3M), and DCU-20 (3.5M)) are investigated to see how architectural differences in the model affect quantitative results. In terms of masking strategy, the proposed BDT mask mostly yields better results than UBD mask in DCU-10 and DCU-16, implying the importance of limiting optimization space with prior knowledge. However, in the case of DCU-20, UBD mask was able to frequently surpass the performance of BDT mask. Intuitively, this indicates that when the number of parameter gets large enough, the model is able to fit the distribution of data well even when the optimization space is not bounded. In terms of the loss function, almost every result shows that optimizing with wSDR loss gives the best result. However, we found out that Spc loss often provides better PESQ results than wSDR loss for DCU-10 and DCU-16 except DCU-20 case where Spc and wSDR gave similar PESQ results. ", + "bbox": [ + 174, + 138, + 825, + 291 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/95a337c1cc3e85f5d350fe100a72dd5dd6266c505b0630ac72473e1f42efdf41.jpg", + "table_caption": [ + "Table 3: Table of quantitative evaluation results from three different settings (cRMCn: Complexvalued output/Complex-valued network, cRMRn: Complex-valued output/Real-valued network, and $\\mathbf { R M } \\mathbb { R } \\mathbf { n }$ : Real-valued output/Real-valued network) to show the appropriateness of using complexvalued networks for speech enhancement. Bold font indicates the best results. " + ], + "table_footnote": [], + "table_body": "
20-layer16-layer10-layer
cRMCncRMRnRMRncRMCncRMRnRMRncRMCncRMRnRMRn
CSIG4.244.214.064.104.063.883.743.713.71
CBAK4.003.923.403.773.753.333.603.593.23
COVL3.693.653.403.523.493.263.223.203.01
PESQ3.133.072.742.932.912.662.722.722.51
SSNR15.9515.549.9014.4414.259.6613.6013.499.43
", + "bbox": [ + 186, + 378, + 812, + 498 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Validation on complex-valued network and mask. In order to show that complex neural networks are effective, we compare evaluation results of DCUnet $( \\mathbb { C } \\mathrm { n } )$ and its corresponding real-valued UNet setting with the same parameter size $( \\mathbb { R } \\mathbf { n } )$ . For the real-valued network, we tested two settings $\\mathbf { c R M } \\mathbb { R } \\mathbf { n }$ and $\\mathbf { R M } \\mathbb { R } \\mathbf { n }$ to show the effectiveness of phase estimation. The first setting takes a complexvalued spectrogram as an input, estimating a complex ratio mask (cRM) with a tanh bound. The second setting takes a magnitude spectrogram as an input, estimating a magnitude ratio mask (RM) with a sigmoid bound. All models were trained with weighted-SDR loss, where the ground truth phase was given while training $\\mathbf { R M } \\mathbb { R } \\mathbf { n }$ . Additionally, all models were trained on different number of parameters (20-layer (3.5M), 16-layer (2.3M), and 10-layer (1.4M)) to show that the results are consistent regardless of model capacity. Detailed network architectures for each model are illustrated in Appendix B. ", + "bbox": [ + 173, + 518, + 825, + 671 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Table 3, evaluation results show that our approach cRMCn makes better results than conventional method $\\mathbf { R M } \\mathbb { R } \\mathbf { n }$ for all cases, showing the effectiveness of phase correction. Also, cRMCn gives better results than $\\mathbf { c R M } \\mathbb { R } \\mathbf { n }$ , which indicates that using complex-valued networks consistently improve the performance of the network. Note that these results are consistent through every evaluation measure and model size. ", + "bbox": [ + 174, + 678, + 823, + 748 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 QUALITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 765, + 405, + 780 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We performed qualitative evaluations by obtaining preference scores between the proposed DCUnet (Large-DCUnet-20) and baseline methods. 15 utterance samples with different noise levels were selected from the test set and used for subjective listening tests. For each noisy sample, all possible six pairs of denoised audio samples from four different algorithms were presented to the participants in a random order, resulting in 90 pairwise comparisons to be made by each subject. For each comparison, participants were presented with three audio samples - original noisy speech and two denoised speech samples by two randomly selected algorithms - and instructed to choose either a preferred sample (score 1) or “can’t decide” (score 0.5). A total of 30 subjects participated in the listening test, and the results are presented in Table 4 and in Table 7. ", + "bbox": [ + 173, + 791, + 825, + 916 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 4: Preference scores of DCUnet compared to other baseline models. The scores were obtained by calculating the relative frequency the subjects prefer one method to the other method. Hard/Medium/Easy denote $2 . 5 / 7 . 5 / 1 7 . 5$ SNR conditions in dB, respectively. All statistics had significance of $\\mathrm { p } { < } 0 . 0 0 1$ . Complete pairwise preference scores are presented in Appendix H. ", + "bbox": [ + 173, + 101, + 823, + 157 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5273a5d39e03236dc3e9d1aa19bbf4195975302772062e2e110f843086e937da.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
HardMediumEasyTotal
DCUnet > Deep Feature Loss0.900.820.690.82
DCUnet > Wavenet0.980.950.750.93
DCUnet > SEGAN0.990.930.710.92
", + "bbox": [ + 290, + 175, + 709, + 247 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/cb0f9851f16949be8934410c702ff74a5beae189f043cb1f8fd000d8e2394515.jpg", + "image_caption": [ + "Figure 4: Scatter plots of estimated cRMs with 9 different mask and loss function configurations for a randomly picked noisy speech signal. Each scatter plot shows the distribution of complex values from an estimated cRM. The leftmost plot is from the cIRM for the given input. We can observe that most real-values are distributed around 0 and 1, while being relatively sparse in between. The configuration that fits the most to this distribution pattern is observed in the red dotted box which is achieved by the combination of our proposed methods (Bounded (tanh) and weighted-SDR). " + ], + "image_footnote": [], + "bbox": [ + 217, + 273, + 784, + 526 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 4 shows that DCUnet clearly outperforms the other methods in terms of preference scores in every SNR condition. These differences are statistically significant as confirmed by pairwise one-tailed t-tests. Furthermore, the difference becomes more obvious as the input SNR condition gets worse, which supports our motivation that accurate phase estimation is even more important under harsh noisy conditions. This is further confirmed by in-depth quantitative analysis of the phase distance as described in Section 5 and Table 5. ", + "bbox": [ + 173, + 647, + 825, + 731 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 IN-DEPTH ANALYSIS ON PHASE ", + "text_level": 1, + "bbox": [ + 176, + 752, + 464, + 767 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we aim to provide constructive insights on phase estimation by analyzing how and why our proposed method is effective. We first visualized estimated complex masks with scatter plots in Figure 4 for each masking method and loss function configuration from Table 2. The plots clearly show that the Bounded (sig-sig) mask by Ephrat et al. fails to reflect the distribution of the complex ideal ratio mask (cIRM), whereas the proposed Bounded (tanh) or Unbounded mask tries to follow the ground truth. One interesting finding is that, while time-domain loss functions such as weighted-SDR or Wave-MSE can capture the distribution of the ground truth, Spectrogram-MSE fails to capture the distribution of the cIRM. More specifically, when we use Spectrogram-MSE, the imaginary values of the predicted cRMs are almost always zero, indicating that it ends up only scalin(e.g., $\\bar { ( } X _ { t , f } ^ { r e a l } + i X _ { t , f } ^ { i m a g } ) ( a + \\bar { i } \\cdot \\bar { 0 } ) = a ( X _ { t , f } ^ { r e a l } + i X _ { t , f } ^ { i m a g } ) )$ he phase of noisy speech with rotations. ", + "bbox": [ + 174, + 782, + 825, + 909 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/2fee5faa352a331c4b09236534209a0c3b7a0321fef337a23dd190b8e99fd896.jpg", + "image_caption": [ + "Figure 5: Illustration of four wave-plot segments of estimated speech with reference clean speech. (a) shows the wave-plot of synthesized speech with clean magnitude speech and noisy phase (CMNP). Three other cases show the wave-plots of estimated speech with different loss functions (b) Spectrogram-MSE, (c) Wave-MSE, and (d) weighted-SDR. " + ], + "image_footnote": [], + "bbox": [ + 178, + 101, + 815, + 295 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 393, + 821, + 428 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In order to demonstrate this effect in an alternate perspective, we also plotted estimated waveforms for each loss function in Figure 5. As one can notice from Figure 5 (c) $\\&$ (d), estimated speech from models optimized with time-domain loss functions are well-aligned with clean speech. However, Figure 5 (a) $\\&$ (b) align poorly to the ground truth while showing almost identical estimations. Note that Figure 5 (a) is the best possible result without estimating the phase of clean speech. This again confirms that optimizing the model with Spectrogram-MSE makes it difficult to learn phase correction, meaning that the model ends up reusing noisy phase just like conventional approaches. ", + "bbox": [ + 173, + 433, + 825, + 531 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "To explicitly support these observations, we would need a quantitative measure for phase estimation. Here, we define the phase distance between target spectrogram $( A )$ and estimated spectrogram $( B )$ as the weighted average of angle between corresponding complex TF bins, where each bin is weighted by the magnitude of target speech $\\big \\langle \\big _ { { A t } , f \\big | } \\big \\rangle$ to emphasize the relative importance of each TF bin. Phase distance is formulated as the following: ", + "bbox": [ + 174, + 536, + 825, + 608 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/6cb32544b464963c73fbc6c5de4309a82e85d001f0b6e0f5b334411ab53bb72e.jpg", + "text": "$$\nP h a s e D i s t ( A , B ) = \\sum _ { t , f } \\frac { \\left| A _ { t , f } \\right| } { \\sum _ { t ^ { \\prime } , f ^ { \\prime } } \\left| A _ { t ^ { \\prime } , f ^ { \\prime } } \\right| } \\angle ( A _ { t , f } , B _ { t , f } )\n$$", + "text_format": "latex", + "bbox": [ + 316, + 613, + 679, + 656 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "where, $\\angle ( A _ { t , f } , B _ { t , f } )$ represents the angle between $A _ { t , f }$ and $B _ { t , f }$ , having a range of [0,180]. ", + "bbox": [ + 173, + 662, + 777, + 679 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The phase distance between clean and noisy speech (PhaseDist $( \\mathbf { C } , \\mathbf { N } ) ,$ ) and the phase distance between clean and estimated speech (PhaseDist $( \\mathbf { C } , \\mathbf { E } ) )$ are presented in Table 5. The results show that the best phase improvement (Phase Improvement $=$ PhaseDist(C, N) − PhaseDist(C, E)) is obtained with wSDR loss under every SNR condition. Also Spc loss gives the worst results, again reinforcing our observation. Analysis between the phase improvement and performance improvement is further discussed in Appendix G. ", + "bbox": [ + 174, + 684, + 825, + 768 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 789, + 318, + 805 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we proposed Deep Complex U-Net which combines two models to deal with complexvalued spectrograms for speech enhancement. In doing so, we designed a new complex-valued masking method optimized with a novel loss function, weighted-SDR loss. Through ablation studies, we showed that the proposed approaches are effective for more precise phase estimation, resulting in state-of-the-art performance for speech enhancement. Furthermore, we conducted both quantitative and qualitative studies and demonstrated that the proposed method is consistently superior to the previously proposed algorithms. ", + "bbox": [ + 174, + 819, + 825, + 917 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/434525d265e9015f41b3ea1276c20639e795f790631568d1425d1cf5cd044643.jpg", + "table_caption": [ + "Table 5: Phase distance and phase improvement under four different SNR conditions. The Spc, Wav, and wSDR column show results from models trained with three different objectives SpectrogramMSE, Wave-MSE, and weighted-SDR, respectively. " + ], + "table_footnote": [], + "table_body": "
SNR (dB)PhaseDist(C, N)PhaseDist(C,E)Phase Improvement
SpcWavwSDRSpcWavwSDR
2.514.521°10.512°9.288°7.807°4.009°5.233°6.714°
7.510.066°7.548°6.919°5.658°2.518°3.147°4.408°
12.57.197°5.455°5.105°4.215°1.742°2.092°2.982°
17.54.853°3.949°3.872°3.151°0.905°0.981°1.702°
", + "bbox": [ + 191, + 170, + 805, + 276 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In the near future, we plan to apply our system to various separation tasks such as speaker separation or music source separation. Another important direction is to extend the proposed model to deal with multichannel audio since accurate estimation of phase is even more critical in multichannel environments (Wang et al., 2018). Apart from separation, our approach can be generalized to various audio-related tasks such as dereverberation, bandwidth extension or phase estimation networks for text-to-speech systems. Taking advantage of sequence modeling, it may also be interesting to find further extensions with complex-valued LSTMs (Arjovsky et al., 2016; Wisdom et al., 2016). ", + "bbox": [ + 174, + 304, + 825, + 402 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 420, + 334, + 433 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "This work was partly supported by Next-Generation Information Computing Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (MSIT; Grant No. NRF-2017M3C4A7078548), partly supported by National Research Foundation of Korea (NRF) funded by the Korea government (MSIT; Grant No. NRF2017R1E1A1A01076284), and partly supported by the Creative Industrial Technology Development Program (10053249) funded by the Ministry of Trade, Industry and Energy (MOTIE, Korea). 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", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A REAL-VALUED CONVOLUTION & COMPLEX-VALUED CONVOLUTION", + "bbox": [ + 178, + 133, + 674, + 147 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we address the difference between the real-valued convolution and the complexvalued convolution. Given a complex-valued convolution filter $W = A + i B$ with real-valued matrices $\\pmb { A }$ and $\\textbf { { B } }$ , the complex-valued convolution can be interpreted as two different real-valued convolution operations with shared parameters, as illustrated in Figure 6 (b). For a fixed number of #Cha $\\AA \\ i n e l \\ p r o d u c t = \\# I n p u t \\ c h a n n e l ( M ) \\times \\# O u t p u t \\ c h a n n e l ( N )$ , the number of parameters of the complex-valued convolution becomes double of that of a real-valued convolution. Considering this fact, we built the pair of a real-valued network and a complex-valued network with the same number of parameters by reducing #Channel product of complex-valued convolution by half for a fair comparison. The detail of models reflecting this configuration is explained in Appendix B. ", + "bbox": [ + 174, + 159, + 825, + 285 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/b2e6920edf71e4738bc877916852adb08f1fa7335b1489df3a69799b1c85cfa3.jpg", + "image_caption": [ + "Figure 6: Illustration of (a) real-valued convolution and (b) complex-valued convolution. " + ], + "image_footnote": [], + "bbox": [ + 194, + 303, + 805, + 556 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B MODEL ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 622, + 372, + 636 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we describe three different model architectures (DCUnet-20 (#params: 3.5M), DCUnet-16 ( $\\#$ params: 2.3M), and DCUnet-10 (#params: 1.4M)) each in complex-valued network setting and real-valued network setting in Figure 7, 8, 9. Both complex-valued network $\\left( \\mathbb { C } \\right)$ and realvalued network $( \\mathbb { R } )$ have the same size of convolution filters with different number of channels to set the parameter equally. The largest model, Large-DCUnet-20, in Table 1 is also described in Figure 10. Every convolution operation is followed by batch normalization and an activation function as described in Figure 11. For the complex-valued network, the complex-valued version of batch normalization and activation function was used following Deep Complex Networks (Trabelsi et al., 2018). Note that in the very last layer of every model the batch normalization and leaky ReLU activation was not used and non-linearity function for mask was applied instead. The real-valued network configuration was not considered in the case of largest model. ", + "bbox": [ + 174, + 647, + 825, + 800 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C PROPERTIES OF WEIGHTED-SDR LOSS", + "text_level": 1, + "bbox": [ + 176, + 818, + 470, + 832 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this section, we summarize the properties of the proposed weighted-SDR loss. First, we show that the range of weighted-SDR loss is bounded and explain the conditions under which the minimum value is obtained. Next, we explain the gradients in the case of noise-only input. ", + "bbox": [ + 176, + 843, + 823, + 886 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/18fe8b0fe350f7c9f1230e291686d7e6d54e6af58fafb00b3cda3cbcd60f0cf1.jpg", + "image_caption": [ + "Figure 7: DCUnet-20 (20-layer): a model with 20 convolutional layers. " + ], + "image_footnote": [], + "bbox": [ + 186, + 132, + 820, + 239 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/e27862aec27194bf33be5f24d446a32a9a598117a513c735591da7ffe7d431cd.jpg", + "image_caption": [ + "Figure 8: DCUnet-16 (16-layer): a model with 16 convolutional layers. " + ], + "image_footnote": [], + "bbox": [ + 246, + 335, + 758, + 443 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/464629eaea84d85782602aef46ea169ce3af4774b1ceaf9ef32fc149e20347f6.jpg", + "image_caption": [ + "Figure 9: DCUnet-10 (10-layer): a model with 10 convolutional layers. " + ], + "image_footnote": [], + "bbox": [ + 331, + 537, + 674, + 646 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/90ee435567d6811d8a1a1579a4e87668013bb42b6677693c90963420730b17f6.jpg", + "image_caption": [ + "Figure 10: Large-DCUnet-20: a model with 20 convolutional layers with more number of channels per layer. " + ], + "image_footnote": [], + "bbox": [ + 186, + 741, + 820, + 849 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/bd47132f14f389a06e1bf72c2858b80ef697cccaa9ea67ac29e7b6ab32779ace.jpg", + "image_caption": [ + "Figure 11: Description of encoder and decoder block. $F _ { f }$ and $F _ { t }$ denote the convolution filter size along the frequency and time axis, respectively. $S _ { f }$ and $\\ddot { S _ { t } }$ denote the stride size of convolution filter along the frequency and time axis, respectively. $O _ { \\mathbb { C } }$ and $O _ { \\mathbb { R } }$ denote the different number of channels in complex-valued network setting and real-valued network setting, respectively. The number of√ channels of $O _ { \\mathbb { R } }$ is set to be roughly $\\sqrt { 2 }$ times the number of channels of $O _ { \\mathbb { C } }$ so that the number of trainable parameters of real-valued network and complex-valued network becomes approximately the same. " + ], + "image_footnote": [], + "bbox": [ + 184, + 104, + 813, + 233 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Let $x$ denotes noisy speech with $T$ time step, $y$ denotes target source and $\\hat { y }$ denotes estimated source. Then, $l o s s _ { w S D R } ( x , y , \\hat { y } )$ is defined as follows: ", + "bbox": [ + 173, + 371, + 823, + 400 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/3fa230897872069440aff3b76eb3045854cb87f1f5559a1a2e6287370d1f431b.jpg", + "text": "$$\n\\mathit { l o s s } _ { \\mathit { w S D R } } ( x , y , \\hat { y } ) = - \\alpha \\frac { < y , \\hat { y } > } { \\| y \\| \\| \\hat { y } \\| } - ( 1 - \\alpha ) \\frac { < x - y , x - \\hat { y } > } { \\| x - y \\| \\| x - \\hat { y } \\| }\n$$", + "text_format": "latex", + "bbox": [ + 285, + 406, + 712, + 440 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where, $\\alpha$ is the energy ratio between target source and noise, i.e., $\\left\\| y \\right\\| ^ { 2 } / ( \\left\\| y \\right\\| ^ { 2 } + \\left\\| x - y \\right\\| ^ { 2 } )$ ", + "bbox": [ + 171, + 448, + 767, + 465 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proposition 1. $l o s s _ { w S D R } ( x , y , \\hat { y } )$ is bounded on $\\left. l - I , I \\right.$ . Moreover, for fixed $y \\ne 0$ and $x - y \\neq 0$ , the minimum value - $\\mathbf { \\nabla } . \\mathit { l }$ can only be attained when ${ \\hat { y } } = y$ , if $x \\neq c y$ for $\\forall c \\in \\mathbb { R }$ . ", + "bbox": [ + 174, + 468, + 823, + 497 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Cauchy-Schwarz inequality states that for $a \\in \\mathbb { R } ^ { T }$ and $b \\in \\mathbb { R } ^ { T }$ , $- \\| a \\| \\| b \\| \\leq < a , b > \\leq$ $\\| a \\| \\| b \\|$ . By this inequality, [-1,1] becomes the range of $l o s s _ { w S D R }$ . To attain the minimum value, the equality condition of the Cauchy-Schwarz inequality must be satisfied. This equality condition is equivalent to $b = 0$ or $a = t b$ , for $\\exists t \\in \\mathbb { R }$ . Applying the equality condition with the assumption $( y \\neq 0 , x - y \\neq 0 )$ to Equation 9 leads to $\\hat { y } = t _ { 1 } y$ and $x - { \\hat { y } } = t _ { 2 } ( x - y )$ , f or $\\exists t _ { 1 } \\in \\mathbb { R }$ and $\\exists t _ { 2 } \\in \\mathbb { R }$ . By adding these two equations, we can get $( 1 - t _ { 1 } ) x = ( t _ { 1 } - t _ { 2 } ) y$ . By the assumption $x \\neq c y$ , which is generally satisfied for large $T$ , we can conclude $t _ { 1 } = 1 , t _ { 1 } = t _ { 2 }$ must be satisfied when the minimum value is attained. □ ", + "bbox": [ + 173, + 511, + 826, + 625 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The following property of the weighted-SDR loss shows that the network can also learn from noiseonly training data. In experiments, we add small number $\\epsilon$ to denominators of Equation 9. Thus for the case of $y = 0$ , Equation 9 becomes ", + "bbox": [ + 173, + 638, + 825, + 681 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/cbf1c5396ad5b5888f7e0fcc8d611367c4b7ddbb3b5890d2d8a141346bfd1c16.jpg", + "text": "$$\n\\mathit { l o s s } _ { w S D R } ( x , 0 , \\hat { y } ) = - \\frac { < x , x - \\hat { y } > } { \\| x \\| \\| x - \\hat { y } \\| + \\epsilon }\n$$", + "text_format": "latex", + "bbox": [ + 364, + 688, + 633, + 722 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proposition 2. When we parameterize ${ \\hat { y } } = g _ { \\boldsymbol { \\theta } } ( { \\boldsymbol { x } } )$ , the $\\begin{array} { r } { l o s s _ { w S D R } ( x , y , g _ { \\theta } ( x ) ) } \\end{array}$ has a non-zero gradient with respect to $\\theta$ even if the target source y is empty. ", + "bbox": [ + 171, + 728, + 823, + 757 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. We can calculate partial derivatives as follows: ", + "bbox": [ + 173, + 772, + 529, + 786 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/607484b75470ad679cd55203bb56e2c358067e2b7035ef5a7c8acbaf96d89f4c.jpg", + "text": "$$\n\\begin{array} { r l } { { \\frac { \\partial l o s s _ { w S D R } ( x , y , g _ { \\theta } ( x ) ) } { \\partial \\theta } \\Big | _ { ( x = x _ { 0 } , y = 0 , \\theta = \\theta _ { 0 } ) } = \\frac { \\partial l o s s _ { w S D R } ( x _ { 0 } , 0 , g _ { \\theta } ( x _ { 0 } ) ) } { \\partial \\theta } \\Big | _ { \\theta = \\theta _ { 0 } } } } \\\\ & { = < \\frac { x _ { 0 } } { \\| x _ { 0 } \\| } , \\frac { \\partial } { \\partial \\theta } \\{ \\frac { x _ { 0 } - g _ { \\theta } ( x _ { 0 } ) } { \\| x _ { 0 } - g _ { \\theta } ( x _ { 0 } ) \\| + \\epsilon / \\| x _ { 0 } \\| } \\} \\Big | _ { \\theta = \\theta _ { 0 } } > } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 792, + 803, + 866 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Thus, the non-zero gradients with respect to $\\theta$ can be back-propagated. ", + "bbox": [ + 173, + 883, + 637, + 900 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D IRREDUCIBLE ERRORS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 362, + 117 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section, we illustrate two possible irreducible errors. Figure 12 (a) shows the irreducible phase error due to lack of phase estimation. Figure 12 (b) shows the irreducible error induced when bounding the range of mask. Not bounding the range of the mask makes the problem well-posed but it may suffer from the wide range of optimization search space because of the lack of prior knowledge on the distribution of cIRM. ", + "bbox": [ + 174, + 128, + 825, + 199 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/fa32b64835c251e121e7b604cd359d7881d74ae27955ab25b6ba847ad36fa868.jpg", + "image_caption": [ + "Figure 12: Two cases of possible irreducible error types in a TF bin in spectrogram. (a) The case where phase is not estimated and phase of mixture is reused. Even if the magnitude estimation is assumed to be perfect, there is still an irreducible phase error between the true source and the estimation. (b) The case where the magnitude of an estimated mask is bounded to 1. This again induces an irreducible error since there can be sources which have a higher magnitude than the magnitude of a mixture. " + ], + "image_footnote": [], + "bbox": [ + 227, + 217, + 771, + 396 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E SCATTER PLOTS OF CIRM ", + "text_level": 1, + "bbox": [ + 174, + 521, + 385, + 536 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The scatter plots of cIRM from training set is shown in Figure 13. We show four different scatter plots according to their SNR values of mixture (0, 5, 10, and 15 (dB)). Each scattered point of cIRM, $M _ { t , f }$ , is defined as follows: ", + "bbox": [ + 174, + 547, + 825, + 588 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/13f43a390d8b84ff18ffdb9fa906c9401c7d40aadc4f8fe940c5e7d8e660f56b.jpg", + "text": "$$\nM _ { t , f } = \\frac { \\left| Y _ { t , f } \\right| } { \\left| X _ { t , f } \\right| } e ^ { i ( \\theta _ { Y _ { t , f } } - \\theta _ { X _ { t , f } } ) }\n$$", + "text_format": "latex", + "bbox": [ + 401, + 588, + 596, + 627 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The scattered points near origin indicate the TF bins where the value of $\\left| Y _ { t , f } \\right|$ is significantly small compared to $\\left| X _ { t , f } \\right|$ . Therefore, those TF bins can be interpreted as the bins dominated with noise rather than source. On the other hand, the scattered points near (1,0) indicates the TF bins where the value of $\\left| Y _ { t , f } \\right|$ is almost the same as $\\left| X _ { t , f } \\right|$ . In this case, those TF bins can be interpreted as the bins dominated with source rather than noise. Therefore, as SNR becomes higher, the amount of TF bins dominated with clean source becomes larger compared to the lower SNR cases, and consequently the portion of real part close to 1 becomes larger as in Figure 13. ", + "bbox": [ + 173, + 637, + 825, + 738 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F VISUALIZATION OF ESTIMATED PHASE ", + "text_level": 1, + "bbox": [ + 176, + 756, + 467, + 770 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this section, we show a supplementary visualization of phase of estimated speech. Although the raw phase information itself does not show a distinctive pattern, the hidden structure can be revealed with group delay, which is the negative derivative of the phase along frequency axis (Yegnanarayana & Murthy, 1992). With this technique, the phase information can be explicitly shown as in Figure 14. Figure 14 (d) shows the group delay of clean speech and the corresponding magnitude is shown in Figure 14 (a). The two representations shows that the group delay of phase has a similar structure to that of magnitude spectrogram. The estimated phase by our model is shown in in Figure 14 (c). While the group delay of noisy speech (Figure 14 (b)) does not show a distinctive harmonic pattern, our estimation show the harmonic pattern similar to the group delay of clean speech, as shown in the yellow boxes in Figure 14 (c) and (d). ", + "bbox": [ + 173, + 781, + 825, + 921 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/ea2a3771e7eb6841552ebc9bd7d5caa6d96f1359539e5a009709fbebfb836a12.jpg", + "image_caption": [ + "Figure 13: Four different scatter plots of cIRM according to the four different SNR values of input mixture in training set, (a) SNR: 0 (dB) (b) SNR: 5 (dB) (c) SNR: 10 (dB) (d) SNR: 15 (dB). " + ], + "image_footnote": [], + "bbox": [ + 261, + 104, + 745, + 503 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/7acbbbf14bf8e233f5bf1c11287177383896acb25b1c6becfca763dae0a51b22.jpg", + "image_caption": [ + "Figure 14: Illustration of phase group delay. The group delay of the estimated phase from our model (c) shows the similar pattern to that of clean speech (d). For better visualization, we only show the TF bins where the magnitude of clean speech spectrogram exceeds certain threshold 0.01. " + ], + "image_footnote": [], + "bbox": [ + 174, + 565, + 820, + 757 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "G IMPORTANCE OF PHASE ESTIMATION ", + "text_level": 1, + "bbox": [ + 176, + 844, + 459, + 858 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In this section, to show the limitation of conventional approach (without phase estimation), we emphasize that the phase estimation is important, especially under low SNR condition (harsh condition). ", + "bbox": [ + 174, + 869, + 823, + 912 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We first make an assumption that the estimation of phase information becomes more important when the given mixture has low SNR. Our reasoning behind this assumption is that if the SNR of a given mixture is low, the irreducible phase error is likely to be greater, hence a more room for improvement with phase estimation as illustrated in Figure 15. This can also be verified in Table 5 columns PhaseDist(C, N) and Phase Improvement where the values of both columns increase as SNR becomes higher. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/12b9d9801754fe5cc7bd817b128fda482a6f9e543085ebc1dfc0d1cc6c2c528f.jpg", + "image_caption": [ + "Figure 15: (a) The case where SNR of given mixture is high. In this case the source is likely to be dominant in the mixture. Therefore it is relatively easier to estimate ground truth source with better precision even when the phase is not estimated. (b) The case where SNR of given mixture is low. In this case the source is not dominant in the mixture. Therefore, the irreducible phase error is likely to be higher in low SNR conditions than higher SNR conditions. Under this circumstance, we assume the lack of phase estimation will result in a particularly bad system performance. " + ], + "image_footnote": [], + "bbox": [ + 230, + 217, + 769, + 397 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To empirically show the importance of phase estimation, we show correlation between phase improvement and performance difference between the conventional method (without phase estimation) and our proposed method (with phase estimation) in Table 6. The performance difference was calculated by simply subtracting the evaluation results of conventional method from the evaluation results of our method with phase estimation. For fair comparison, both conventional method $( \\mathbf { R M R } \\mathbf { n } )$ and proposed method (cRMCn) were set to have the same number of parameters. Also, both models were trained with weighted-SDR loss. The results show that when the SNR is low, both the phase improvement and the performance difference are relatively higher than the results from higher SNR conditions. Furthermore, almost all results show an incremental increase of phase improvement and performance difference as the SNR decreases, which agrees on our assumption. Therefore we believe that phase estimation is important especially in harsh noisy conditions (low SNR conditions). ", + "bbox": [ + 174, + 506, + 825, + 659 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/aa63049b3bde5b0504c6c4d8fe9b9d1cab0ac0732574e53758ecb52b202f5dd4.jpg", + "table_caption": [ + "Table 6: The performance difference between conventional method (without phase estimation) and our method (with phase estimation). The performance difference is presented with four different SNR values of mixture in test set. " + ], + "table_footnote": [], + "table_body": "
SNR (dB)Phase ImprovementPerformance Difference
PESQCSIGCBAKCOVLSSNR
2.56.714°0.060.440.140.215.32
7.54.408°0.030.420.110.195.21
12.52.982°0.050.390.110.184.96
17.51.702°0.050.300.090.133.93
", + "bbox": [ + 217, + 732, + 779, + 838 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "H COMPLETE QUALITATIVE RESULTS", + "text_level": 1, + "bbox": [ + 176, + 104, + 444, + 117 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/e7e6a73cce586cf7ab1c97a6670f54ce31e07a29b03523012e42b707035ecd48.jpg", + "table_caption": [ + "Table 7: Pairwise preference scores of four models including DCUnet. The scores are obtained by calculating the relative frequency the subjects prefer one method to the other method. Hard/Medium/Easy denote 2.5/7.5/17.5 SNR conditions in dB, respectively. Significance for each statistic is also described (n.s.: not significant, $\\ast \\colon \\mathsf { p } { < } 0 . 0 5$ , $^ { * * }$ : $\\mathrm { p { < } 0 . 0 1 }$ , $^ { \\ast \\ast \\ast }$ : $\\mathrm { p { < } 0 . 0 0 1 }$ ). " + ], + "table_footnote": [], + "table_body": "
HardMedium EasyTotal
DCUnet > Deep Feature Loss0.90 (***）0.82 (***)0.69 (***)0.82 (***)
DCUnet > Wavenet0.98 (***)0.95 (***)0.75 (***)0.93 (***)
DCUnet > SEGAN0.99 (***)0.93 (***)0.71 (***)0.92 (***)
Deep Feature Loss > Wavenet0.67 (***)0.83 (***)0.48 (n.s.)0.74 (***)
Deep Feature Loss > SEGAN0.72 (***)0.81 (***)0.40 (**)0.73 (***)
SEGAN > Wavenet0.53 (n.s.)0.56(**)
0.58 (**)0.50 (n.s.)
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Spectrograms are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "represented as complex matrices, which are normally decomposed into magnitude and phase com-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "ponents to be used in real-valued networks. In tasks involving audio signal reconstruction, such as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "speech enhancement, it is ideal to perform correct estimation of both components. Unfortunately,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "complex-valued phase has been often neglected due to the difficulty of its estimation. This has led", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 648, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 506, + 664 + ], + "score": 1.0, + "content": "to the situation where most approaches focus only on the estimation of a magnitude spectrogram", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "while reusing noisy phase information (Huang et al., 2014; Xu et al., 2015; Grais et al., 2016; Nu-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "graha et al., 2016; Takahashi et al., 2018b). 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This is due to the difficulty of estimating the phase", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 315 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 315 + ], + "score": 1.0, + "content": "of clean speech. To improve speech enhancement performance, we tackle the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 469, + 325 + ], + "score": 1.0, + "content": "phase estimation problem in three ways. First, we propose Deep Complex U-Net,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 470, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 470, + 336 + ], + "score": 1.0, + "content": "an advanced U-Net structured model incorporating well-defined complex-valued", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 470, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 347 + ], + "score": 1.0, + "content": "building blocks to deal with complex-valued spectrograms. Second, we propose a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "score": 1.0, + "content": "polar coordinate-wise complex-valued masking method to reflect the distribution", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 356, + 469, + 368 + ], + "spans": [ + { + "bbox": [ + 142, + 356, + 469, + 368 + ], + "score": 1.0, + "content": "of complex ideal ratio masks. Third, we define a novel loss function, weighted", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 470, + 379 + ], + "score": 1.0, + "content": "source-to-distortion ratio (wSDR) loss, which is designed to directly correlate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 378, + 470, + 389 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 470, + 389 + ], + "score": 1.0, + "content": "with a quantitative evaluation measure. Our model was evaluated on a mixture of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 469, + 402 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 469, + 402 + ], + "score": 1.0, + "content": "the Voice Bank corpus and DEMAND database, which has been widely used by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 400, + 470, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 400, + 470, + 412 + ], + "score": 1.0, + "content": "many deep learning models for speech enhancement. Ablation experiments were", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 411, + 470, + 424 + ], + "spans": [ + { + "bbox": [ + 141, + 411, + 470, + 424 + ], + "score": 1.0, + "content": "conducted on the mixed dataset showing that all three proposed approaches are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 422, + 469, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 422, + 469, + 434 + ], + "score": 1.0, + "content": "empirically valid. Experimental results show that the proposed method achieves", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "score": 1.0, + "content": "state-of-the-art performance in all metrics, outperforming previous approaches by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 444, + 209, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 209, + 457 + ], + "score": 1.0, + "content": "a large margin1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 18, + "bbox_fs": [ + 141, + 269, + 470, + 457 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 476, + 206, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 208, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 208, + 491 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 504, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Speech enhancement is one of the most important and challenging tasks in speech applications", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "where the goal is to separate clean speech from noise when noisy speech is given as an input.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "As a fundamental component for speech-related systems, the applications of speech enhancement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "vary from speech recognition front-end modules to hearing aid systems for the hearing-impaired", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 343, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 343, + 559 + ], + "score": 1.0, + "content": "(Erdogan et al., 2015; Weninger et al., 2015; Wang, 2017).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 501, + 505, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "Due to recent advances in deep learning, the speech enhancement task has been able to reach high", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "levels in performance through significant improvements. When using audio signals with deep learn-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 582, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 505, + 598 + ], + "score": 1.0, + "content": "ing models, it has been a common practice to transform a time-domain waveform to a time-frequency", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "(TF) representation (i.e. spectrograms) via short-time-Fourier-transform (STFT). Spectrograms are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "represented as complex matrices, which are normally decomposed into magnitude and phase com-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "ponents to be used in real-valued networks. In tasks involving audio signal reconstruction, such as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "speech enhancement, it is ideal to perform correct estimation of both components. Unfortunately,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "complex-valued phase has been often neglected due to the difficulty of its estimation. This has led", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 648, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 506, + 664 + ], + "score": 1.0, + "content": "to the situation where most approaches focus only on the estimation of a magnitude spectrogram", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "while reusing noisy phase information (Huang et al., 2014; Xu et al., 2015; Grais et al., 2016; Nu-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "graha et al., 2016; Takahashi et al., 2018b). However, reusing phase from noisy speech has clear", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "limitations, particularly under extremely noisy conditions, in other words, when signal-to-noise ra-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 693, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 505, + 707 + ], + "score": 1.0, + "content": "tio (SNR) is low. This can be easily verified by simply using the magnitude spectrogram of clean", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "speech with the phase spectrogram of noisy speech to reconstruct clean speech, as illustrated in Fig-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "ure 1. We can clearly see that the difference between clean and estimated speech signals gets larger", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 222, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 222, + 117 + ], + "score": 1.0, + "content": "as the input SNR gets lower.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 562, + 506, + 707 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "speech with the phase spectrogram of noisy speech to reconstruct clean speech, as illustrated in Fig-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "ure 1. We can clearly see that the difference between clean and estimated speech signals gets larger", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 222, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 222, + 117 + ], + "score": 1.0, + "content": "as the input SNR gets lower.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "image", + "bbox": [ + 126, + 128, + 488, + 199 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 128, + 488, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 128, + 488, + 199 + ], + "spans": [ + { + "bbox": [ + 126, + 128, + 488, + 199 + ], + "score": 0.964, + "type": "image", + "image_path": "44281215127d595e91a725f5d3e2398f9d143edac8059550ca1138d5b01daf99.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 126, + 128, + 488, + 151.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 126, + 151.66666666666666, + 488, + 175.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 126, + 175.33333333333331, + 488, + 198.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 140, + 210, + 470, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 139, + 209, + 471, + 223 + ], + "spans": [ + { + "bbox": [ + 139, + 209, + 471, + 223 + ], + "score": 1.0, + "content": "Figure 1: Effects of reusing phase of noisy speech under different SNR conditions.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + } + ], + "index": 5.0 + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "score": 1.0, + "content": "A popular approach to speech enhancement is to optimize a mask which produces a spectrogram", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "of clean speech when applied to noisy input audio. One of the first mask-based attempts to perform", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "score": 1.0, + "content": "the task by incorporating phase information was the proposal of the phase-sensitive mask (PSM)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 266, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 281 + ], + "score": 1.0, + "content": "(Erdogan et al., 2015). 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While iterative methods such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "as the Griffin-Lim algorithm and its variants (Griffin & Lim, 1984; Perraudin et al., 2013) aimed", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "to address this problem, neural network-based approaches are recently attracting attention as non-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 645, + 191, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 191, + 658 + ], + "score": 1.0, + "content": "iterative alternatives.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 603, + 505, + 658 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 505, + 729 + ], + "lines": [ + { + "bbox": [ + 106, + 663, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 505, + 674 + ], + "score": 1.0, + "content": "One major approach is to use an end-to-end model that takes audio as raw waveform inputs without", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "using any explicit time-frequency (TF) representation computed via STFT (Pascual et al., 2017;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 684, + 504, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 504, + 696 + ], + "score": 1.0, + "content": "Rethage et al., 2018; Stoller et al., 2018; Germain et al., 2018). Since raw waveforms inherently", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 696, + 505, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 505, + 707 + ], + "score": 1.0, + "content": "contain phase information, it is expected to achieve phase estimation naturally. Another method is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 707, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 505, + 719 + ], + "score": 1.0, + "content": "to estimate magnitude and phase using two separate neural network modules which serially estimate", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 717, + 505, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 505, + 730 + ], + "score": 1.0, + "content": "magnitude and phase (Afouras et al., 2018; Takahashi et al., 2018a). In this framework, the phase", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "estimation module uses noisy phase with predicted magnitude to estimate phase of clean speech.", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "There is also a recent study which proposed to use additional layers with trainable discrete values", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 270, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 270, + 117 + ], + "score": 1.0, + "content": "for phase estimation (Roux et al., 2018).", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 663, + 505, + 730 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "estimation module uses noisy phase with predicted magnitude to estimate phase of clean speech.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "There is also a recent study which proposed to use additional layers with trainable discrete values", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 270, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 270, + 117 + ], + "score": 1.0, + "content": "for phase estimation (Roux et al., 2018).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "A more straightforward method would be to jointly estimate magnitude and phase by using a con-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "tinuous complex-valued ratio mask (cRM). Previous studies tried this joint estimation approach", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "bounding the range of the cRM (Williamson et al., 2016; Ephrat et al., 2018). Despite the advan-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "tages of the cRM approach, previously proposed methods had limitations with regard to the loss", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "function and the range of the mask which we will be returning with more details in Section 3 along", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "with our proposed methods to alleviate these issues. As a natural extension to the works above, some", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "studies have also undergone to examine whether complex-valued networks are useful when dealing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "with intrinsically complex-valued data. In the series of two works, complex-valued networks were", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "shown to help singing voice separation performance with both fully connected neural networks and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "recurrent neural networks (Lee et al., 2017a;b). However, the approaches were limited as it ended", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "up only switching the real-valued network into a complex-valued counterpart and leaving the other", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "deep learning building blocks such as weight initialization and normalization technique in a real-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "valued manner. Also, the works do not show whether the phase was actually well estimated either", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 466, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 466, + 277 + ], + "score": 1.0, + "content": "quantitatively or qualitatively, only ending up showing that there was a performance gain.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 292, + 326, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 327, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 327, + 306 + ], + "score": 1.0, + "content": "3 PHASE-AWARE SPEECH ENHANCEMENT", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "In this section we will provide details on our approach, starting with our proposed model Deep", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "Complex U-Net, followed by the masking framework based on the model. Finally, we will introduce", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 339, + 498, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 498, + 351 + ], + "score": 1.0, + "content": "a new loss function to optimize our model, which takes a critical role for proper phase estimation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "Before getting into details, here are some notations used throughout the paper. The input mixture", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 504, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 133, + 379 + ], + "score": 1.0, + "content": "signal", + "type": "text" + }, + { + "bbox": [ + 133, + 366, + 237, + 378 + ], + "score": 0.92, + "content": "\\bar { x ( n ) } \\overset { \\cdot } { = } y ( n ) + z ( n ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 366, + 464, + 379 + ], + "score": 1.0, + "content": "is assumed to be a linear sum of the clean speech signal", + "type": "text" + }, + { + "bbox": [ + 464, + 366, + 504, + 378 + ], + "score": 0.92, + "content": "y ( n ) \\in \\mathbb { R }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 148, + 390 + ], + "score": 1.0, + "content": "and noise", + "type": "text" + }, + { + "bbox": [ + 149, + 378, + 190, + 389 + ], + "score": 0.9, + "content": "z ( n ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 377, + 349, + 390 + ], + "score": 1.0, + "content": ", where estimated speech is denoted as", + "type": "text" + }, + { + "bbox": [ + 349, + 377, + 391, + 389 + ], + "score": 0.92, + "content": "\\hat { y } ( n ) \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 377, + 505, + 390 + ], + "score": 1.0, + "content": ". Each of the corresponding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 170, + 402 + ], + "score": 1.0, + "content": "time-frequency", + "type": "text" + }, + { + "bbox": [ + 171, + 389, + 192, + 400 + ], + "score": 0.91, + "content": "( t , f )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 387, + 391, + 402 + ], + "score": 1.0, + "content": "representations computed by STFT is denoted as", + "type": "text" + }, + { + "bbox": [ + 391, + 388, + 505, + 401 + ], + "score": 0.26, + "content": "X _ { t , f } \\in \\mathbb { C } , Y _ { t , f } \\in \\bar { \\mathbb { C } } , Z _ { t , f } \\in", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 115, + 412 + ], + "score": 0.69, + "content": "\\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 401, + 136, + 414 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 136, + 399, + 175, + 414 + ], + "score": 0.94, + "content": "\\hat { Y } _ { t , f } \\in \\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 401, + 355, + 414 + ], + "score": 1.0, + "content": ". The ground truth mask cIRM is denoted as", + "type": "text" + }, + { + "bbox": [ + 356, + 401, + 398, + 414 + ], + "score": 0.93, + "content": "M _ { t , f } \\in \\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "and the estimated cRM is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 302, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 151, + 429 + ], + "score": 1.0, + "content": "denoted as", + "type": "text" + }, + { + "bbox": [ + 151, + 414, + 192, + 427 + ], + "score": 0.92, + "content": "\\hat { M } _ { t , f } \\in \\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 412, + 222, + 429 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 223, + 414, + 297, + 427 + ], + "score": 0.93, + "content": "M _ { t , f } = Y _ { t , f } / X _ { t , f }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 412, + 302, + 429 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 439, + 233, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 235, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 235, + 453 + ], + "score": 1.0, + "content": "3.1 DEEP COMPLEX U-NET", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "image", + "bbox": [ + 117, + 463, + 495, + 524 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 463, + 495, + 524 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 463, + 495, + 524 + ], + "spans": [ + { + "bbox": [ + 117, + 463, + 495, + 524 + ], + "score": 0.959, + "type": "image", + "image_path": "97b1a0b158f74dedaa4e3df7048d9ac98c9b57d80f09e60aa2417cc5b10ed965.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 117, + 463, + 495, + 483.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 117, + 483.3333333333333, + 495, + 503.66666666666663 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 117, + 503.66666666666663, + 495, + 524.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 162, + 537, + 447, + 549 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 163, + 536, + 448, + 550 + ], + "spans": [ + { + "bbox": [ + 163, + 536, + 448, + 550 + ], + "score": 1.0, + "content": "Figure 2: Illustration of speech enhancement framework with DCUnet.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + } + ], + "index": 30.0 + }, + { + "type": "text", + "bbox": [ + 106, + 563, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "The U-Net structure is a well known architecture composed as a convolutional autoencoder with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "skip-connections, originally proposed for medical imaging in computer vision community (Ron-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "neberger et al., 2015). Furthermore, the use of real-valued U-Net has been shown to be also effective", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 608 + ], + "score": 1.0, + "content": "in many recent audio source separation tasks such as music source separation (Jansson et al., 2017;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 622 + ], + "score": 1.0, + "content": "Stoller et al., 2018; Takahashi et al., 2018b), and speech enhancement (Pascual et al., 2017). 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In this section, we will describe how U-Net is modified using the complex build-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 640, + 334, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 334, + 653 + ], + "score": 1.0, + "content": "ing blocks originally proposed by (Trabelsi et al., 2018).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 426, + 670 + ], + "score": 1.0, + "content": "Complex-valued Building Blocks. 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Details are illustrated in Appendix A.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "A more straightforward method would be to jointly estimate magnitude and phase by using a con-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "tinuous complex-valued ratio mask (cRM). 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As a natural extension to the works above, some", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "studies have also undergone to examine whether complex-valued networks are useful when dealing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "with intrinsically complex-valued data. 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Also, the works do not show whether the phase was actually well estimated either", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 466, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 466, + 277 + ], + "score": 1.0, + "content": "quantitatively or qualitatively, only ending up showing that there was a performance gain.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 120, + 506, + 277 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 292, + 326, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 327, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 327, + 306 + ], + "score": 1.0, + "content": "3 PHASE-AWARE SPEECH ENHANCEMENT", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "In this section we will provide details on our approach, starting with our proposed model Deep", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "Complex U-Net, followed by the masking framework based on the model. 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Note that all experiments performed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 569, + 295, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 295, + 581 + ], + "score": 1.0, + "content": "in Section 4 are done with these modifications.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 593, + 369, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 371, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 371, + 606 + ], + "score": 1.0, + "content": "3.2 COMPLEX-VALUED MASKING ON POLAR COORDINATES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 614, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "As our proposed model can handle complex values, we aim to estimate cRM for speech enhance-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "score": 1.0, + "content": "ment. 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Note that all experiments performed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 569, + 295, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 295, + 581 + ], + "score": 1.0, + "content": "in Section 4 are done with these modifications.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 448, + 506, + 581 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 593, + 369, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 371, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 371, + 606 + ], + "score": 1.0, + "content": "3.2 COMPLEX-VALUED MASKING ON POLAR COORDINATES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 614, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "As our proposed model can handle complex values, we aim to estimate cRM for speech enhance-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "score": 1.0, + "content": "ment. Although it is possible to directly estimate the spectrogram of a clean source signal, it has", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "been shown that better performance can be achieved by applying a weighting mask to the mixture", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "score": 1.0, + "content": "spectrogram (Wang et al., 2014). One thing to note is that real-valued ratio masks (RM) only change", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "the scale of the magnitude without changing phase, resulting in irreducible errors as illustrated in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "Appendix D. On the other hand, cRM also perform a rotation on the polar coordinates, allowing to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 680, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 394, + 695 + ], + "score": 1.0, + "content": "correct phase errors. In other words, the estimated speech spectrogram", + "type": "text" + }, + { + "bbox": [ + 394, + 680, + 412, + 694 + ], + "score": 0.93, + "content": "\\hat { Y } _ { t , f }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 680, + 505, + 695 + ], + "score": 1.0, + "content": "is computed by multi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 693, + 406, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 213, + 708 + ], + "score": 1.0, + "content": "plying the estimated mask", + "type": "text" + }, + { + "bbox": [ + 213, + 693, + 235, + 707 + ], + "score": 0.93, + "content": "\\hat { M } _ { t , f }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 694, + 338, + 708 + ], + "score": 1.0, + "content": "on the input spectrogram", + "type": "text" + }, + { + "bbox": [ + 338, + 695, + 358, + 707 + ], + "score": 0.9, + "content": "X _ { t , f }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 694, + 406, + 708 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 614, + 506, + 708 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 198, + 90, + 412, + 111 + ], + "lines": [ + { + "bbox": [ + 198, + 90, + 412, + 111 + ], + "spans": [ + { + "bbox": [ + 198, + 90, + 412, + 111 + ], + "score": 0.93, + "content": "\\hat { Y } _ { t , f } = \\hat { M } _ { t , f } \\cdot X _ { t , f } = \\Big | \\hat { M } _ { t , f } \\Big | \\cdot \\big | X _ { t , f } \\big | \\cdot e ^ { i ( \\theta _ { \\hat { M } _ { t , f } } + \\theta _ { X _ { t , f } } ) }", + "type": "interline_equation", + "image_path": "e0e866a8259332e59e6703d58ddaf55f803864086b94b8f29bfb250427227ebc.jpg" + } + ] + } + ], + "index": 0, + "virtual_lines": [ + { + "bbox": [ + 198, + 90, + 412, + 111 + ], + "spans": [], + "index": 0 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 118, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 117, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 132 + ], + "score": 1.0, + "content": "In this state, the real and imaginary values of the estimated cRM is unbounded. Although estimating", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 372, + 142 + ], + "score": 1.0, + "content": "an unbounded mask makes the problem well-posed (see Appendix", + "type": "text" + }, + { + "bbox": [ + 372, + 129, + 382, + 139 + ], + "score": 0.26, + "content": "\\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "for more information), we can", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 140, + 483, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 483, + 154 + ], + "score": 1.0, + "content": "imagine the difficulty of optimizing from an infinite search space compared to a bounded one.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 505, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 505, + 169 + ], + "score": 1.0, + "content": "Therefore, a few techniques have been tried to bound the range of cRM. For example, Williamson", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "et al. tried to directly optimize a complex mask into a cIRM compressed to a heuristic bound", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 179, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 505, + 191 + ], + "score": 1.0, + "content": "(Williamson et al., 2016). However, this method was limited since it was only able to succeed in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "score": 1.0, + "content": "training the model by computing the error between cIRM and the predicted cRM which often leads", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "to a degradation of performance (Wang et al., 2014; Yu et al., 2017). More recently, Ephrat et al.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "proposed a rectangular coordinate-wise cRM made with sigmoid compressions onto each of the real", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "and imaginary parts of the output of the model (Ephrat et al., 2018). After then MSE between clean", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 135, + 248 + ], + "score": 1.0, + "content": "source", + "type": "text" + }, + { + "bbox": [ + 135, + 235, + 145, + 246 + ], + "score": 0.69, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 235, + 232, + 248 + ], + "score": 1.0, + "content": "and estimated source", + "type": "text" + }, + { + "bbox": [ + 233, + 234, + 243, + 245 + ], + "score": 0.85, + "content": "\\hat { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "was computed in STFT-domain to train the model. 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Second, this approach results in a cRM with a restricted rotation range of", + "type": "text" + }, + { + "bbox": [ + 464, + 268, + 475, + 279 + ], + "score": 0.86, + "content": "0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 267, + 487, + 281 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 488, + 268, + 504, + 279 + ], + "score": 0.85, + "content": "9 0 °", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 279, + 357, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 357, + 292 + ], + "score": 1.0, + "content": "(only clock-wise), which makes it hard to correct noisy phase.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "To alleviate these problems, we propose a polar coordinate-wise cRM method that imposes non-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "score": 1.0, + "content": "linearity only on the magnitude part. 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More formally, let", + "type": "text" + }, + { + "bbox": [ + 365, + 340, + 381, + 352 + ], + "score": 0.91, + "content": "g ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "be our neural network and the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 499, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 165, + 366 + ], + "score": 1.0, + "content": "output of it be", + "type": "text" + }, + { + "bbox": [ + 165, + 352, + 230, + 365 + ], + "score": 0.94, + "content": "O _ { t , f } = g ( X _ { t , f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 352, + 380, + 366 + ], + "score": 1.0, + "content": ". 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However, this method was limited since it was only able to succeed in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "score": 1.0, + "content": "training the model by computing the error between cIRM and the predicted cRM which often leads", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "to a degradation of performance (Wang et al., 2014; Yu et al., 2017). 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After then MSE between clean", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 135, + 248 + ], + "score": 1.0, + "content": "source", + "type": "text" + }, + { + "bbox": [ + 135, + 235, + 145, + 246 + ], + "score": 0.69, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 235, + 232, + 248 + ], + "score": 1.0, + "content": "and estimated source", + "type": "text" + }, + { + "bbox": [ + 233, + 234, + 243, + 245 + ], + "score": 0.85, + "content": "\\hat { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "was computed in STFT-domain to train the model. 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Second, this approach results in a cRM with a restricted rotation range of", + "type": "text" + }, + { + "bbox": [ + 464, + 268, + 475, + 279 + ], + "score": 0.86, + "content": "0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 267, + 487, + 281 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 488, + 268, + 504, + 279 + ], + "score": 0.85, + "content": "9 0 °", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 279, + 357, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 357, + 292 + ], + "score": 1.0, + "content": "(only clock-wise), which makes it hard to correct noisy phase.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 156, + 506, + 292 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "To alleviate these problems, we propose a polar coordinate-wise cRM method that imposes non-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "score": 1.0, + "content": "linearity only on the magnitude part. 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The corresponding phase mask is naturally obtained by dividing the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 364, + 353 + ], + "score": 1.0, + "content": "output of the model with the magnitude of it. 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However, it has been reported that optimiz-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "ing the model with MSE in complex STFT-domain fails in phase estimation due to the randomness", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 504, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 504, + 532 + ], + "score": 1.0, + "content": "in phase structure (Williamson et al., 2016). As an alternative, it is possible to use a loss function de-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "fined in the time-domain instead, as raw waveforms contain inherent phase information. While MSE", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "on waveforms can be an easy solution, we can expect it to be more effective if the loss function is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 553, + 447, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 447, + 564 + ], + "score": 1.0, + "content": "directly correlated with well-known evaluation measures defined in the time-domain.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 485, + 505, + 564 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "Here, we propose an improved loss function weighted-SDR loss by building upon a previous work", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "which attempts to optimize a standard quality measure, source-to-distortion ratio (SDR) (Venkatara-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 590, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 289, + 605 + ], + "score": 1.0, + "content": "mani et al., 2017). The original loss function", + "type": "text" + }, + { + "bbox": [ + 289, + 592, + 324, + 603 + ], + "score": 0.89, + "content": "l o s s _ { V e n }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 590, + 505, + 605 + ], + "score": 1.0, + "content": "suggested by Venkataramani et al. is formu-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 309, + 615 + ], + "score": 1.0, + "content": "lated upon the observation from Equation 4, where", + "type": "text" + }, + { + "bbox": [ + 309, + 605, + 316, + 614 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 603, + 433, + 615 + ], + "score": 1.0, + "content": "is the clean source signal and", + "type": "text" + }, + { + "bbox": [ + 434, + 603, + 440, + 614 + ], + "score": 0.83, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "is the estimated", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 613, + 425, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 425, + 626 + ], + "score": 1.0, + "content": "source signal. In practice, the negative reciprocal is optimized as in Equation 5.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 569, + 505, + 626 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 643, + 444, + 702 + ], + "lines": [ + { + "bbox": [ + 166, + 643, + 444, + 702 + ], + "spans": [ + { + "bbox": [ + 166, + 643, + 444, + 702 + ], + "score": 0.87, + "content": "\\begin{array} { l } { \\displaystyle \\underset { \\hat { y } } { \\mathrm { m a x } } \\ S D R ( y , \\hat { y } ) : = \\operatorname* { m a x } _ { \\hat { y } } \\frac { < y , \\hat { y } > ^ { 2 } } { | | y | | ^ { 2 } | | \\hat { y } | | ^ { 2 } - < y , \\hat { y } > ^ { 2 } } \\propto \\operatorname* { m i n } _ { \\hat { y } } \\frac { | | \\hat { y } | | ^ { 2 } } { < y , \\hat { y } > ^ { 2 } } } \\\\ { l o s s _ { V e n } ( y , \\hat { y } ) : = - \\frac { < y , \\hat { y } > ^ { 2 } } { | | \\hat { y } | | ^ { 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "2559934564a994f57fa1bec86f7bcc34e59b93547039804f003197f883ea7b15.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 166, + 643, + 444, + 662.6666666666666 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 166, + 662.6666666666666, + 444, + 682.3333333333333 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 166, + 682.3333333333333, + 444, + 701.9999999999999 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 707, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 707, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 505, + 721 + ], + "score": 1.0, + "content": "Although using Equation 5 works as a loss function, there are a few critical flaws in the design.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 718, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 235, + 734 + ], + "score": 1.0, + "content": "First, the lower bound becomes", + "type": "text" + }, + { + "bbox": [ + 236, + 718, + 264, + 732 + ], + "score": 0.92, + "content": "- \\| y \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 718, + 393, + 734 + ], + "score": 1.0, + "content": ", which depends on the value of", + "type": "text" + }, + { + "bbox": [ + 393, + 722, + 400, + 732 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 718, + 505, + 734 + ], + "score": 1.0, + "content": "causing fluctuation in the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 313, + 96 + ], + "score": 1.0, + "content": "loss values when training. Second, when the target", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 313, + 84, + 321, + 94 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 321, + 82, + 380, + 96 + ], + "score": 1.0, + "content": "is empty (i.e.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 380, + 83, + 407, + 94 + ], + "score": 0.89, + "content": "y = 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 407, + 82, + 506, + 96 + ], + "score": 1.0, + "content": ") the loss becomes zero,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "preventing the model to learn from noisy-only data due to zero gradients. Finally, the loss function", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 459, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 364, + 117 + ], + "score": 1.0, + "content": "is not scale sensitive, meaning that the loss value is the same for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 364, + 105, + 371, + 116 + ], + "score": 0.85, + "content": "\\hat { y }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 371, + 105, + 389, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 389, + 105, + 400, + 116 + ], + "score": 0.83, + "content": "c \\hat { y }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 400, + 105, + 430, + 117 + ], + "score": 1.0, + "content": ", where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 430, + 105, + 455, + 115 + ], + "score": 0.9, + "content": "c \\in \\mathbb { R }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 455, + 105, + 459, + 117 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 707, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 313, + 96 + ], + "score": 1.0, + "content": "loss values when training. Second, when the target", + "type": "text" + }, + { + "bbox": [ + 313, + 84, + 321, + 94 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 82, + 380, + 96 + ], + "score": 1.0, + "content": "is empty (i.e.,", + "type": "text" + }, + { + "bbox": [ + 380, + 83, + 407, + 94 + ], + "score": 0.89, + "content": "y = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 82, + 506, + 96 + ], + "score": 1.0, + "content": ") the loss becomes zero,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "preventing the model to learn from noisy-only data due to zero gradients. Finally, the loss function", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 459, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 364, + 117 + ], + "score": 1.0, + "content": "is not scale sensitive, meaning that the loss value is the same for", + "type": "text" + }, + { + "bbox": [ + 364, + 105, + 371, + 116 + ], + "score": 0.85, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 105, + 389, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 389, + 105, + 400, + 116 + ], + "score": 0.83, + "content": "c \\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 105, + 430, + 117 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 430, + 105, + 455, + 115 + ], + "score": 0.9, + "content": "c \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 105, + 459, + 117 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "score": 1.0, + "content": "To resolve these issues, we redesigned the loss function by giving several modifications to Equation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 426, + 145 + ], + "score": 1.0, + "content": "5. First, we made the lower bound of the loss function independent to the source", + "type": "text" + }, + { + "bbox": [ + 426, + 134, + 433, + 144 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "by restoring back", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 145, + 159 + ], + "score": 1.0, + "content": "the term", + "type": "text" + }, + { + "bbox": [ + 146, + 143, + 167, + 157 + ], + "score": 0.92, + "content": "\\left\\| y \\right\\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 144, + 506, + 159 + ], + "score": 1.0, + "content": "and applying square root as in Equation 6. This makes the loss function bounded", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 498, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 498, + 168 + ], + "score": 1.0, + "content": "within the range [-1, 1] and also be more phase sensitive, as inverted phase gets penalized as well.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 173, + 404, + 206 + ], + "lines": [ + { + "bbox": [ + 207, + 173, + 404, + 206 + ], + "spans": [ + { + "bbox": [ + 207, + 173, + 404, + 206 + ], + "score": 0.94, + "content": "\\mathit { l o s s } _ { \\mathit { S D R } } ( y , \\hat { y } ) : = - \\sqrt { - \\frac { \\mathit { l o s s } _ { \\mathit { V e n } } } { \\left\\| y \\right\\| ^ { 2 } } } = - \\frac { < y , \\hat { y } > } { \\left\\| y \\right\\| \\left\\| \\hat { y } \\right\\| }", + "type": "interline_equation", + "image_path": "5398494a6a602e0124002b55f96e52bdc1b95abc8c2b58f9229aefd1eb355ea5.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 207, + 173, + 404, + 189.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 207, + 189.5, + 404, + 206.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "Expecting to be complementary to source prediction and to propagate errors for noise-only samples,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 259, + 235 + ], + "score": 1.0, + "content": "we also added a noise prediction term", + "type": "text" + }, + { + "bbox": [ + 259, + 222, + 318, + 234 + ], + "score": 0.91, + "content": "l o s s _ { S D R } ( z , \\hat { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 222, + 506, + 235 + ], + "score": 1.0, + "content": ". To properly balance the contributions of each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 247 + ], + "score": 1.0, + "content": "loss term and solve the scale insensitivity problem, we weighted each term proportional to the energy", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 428, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 428, + 256 + ], + "score": 1.0, + "content": "of each signal. The final form of the suggested weighted-SDR loss is as follows:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 261, + 437, + 275 + ], + "lines": [ + { + "bbox": [ + 174, + 261, + 437, + 275 + ], + "spans": [ + { + "bbox": [ + 174, + 261, + 437, + 275 + ], + "score": 0.82, + "content": "l o s s _ { w S D R } ( x , y , \\hat { y } ) : = \\alpha l o s s _ { S D R } ( y , \\hat { y } ) + ( 1 - \\alpha ) l o s s _ { S D R } ( z , \\hat { z } )", + "type": "interline_equation", + "image_path": "c823f51cb1dd7e0a902278eb0f9fd88e0a81a4612fcc3dae96decc4d67abe56d.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 174, + 261, + 437, + 275 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 136, + 294 + ], + "score": 1.0, + "content": "where,", + "type": "text" + }, + { + "bbox": [ + 136, + 281, + 176, + 293 + ], + "score": 0.92, + "content": "\\hat { z } = x - \\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 280, + 266, + 294 + ], + "score": 1.0, + "content": "is estimated noise and", + "type": "text" + }, + { + "bbox": [ + 266, + 280, + 372, + 293 + ], + "score": 0.92, + "content": "\\alpha = | | y | | ^ { 2 } / ( | | y | | ^ { 2 } + | | z | | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "is the energy ratio between clean", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 135, + 304 + ], + "score": 1.0, + "content": "speech", + "type": "text" + }, + { + "bbox": [ + 136, + 294, + 142, + 304 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 291, + 182, + 304 + ], + "score": 1.0, + "content": "and noise", + "type": "text" + }, + { + "bbox": [ + 183, + 294, + 189, + 302 + ], + "score": 0.7, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 291, + 505, + 304 + ], + "score": 1.0, + "content": ". Note that although weighted SDR loss is a time-domain loss function, it can be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "backpropagated through our framework. Specifically, STFT and ISTFT operations are implemented", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 314, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 504, + 326 + ], + "score": 1.0, + "content": "as 1-D convolution and deconvolution layers consisting of fixed filters initialized with the discrete", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 502, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 502, + 338 + ], + "score": 1.0, + "content": "Fourier transform matrix. The detailed properties of the proposed loss function are in Appendix C.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 352, + 200, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 201, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 201, + 366 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 108, + 377, + 219, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 220, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 220, + 389 + ], + "score": 1.0, + "content": "4.1 EXPERIMENT SETUP", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Dataset. For all experiments, we used the same experimental setups as previous works in order", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "to perform direct performance comparison (Pascual et al., 2017; Rethage et al., 2018; Soni et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "2018; Germain et al., 2018). 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The final form of the suggested weighted-SDR loss is as follows:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 211, + 506, + 256 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 261, + 437, + 275 + ], + "lines": [ + { + "bbox": [ + 174, + 261, + 437, + 275 + ], + "spans": [ + { + "bbox": [ + 174, + 261, + 437, + 275 + ], + "score": 0.82, + "content": "l o s s _ { w S D R } ( x , y , \\hat { y } ) : = \\alpha l o s s _ { S D R } ( y , \\hat { y } ) + ( 1 - \\alpha ) l o s s _ { S D R } ( z , \\hat { z } )", + "type": "interline_equation", + "image_path": "c823f51cb1dd7e0a902278eb0f9fd88e0a81a4612fcc3dae96decc4d67abe56d.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 174, + 261, + 437, + 275 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 136, + 294 + ], + "score": 1.0, + "content": "where,", + "type": "text" + }, + { + "bbox": [ + 136, + 281, + 176, + 293 + ], + "score": 0.92, + "content": "\\hat { z } = x - \\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 280, + 266, + 294 + ], + "score": 1.0, + "content": "is estimated noise and", + "type": "text" + }, + { + "bbox": [ + 266, + 280, + 372, + 293 + ], + "score": 0.92, + "content": "\\alpha = | | y | | ^ { 2 } / ( | | y | | ^ { 2 } + | | z | | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "is the energy ratio between clean", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 135, + 304 + ], + "score": 1.0, + "content": "speech", + "type": "text" + }, + { + "bbox": [ + 136, + 294, + 142, + 304 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 291, + 182, + 304 + ], + "score": 1.0, + "content": "and noise", + "type": "text" + }, + { + "bbox": [ + 183, + 294, + 189, + 302 + ], + "score": 0.7, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 291, + 505, + 304 + ], + "score": 1.0, + "content": ". 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Note that the speaker and noise classes were uniquely selected for the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 194, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 194, + 532 + ], + "score": 1.0, + "content": "training and test sets.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 397, + 506, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 535, + 505, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "Pre-processing. The original raw waveforms were first downsampled from 48kHz to 16kHz. For the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "actual model input, complex-valued spectrograms were obtained from the downsampled waveforms", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 558, + 367, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 367, + 570 + ], + "score": 1.0, + "content": "via STFT with a 64ms sized Hann window and 16ms hop length.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 535, + 505, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 503, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "Implementation. All experiments were implemented and fine-tuned with NAVER Smart Machine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 585, + 367, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 367, + 597 + ], + "score": 1.0, + "content": "Learning (NSML) platform (Sung et al., 2017; Kim et al., 2018).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 574, + 505, + 597 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 610, + 231, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 232, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 232, + 623 + ], + "score": 1.0, + "content": "4.2 COMPARISON RESULTS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 630, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 644 + ], + "score": 1.0, + "content": "In this subsection, we compare overall speech enhancement performance of our method with pre-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 642, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 506, + 654 + ], + "score": 1.0, + "content": "viously proposed algorithms. As a baseline approach, Wiener filtering (Wiener) with a priori noise", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "score": 1.0, + "content": "SNR estimation was used, along with recent deep-learning based models which are briefly described", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "as the following: SEGAN: a time-domain U-Net model optimized with generative adversarial net-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "works. Wavenet: a time-domain non-causal dilated wavenet-based network. MMSE-GAN: a time-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 686, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 698 + ], + "score": 1.0, + "content": "frequency masking-based method with modified adversarial training method. 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CSIGCBAKCOVLPESQSSNR
Wiener (Scalart et al., 1996)3.232.682.672.225.07
SEGAN (Pascual et al., 2017)3.482.942.802.167.73
Wavenet (Rethage et al., 2018)3.623.232.9811
MMSE-GAN (Soni et al., 2018)3.803.123.142.53-
Deep Feature Loss (Germain et al., 2018)3.863.333.2211
DCUnet-20 (ours)4.244.003.693.1315.95
Large-DCUnet-20 (ours)4.344.103.813.2416.85
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CSIGCBAKCOVLPESQSSNR
DCU-10|Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR||Spc Wav wSDR|SpcWav wSDR
UBD|3.51 3.713.72 |3.50 3.523.56|3.13 3.153.19|2.78 2.61 2.67[11.86 13.1813.37
BDSS2.94 3.193.21 3.43 3.353.392.81 2.852.892.79 2.61 2.6711.61 12.30 12.30
BDT3.30 3.743.723.52 3.57 3.603.02 3.193.222.79 2.66 2.7212.30 13.5413.60
DCU-16| |SpcWav wSDR||Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD3.97 4.004.07 13.66 3.723.753.46 3.433.482.96 2.85 2.9312.75 14.2514.44
BDSS3.63 3.573.72 3.59 3.513.563.28 3.153.272.98 2.82 2.8712.05 12.8912.89
BDT3.97 4.034.10 3.70 3.743.773.49 3.453.523.01 2.88 2.9312.96 14.3914.44
DCU-20| |Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD|4.02 4.154.25 3.88 3.934.01|3.57 3.593.70[3.11 3.01 3.1314.81 16.0316.36
BDSS3.77 3.743.93 3.68 3.653.723.40 3.333.473.06 2.98 3.0512.80 13.6613.68
BDT4.02 4.184.24 3.87 3.954.003.58 3.643.693.13 3.08 3.1314.59 15.85 15.95
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CSIGCBAKCOVLPESQSSNR
Wiener (Scalart et al., 1996)3.232.682.672.225.07
SEGAN (Pascual et al., 2017)3.482.942.802.167.73
Wavenet (Rethage et al., 2018)3.623.232.9811
MMSE-GAN (Soni et al., 2018)3.803.123.142.53-
Deep Feature Loss (Germain et al., 2018)3.863.333.2211
DCUnet-20 (ours)4.244.003.693.1315.95
Large-DCUnet-20 (ours)4.344.103.813.2416.85
", + "type": "table", + "image_path": "b5aaf5e7d10cde49b1ad7a5468086a135d36b1fa4f1e84dbf703db0caec97890.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 119, + 160, + 493, + 194.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 119, + 194.0, + 493, + 228.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 119, + 228.0, + 493, + 262.0 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 4.75 + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "For comparison, we used the configuration of using a 20-layer Deep Complex U-Net (DCUnet-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "20) to estimate a tanh bounded cRM, optimized with weighted-SDR loss. As a showcase for the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "potential of our approach, we also show results from a larger DCUnet-20 (Large-DCUnet-20) which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "has more channels in each layer. Both architectures are specified in detail in Appendix B. Results", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "show that our proposed method outperforms the previous state-of-the-art methods with respect to all", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 356 + ], + "score": 1.0, + "content": "metrics by a large margin. Additionally, we can also see that larger models yield better performance.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "We see the reason to this significant improvement coming from the phase estimation quality of our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 365, + 324, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 324, + 376 + ], + "score": 1.0, + "content": "method, which we plan to investigate in later sections.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 288, + 505, + 376 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 389, + 216, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 217, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 217, + 401 + ], + "score": 1.0, + "content": "4.3 ABLATION STUDIES", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "table", + "bbox": [ + 106, + 472, + 513, + 653 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 412, + 506, + 457 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "Table 2: Table of quantitative evaluation results with corresponding mask and loss function in three", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 506, + 435 + ], + "score": 1.0, + "content": "different model configurations (DCU-10, DCU-16 and DCU-20). The bold font indicates the best", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "loss function when fixing the masking method. The underline indicates the best masking method", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 446, + 227, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 227, + 457 + ], + "score": 1.0, + "content": "when fixing the loss function.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 472, + 513, + 653 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 472, + 513, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 513, + 653 + ], + "score": 0.986, + "html": "
CSIGCBAKCOVLPESQSSNR
DCU-10|Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR||Spc Wav wSDR|SpcWav wSDR
UBD|3.51 3.713.72 |3.50 3.523.56|3.13 3.153.19|2.78 2.61 2.67[11.86 13.1813.37
BDSS2.94 3.193.21 3.43 3.353.392.81 2.852.892.79 2.61 2.6711.61 12.30 12.30
BDT3.30 3.743.723.52 3.57 3.603.02 3.193.222.79 2.66 2.7212.30 13.5413.60
DCU-16| |SpcWav wSDR||Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD3.97 4.004.07 13.66 3.723.753.46 3.433.482.96 2.85 2.9312.75 14.2514.44
BDSS3.63 3.573.72 3.59 3.513.563.28 3.153.272.98 2.82 2.8712.05 12.8912.89
BDT3.97 4.034.10 3.70 3.743.773.49 3.453.523.01 2.88 2.9312.96 14.3914.44
DCU-20| |Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD|4.02 4.154.25 3.88 3.934.01|3.57 3.593.70[3.11 3.01 3.1314.81 16.0316.36
BDSS3.77 3.743.93 3.68 3.653.723.40 3.333.473.06 2.98 3.0512.80 13.6613.68
BDT4.02 4.184.24 3.87 3.954.003.58 3.643.693.13 3.08 3.1314.59 15.85 15.95
", + "type": "table", + "image_path": "02806868a0fe890929ffbd9c509f1645d058bb8406bf26c42280a781ff92f1d8.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 106, + 472, + 513, + 532.3333333333334 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 106, + 532.3333333333334, + 513, + 592.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 106, + 592.6666666666667, + 513, + 653.0000000000001 + ], + "spans": [], + "index": 24 + } + ] + } + ], + "index": 21.25 + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "Masking strategy and loss functions. In this experiment, we show the evaluation results on how", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "the various masking strategies and loss functions affect the performance of speech enhancement.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 693, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 506, + 706 + ], + "score": 1.0, + "content": "Three masking strategies (Unbounded (UBD); Ephrat et al.: Bounded (sig-sig) (BDSS); Bounded", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 704, + 505, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 505, + 717 + ], + "score": 1.0, + "content": "(tanh) (BDT)) are compared to see the effect of each different masking method. Then, to compare", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "our proposed loss function (weighted-SDR) with MSE, we compared two MSE in STFT-domain and", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "time-domain (Spectrogram-MSE (Spc); Wave-MSE (Wav); and proposed: weighted-SDR (wSDR)).", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 670, + 506, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "our proposed loss function (weighted-SDR) with MSE, we compared two MSE in STFT-domain and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "time-domain (Spectrogram-MSE (Spc); Wave-MSE (Wav); and proposed: weighted-SDR (wSDR)).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "Table 2 shows the jointly combined results on varied masking strategies and loss functions, where", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "three models (DCU-10 (1.4M), DCU-16 (2.3M), and DCU-20 (3.5M)) are investigated to see how", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "architectural differences in the model affect quantitative results. In terms of masking strategy, the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "proposed BDT mask mostly yields better results than UBD mask in DCU-10 and DCU-16, imply-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "ing the importance of limiting optimization space with prior knowledge. However, in the case of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "score": 1.0, + "content": "DCU-20, UBD mask was able to frequently surpass the performance of BDT mask. Intuitively, this", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "indicates that when the number of parameter gets large enough, the model is able to fit the distri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 504, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 504, + 200 + ], + "score": 1.0, + "content": "bution of data well even when the optimization space is not bounded. In terms of the loss function,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "almost every result shows that optimizing with wSDR loss gives the best result. However, we found", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 221 + ], + "score": 1.0, + "content": "out that Spc loss often provides better PESQ results than wSDR loss for DCU-10 and DCU-16", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 390, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 390, + 233 + ], + "score": 1.0, + "content": "except DCU-20 case where Spc and wSDR gave similar PESQ results.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7 + }, + { + "type": "table", + "bbox": [ + 114, + 300, + 497, + 395 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 241, + 505, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "Table 3: Table of quantitative evaluation results from three different settings (cRMCn: Complex-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "valued output/Complex-valued network, cRMRn: Complex-valued output/Real-valued network, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 135, + 274 + ], + "score": 0.58, + "content": "\\mathbf { R M } \\mathbb { R } \\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 263, + 505, + 276 + ], + "score": 1.0, + "content": ": Real-valued output/Real-valued network) to show the appropriateness of using complex-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 275, + 417, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 417, + 286 + ], + "score": 1.0, + "content": "valued networks for speech enhancement. Bold font indicates the best results.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "table_body", + "bbox": [ + 114, + 300, + 497, + 395 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 300, + 497, + 395 + ], + "spans": [ + { + "bbox": [ + 114, + 300, + 497, + 395 + ], + "score": 0.985, + "html": "
20-layer16-layer10-layer
cRMCncRMRnRMRncRMCncRMRnRMRncRMCncRMRnRMRn
CSIG4.244.214.064.104.063.883.743.713.71
CBAK4.003.923.403.773.753.333.603.593.23
COVL3.693.653.403.523.493.263.223.203.01
PESQ3.133.072.742.932.912.662.722.722.51
SSNR15.9515.549.9014.4414.259.6613.6013.499.43
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In order to show that complex neural networks", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 335, + 434 + ], + "score": 1.0, + "content": "are effective, we compare evaluation results of DCUnet", + "type": "text" + }, + { + "bbox": [ + 336, + 423, + 353, + 433 + ], + "score": 0.66, + "content": "( \\mathbb { C } \\mathrm { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "and its corresponding real-valued U-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 274, + 447 + ], + "score": 1.0, + "content": "Net setting with the same parameter size", + "type": "text" + }, + { + "bbox": [ + 274, + 434, + 292, + 444 + ], + "score": 0.74, + "content": "( \\mathbb { R } \\mathbf { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 432, + 505, + 447 + ], + "score": 1.0, + "content": ". For the real-valued network, we tested two settings", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 140, + 455 + ], + "score": 0.37, + "content": "\\mathbf { c R M } \\mathbb { R } \\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 444, + 157, + 457 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 444, + 187, + 455 + ], + "score": 0.67, + "content": "\\mathbf { R M } \\mathbb { R } \\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "to show the effectiveness of phase estimation. The first setting takes a complex-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "valued spectrogram as an input, estimating a complex ratio mask (cRM) with a tanh bound. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "second setting takes a magnitude spectrogram as an input, estimating a magnitude ratio mask (RM)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "with a sigmoid bound. All models were trained with weighted-SDR loss, where the ground truth", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 235, + 500 + ], + "score": 1.0, + "content": "phase was given while training", + "type": "text" + }, + { + "bbox": [ + 235, + 488, + 264, + 499 + ], + "score": 0.4, + "content": "\\mathbf { R M } \\mathbb { R } \\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 488, + 506, + 500 + ], + "score": 1.0, + "content": ". Additionally, all models were trained on different number", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "of parameters (20-layer (3.5M), 16-layer (2.3M), and 10-layer (1.4M)) to show that the results are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "consistent regardless of model capacity. 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Bold font indicates the best results.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "table_body", + "bbox": [ + 114, + 300, + 497, + 395 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 300, + 497, + 395 + ], + "spans": [ + { + "bbox": [ + 114, + 300, + 497, + 395 + ], + "score": 0.985, + "html": "
20-layer16-layer10-layer
cRMCncRMRnRMRncRMCncRMRnRMRncRMCncRMRnRMRn
CSIG4.244.214.064.104.063.883.743.713.71
CBAK4.003.923.403.773.753.333.603.593.23
COVL3.693.653.403.523.493.263.223.203.01
PESQ3.133.072.742.932.912.662.722.722.51
SSNR15.9515.549.9014.4414.259.6613.6013.499.43
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HardMediumEasyTotal
DCUnet > Deep Feature Loss0.900.820.690.82
DCUnet > Wavenet0.980.950.750.93
DCUnet > SEGAN0.990.930.710.92
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HardMediumEasyTotal
DCUnet > Deep Feature Loss0.900.820.690.82
DCUnet > Wavenet0.980.950.750.93
DCUnet > SEGAN0.990.930.710.92
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The plots", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "clearly show that the Bounded (sig-sig) mask by Ephrat et al. fails to reflect the distribution of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "complex ideal ratio mask (cIRM), whereas the proposed Bounded (tanh) or Unbounded mask tries", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "to follow the ground truth. One interesting finding is that, while time-domain loss functions such", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "as weighted-SDR or Wave-MSE can capture the distribution of the ground truth, Spectrogram-MSE", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 697, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 504, + 711 + ], + "score": 1.0, + "content": "fails to capture the distribution of the cIRM. More specifically, when we use Spectrogram-MSE,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "the imaginary values of the predicted cRMs are almost always zero, indicating that it ends up only", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 102, + 312, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 102, + 312, + 129, + 344 + ], + "score": 1.0, + "content": "scalin(e.g.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 130, + 324, + 343, + 340 + ], + "score": 0.89, + "content": "\\bar { ( } X _ { t , f } ^ { r e a l } + i X _ { t , f } ^ { i m a g } ) ( a + \\bar { i } \\cdot \\bar { 0 } ) = a ( X _ { t , f } ^ { r e a l } + i X _ { t , f } ^ { i m a g } ) )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 344, + 312, + 505, + 344 + ], + "score": 1.0, + "content": "he phase of noisy speech with rotations.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 621, + 506, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 80, + 499, + 234 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 80, + 499, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 80, + 499, + 234 + ], + "spans": [ + { + "bbox": [ + 109, + 80, + 499, + 234 + ], + "score": 0.971, + "type": "image", + "image_path": "2fee5faa352a331c4b09236534209a0c3b7a0321fef337a23dd190b8e99fd896.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 80, + 499, + 131.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 131.33333333333334, + 499, + 182.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 182.66666666666669, + 499, + 234.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 246, + 505, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 246, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 504, + 259 + ], + "score": 1.0, + "content": "Figure 5: Illustration of four wave-plot segments of estimated speech with reference clean speech.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "(a) shows the wave-plot of synthesized speech with clean magnitude speech and noisy phase", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "(CMNP). 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This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "again confirms that optimizing the model with Spectrogram-MSE makes it difficult to learn phase", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 408, + 500, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 500, + 423 + ], + "score": 1.0, + "content": "correction, meaning that the model ends up reusing noisy phase just like conventional approaches.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "To explicitly support these observations, we would need a quantitative measure for phase estimation.", + "type": "text" + } + ], + "index": 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The results show that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 303, + 577 + ], + "score": 1.0, + "content": "the best phase improvement (Phase Improvement", + "type": "text" + }, + { + "bbox": [ + 303, + 566, + 312, + 574 + ], + "score": 0.77, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "PhaseDist(C, N) − PhaseDist(C, E)) is obtained", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 573, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 590 + ], + "score": 1.0, + "content": "with wSDR loss under every SNR condition. Also Spc loss gives the worst results, again reinforcing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "our observation. Analysis between the phase improvement and performance improvement is further", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 597, + 212, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 212, + 610 + ], + "score": 1.0, + "content": "discussed in Appendix G.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 625, + 195, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 198, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 198, + 641 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 727 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "In this paper, we proposed Deep Complex U-Net which combines two models to deal with complex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "valued spectrograms for speech enhancement. 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Furthermore, we conducted both quantitative", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 705, + 505, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 505, + 717 + ], + "score": 1.0, + "content": "and qualitative studies and demonstrated that the proposed method is consistently superior to the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 716, + 237, + 729 + ], + "spans": [ + { + "bbox": [ + 105, + 716, + 237, + 729 + ], + "score": 1.0, + "content": "previously proposed algorithms.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 80, + 499, + 234 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 80, + 499, + 234 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 80, + 499, + 234 + ], + "spans": [ + { + "bbox": [ + 109, + 80, + 499, + 234 + ], + "score": 0.971, + "type": "image", + "image_path": "2fee5faa352a331c4b09236534209a0c3b7a0321fef337a23dd190b8e99fd896.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 80, + 499, + 131.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 131.33333333333334, + 499, + 182.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 182.66666666666669, + 499, + 234.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 246, + 505, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 246, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 504, + 259 + ], + "score": 1.0, + "content": "Figure 5: Illustration of four wave-plot segments of estimated speech with reference clean speech.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "(a) shows the wave-plot of synthesized speech with clean magnitude speech and noisy phase", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "(CMNP). Three other cases show the wave-plots of estimated speech with different loss functions", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 279, + 360, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 360, + 292 + ], + "score": 1.0, + "content": "(b) Spectrogram-MSE, (c) Wave-MSE, and (d) weighted-SDR.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 503, + 339 + ], + "lines": [], + "index": 7, + "bbox_fs": [ + 102, + 312, + 505, + 344 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 343, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "In order to demonstrate this effect in an alternate perspective, we also plotted estimated waveforms", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 386, + 367 + ], + "score": 1.0, + "content": "for each loss function in Figure 5. 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This", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "again confirms that optimizing the model with Spectrogram-MSE makes it difficult to learn phase", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 408, + 500, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 500, + 423 + ], + "score": 1.0, + "content": "correction, meaning that the model ends up reusing noisy phase just like conventional approaches.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 343, + 506, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "To explicitly support these observations, we would need a quantitative measure for phase 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In Acoustics, Speech and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 116, + 720, + 341, + 733 + ], + "spans": [ + { + "bbox": [ + 116, + 720, + 341, + 733 + ], + "score": 1.0, + "content": "Signal Processing (ICASSP), pp. 241–245. IEEE, 2017.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47, + "bbox_fs": [ + 106, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 161, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 109, + 106, + 413, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 415, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 415, + 119 + ], + "score": 1.0, + "content": "A REAL-VALUED CONVOLUTION & COMPLEX-VALUED CONVOLUTION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 504, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 504, + 137 + ], + "score": 1.0, + "content": "In this section, we address the difference between the real-valued convolution and the complex-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 367, + 150 + ], + "score": 1.0, + "content": "valued convolution. Given a complex-valued convolution filter", + "type": "text" + }, + { + "bbox": [ + 367, + 138, + 434, + 148 + ], + "score": 0.91, + "content": "W = A + i B", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "with real-valued", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 143, + 162 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 149, + 153, + 159 + ], + "score": 0.68, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 148, + 172, + 162 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 149, + 182, + 159 + ], + "score": 0.75, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 148, + 506, + 162 + ], + "score": 1.0, + "content": ", the complex-valued convolution can be interpreted as two different real-valued", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "convolution operations with shared parameters, as illustrated in Figure 6 (b). For a fixed number of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 136, + 183 + ], + "score": 1.0, + "content": "#Cha", + "type": "text" + }, + { + "bbox": [ + 137, + 171, + 398, + 182 + ], + "score": 0.32, + "content": "\\AA \\ i n e l \\ p r o d u c t = \\# I n p u t \\ c h a n n e l ( M ) \\times \\# O u t p u t \\ c h a n n e l ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 171, + 505, + 183 + ], + "score": 1.0, + "content": ", the number of parameters", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "of the complex-valued convolution becomes double of that of a real-valued convolution. Considering", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "this fact, we built the pair of a real-valued network and a complex-valued network with the same", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "number of parameters by reducing #Channel product of complex-valued convolution by half for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 491, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 491, + 227 + ], + "score": 1.0, + "content": "a fair comparison. The detail of models reflecting this configuration is explained in Appendix B.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "image", + "bbox": [ + 119, + 240, + 493, + 441 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 240, + 493, + 441 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 240, + 493, + 441 + ], + "spans": [ + { + "bbox": [ + 119, + 240, + 493, + 441 + ], + "score": 0.947, + "type": "image", + "image_path": "b2e6920edf71e4738bc877916852adb08f1fa7335b1489df3a69799b1c85cfa3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 119, + 240, + 493, + 307.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 119, + 307.0, + 493, + 374.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 119, + 374.0, + 493, + 441.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 126, + 461, + 482, + 473 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 460, + 483, + 474 + ], + "spans": [ + { + "bbox": [ + 127, + 460, + 483, + 474 + ], + "score": 1.0, + "content": "Figure 6: Illustration of (a) real-valued convolution and (b) complex-valued convolution.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + } + ], + "index": 13.0 + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 228, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 229, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 229, + 505 + ], + "score": 1.0, + "content": "B MODEL ARCHITECTURE", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "In this section, we describe three different model architectures (DCUnet-20 (#params: 3.5M),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 159, + 537 + ], + "score": 1.0, + "content": "DCUnet-16 (", + "type": "text" + }, + { + "bbox": [ + 159, + 525, + 168, + 536 + ], + "score": 0.38, + "content": "\\#", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "params: 2.3M), and DCUnet-10 (#params: 1.4M)) each in complex-valued network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 452, + 548 + ], + "score": 1.0, + "content": "setting and real-valued network setting in Figure 7, 8, 9. Both complex-valued network", + "type": "text" + }, + { + "bbox": [ + 453, + 536, + 467, + 546 + ], + "score": 0.54, + "content": "\\left( \\mathbb { C } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "and real-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 172, + 558 + ], + "score": 1.0, + "content": "valued network", + "type": "text" + }, + { + "bbox": [ + 172, + 547, + 186, + 557 + ], + "score": 0.64, + "content": "( \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "have the same size of convolution filters with different number of channels to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "score": 1.0, + "content": "set the parameter equally. The largest model, Large-DCUnet-20, in Table 1 is also described in Fig-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "ure 10. Every convolution operation is followed by batch normalization and an activation function", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "as described in Figure 11. For the complex-valued network, the complex-valued version of batch", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "normalization and activation function was used following Deep Complex Networks (Trabelsi et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "2018). Note that in the very last layer of every model the batch normalization and leaky ReLU", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "activation was not used and non-linearity function for mask was applied instead. The real-valued", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 624, + 390, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 390, + 635 + ], + "score": 1.0, + "content": "network configuration was not considered in the case of largest model.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 648, + 288, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 289, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 289, + 660 + ], + "score": 1.0, + "content": "C PROPERTIES OF WEIGHTED-SDR LOSS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 668, + 504, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "In this section, we summarize the properties of the proposed weighted-SDR loss. First, we show that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "score": 1.0, + "content": "the range of weighted-SDR loss is bounded and explain the conditions under which the minimum", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 690, + 428, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 428, + 703 + ], + "score": 1.0, + "content": "value is obtained. Next, we explain the gradients in the case of noise-only input.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 161, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 109, + 106, + 413, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 415, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 415, + 119 + ], + "score": 1.0, + "content": "A REAL-VALUED CONVOLUTION & COMPLEX-VALUED CONVOLUTION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 106, + 415, + 119 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 504, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 504, + 137 + ], + "score": 1.0, + "content": "In this section, we address the difference between the real-valued convolution and the complex-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 367, + 150 + ], + "score": 1.0, + "content": "valued convolution. Given a complex-valued convolution filter", + "type": "text" + }, + { + "bbox": [ + 367, + 138, + 434, + 148 + ], + "score": 0.91, + "content": "W = A + i B", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "with real-valued", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 143, + 162 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 149, + 153, + 159 + ], + "score": 0.68, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 148, + 172, + 162 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 172, + 149, + 182, + 159 + ], + "score": 0.75, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 148, + 506, + 162 + ], + "score": 1.0, + "content": ", the complex-valued convolution can be interpreted as two different real-valued", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "convolution operations with shared parameters, as illustrated in Figure 6 (b). For a fixed number of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 136, + 183 + ], + "score": 1.0, + "content": "#Cha", + "type": "text" + }, + { + "bbox": [ + 137, + 171, + 398, + 182 + ], + "score": 0.32, + "content": "\\AA \\ i n e l \\ p r o d u c t = \\# I n p u t \\ c h a n n e l ( M ) \\times \\# O u t p u t \\ c h a n n e l ( N )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 171, + 505, + 183 + ], + "score": 1.0, + "content": ", the number of parameters", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "of the complex-valued convolution becomes double of that of a real-valued convolution. Considering", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "this fact, we built the pair of a real-valued network and a complex-valued network with the same", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "number of parameters by reducing #Channel product of complex-valued convolution by half for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 491, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 491, + 227 + ], + "score": 1.0, + "content": "a fair comparison. The detail of models reflecting this configuration is explained in Appendix B.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 127, + 506, + 227 + ] + }, + { + "type": "image", + "bbox": [ + 119, + 240, + 493, + 441 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 240, + 493, + 441 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 240, + 493, + 441 + ], + "spans": [ + { + "bbox": [ + 119, + 240, + 493, + 441 + ], + "score": 0.947, + "type": "image", + "image_path": "b2e6920edf71e4738bc877916852adb08f1fa7335b1489df3a69799b1c85cfa3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 119, + 240, + 493, + 307.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 119, + 307.0, + 493, + 374.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 119, + 374.0, + 493, + 441.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 126, + 461, + 482, + 473 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 460, + 483, + 474 + ], + "spans": [ + { + "bbox": [ + 127, + 460, + 483, + 474 + ], + "score": 1.0, + "content": "Figure 6: Illustration of (a) real-valued convolution and (b) complex-valued convolution.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + } + ], + "index": 13.0 + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 228, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 229, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 229, + 505 + ], + "score": 1.0, + "content": "B MODEL ARCHITECTURE", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "In this section, we describe three different model architectures (DCUnet-20 (#params: 3.5M),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 159, + 537 + ], + "score": 1.0, + "content": "DCUnet-16 (", + "type": "text" + }, + { + "bbox": [ + 159, + 525, + 168, + 536 + ], + "score": 0.38, + "content": "\\#", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "params: 2.3M), and DCUnet-10 (#params: 1.4M)) each in complex-valued network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 452, + 548 + ], + "score": 1.0, + "content": "setting and real-valued network setting in Figure 7, 8, 9. Both complex-valued network", + "type": "text" + }, + { + "bbox": [ + 453, + 536, + 467, + 546 + ], + "score": 0.54, + "content": "\\left( \\mathbb { C } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "and real-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 172, + 558 + ], + "score": 1.0, + "content": "valued network", + "type": "text" + }, + { + "bbox": [ + 172, + 547, + 186, + 557 + ], + "score": 0.64, + "content": "( \\mathbb { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "have the same size of convolution filters with different number of channels to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 571 + ], + "score": 1.0, + "content": "set the parameter equally. The largest model, Large-DCUnet-20, in Table 1 is also described in Fig-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 581 + ], + "score": 1.0, + "content": "ure 10. Every convolution operation is followed by batch normalization and an activation function", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "as described in Figure 11. 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The real-valued", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 624, + 390, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 390, + 635 + ], + "score": 1.0, + "content": "network configuration was not considered in the case of largest model.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 513, + 506, + 635 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 648, + 288, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 289, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 289, + 660 + ], + "score": 1.0, + "content": "C PROPERTIES OF WEIGHTED-SDR LOSS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 668, + 504, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "In this section, we summarize the properties of the proposed weighted-SDR loss. 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Cauchy-Schwarz inequality states that for", + "type": "text" + }, + { + "bbox": [ + 309, + 405, + 343, + 416 + ], + "score": 0.93, + "content": "a \\in \\mathbb { R } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 404, + 363, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 363, + 405, + 397, + 417 + ], + "score": 0.87, + "content": "b \\in \\mathbb { R } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 404, + 399, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 399, + 405, + 505, + 418 + ], + "score": 0.91, + "content": "- \\| a \\| \\| b \\| \\leq < a , b > \\leq", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 135, + 429 + ], + "score": 0.9, + "content": "\\| a \\| \\| b \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 417, + 334, + 430 + ], + "score": 1.0, + "content": ". By this inequality, [-1,1] becomes the range of", + "type": "text" + }, + { + "bbox": [ + 335, + 418, + 377, + 429 + ], + "score": 0.88, + "content": "l o s s _ { w S D R }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 417, + 506, + 430 + ], + "score": 1.0, + "content": ". To attain the minimum value,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "the equality condition of the Cauchy-Schwarz inequality must be satisfied. This equality condition", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 170, + 452 + ], + "score": 1.0, + "content": "is equivalent to", + "type": "text" + }, + { + "bbox": [ + 171, + 439, + 196, + 450 + ], + "score": 0.9, + "content": "b = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 439, + 208, + 452 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 208, + 439, + 237, + 450 + ], + "score": 0.9, + "content": "a = t b", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 439, + 255, + 452 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 256, + 439, + 286, + 450 + ], + "score": 0.91, + "content": "\\exists t \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 439, + 505, + 452 + ], + "score": 1.0, + "content": ". Applying the equality condition with the assumption", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 108, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 108, + 450, + 181, + 462 + ], + "score": 0.89, + "content": "( y \\neq 0 , x - y \\neq 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 450, + 270, + 463 + ], + "score": 1.0, + "content": "to Equation 9 leads to", + "type": "text" + }, + { + "bbox": [ + 271, + 451, + 304, + 461 + ], + "score": 0.91, + "content": "\\hat { y } = t _ { 1 } y", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 450, + 321, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 450, + 393, + 461 + ], + "score": 0.91, + "content": "x - { \\hat { y } } = t _ { 2 } ( x - y )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 450, + 415, + 463 + ], + "score": 1.0, + "content": ", f or", + "type": "text" + }, + { + "bbox": [ + 415, + 450, + 450, + 461 + ], + "score": 0.91, + "content": "\\exists t _ { 1 } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 450, + 467, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 467, + 450, + 501, + 461 + ], + "score": 0.91, + "content": "\\exists t _ { 2 } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 450, + 506, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 286, + 474 + ], + "score": 1.0, + "content": "By adding these two equations, we can get", + "type": "text" + }, + { + "bbox": [ + 286, + 461, + 385, + 473 + ], + "score": 0.91, + "content": "( 1 - t _ { 1 } ) x = ( t _ { 1 } - t _ { 2 } ) y", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 460, + 468, + 474 + ], + "score": 1.0, + "content": ". By the assumption", + "type": "text" + }, + { + "bbox": [ + 469, + 462, + 501, + 472 + ], + "score": 0.89, + "content": "x \\neq c y", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 460, + 506, + 474 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 253, + 484 + ], + "score": 1.0, + "content": "which is generally satisfied for large", + "type": "text" + }, + { + "bbox": [ + 254, + 473, + 262, + 482 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 471, + 334, + 484 + ], + "score": 1.0, + "content": ", we can conclude", + "type": "text" + }, + { + "bbox": [ + 335, + 473, + 396, + 483 + ], + "score": 0.91, + "content": "t _ { 1 } = 1 , t _ { 1 } = t _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "must be satisfied when the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 217, + 494 + ], + "score": 1.0, + "content": "minimum value is attained.", + "type": "text" + }, + { + "bbox": [ + 494, + 483, + 505, + 494 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "The following property of the weighted-SDR loss shows that the network can also learn from noise-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 335, + 530 + ], + "score": 1.0, + "content": "only training data. In experiments, we add small number", + "type": "text" + }, + { + "bbox": [ + 335, + 520, + 340, + 528 + ], + "score": 0.78, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "to denominators of Equation 9. Thus for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 528, + 264, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 151, + 542 + ], + "score": 1.0, + "content": "the case of", + "type": "text" + }, + { + "bbox": [ + 152, + 529, + 176, + 540 + ], + "score": 0.9, + "content": "y = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 528, + 264, + 542 + ], + "score": 1.0, + "content": ", Equation 9 becomes", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 545, + 388, + 572 + ], + "lines": [ + { + "bbox": [ + 223, + 545, + 388, + 572 + ], + "spans": [ + { + "bbox": [ + 223, + 545, + 388, + 572 + ], + "score": 0.94, + "content": "\\mathit { l o s s } _ { w S D R } ( x , 0 , \\hat { y } ) = - \\frac { < x , x - \\hat { y } > } { \\| x \\| \\| x - \\hat { y } \\| + \\epsilon }", + "type": "interline_equation", + "image_path": "cbf1c5396ad5b5888f7e0fcc8d611367c4b7ddbb3b5890d2d8a141346bfd1c16.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 223, + 545, + 388, + 572 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 577, + 504, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 266, + 591 + ], + "score": 1.0, + "content": "Proposition 2. When we parameterize", + "type": "text" + }, + { + "bbox": [ + 266, + 577, + 309, + 589 + ], + "score": 0.94, + "content": "{ \\hat { y } } = g _ { \\boldsymbol { \\theta } } ( { \\boldsymbol { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 575, + 329, + 591 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 329, + 577, + 421, + 589 + ], + "score": 0.9, + "content": "\\begin{array} { r } { l o s s _ { w S D R } ( x , y , g _ { \\theta } ( x ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 575, + 505, + 591 + ], + "score": 1.0, + "content": "has a non-zero gra-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 587, + 338, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 189, + 602 + ], + "score": 1.0, + "content": "dient with respect to", + "type": "text" + }, + { + "bbox": [ + 190, + 589, + 196, + 598 + ], + "score": 0.72, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 587, + 338, + 602 + ], + "score": 1.0, + "content": "even if the target source y is empty.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 324, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 325, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 325, + 624 + ], + "score": 1.0, + "content": "Proof. We can calculate partial derivatives as follows:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 628, + 492, + 686 + ], + "lines": [ + { + "bbox": [ + 118, + 628, + 492, + 686 + ], + "spans": [ + { + "bbox": [ + 118, + 628, + 492, + 686 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { { \\frac { \\partial l o s s _ { w S D R } ( x , y , g _ { \\theta } ( x ) ) } { \\partial \\theta } \\Big | _ { ( x = x _ { 0 } , y = 0 , \\theta = \\theta _ { 0 } ) } = \\frac { \\partial l o s s _ { w S D R } ( x _ { 0 } , 0 , g _ { \\theta } ( x _ { 0 } ) ) } { \\partial \\theta } \\Big | _ { \\theta = \\theta _ { 0 } } } } \\\\ & { = < \\frac { x _ { 0 } } { \\| x _ { 0 } \\| } , \\frac { \\partial } { \\partial \\theta } \\{ \\frac { x _ { 0 } - g _ { \\theta } ( x _ { 0 } ) } { \\| x _ { 0 } - g _ { \\theta } ( x _ { 0 } ) \\| + \\epsilon / \\| x _ { 0 } \\| } \\} \\Big | _ { \\theta = \\theta _ { 0 } } > } 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\\alpha \\frac { < y , \\hat { y } > } { \\| y \\| \\| \\hat { y } \\| } - ( 1 - \\alpha ) \\frac { < x - y , x - \\hat { y } > } { \\| x - y \\| \\| x - \\hat { y } \\| }", + "type": "interline_equation", + "image_path": "3fa230897872069440aff3b76eb3045854cb87f1f5559a1a2e6287370d1f431b.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 175, + 322, + 436, + 349 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 355, + 470, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 468, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 136, + 371 + ], + "score": 1.0, + "content": "where,", + "type": "text" + }, + { + "bbox": [ + 136, + 359, + 144, + 366 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 353, + 366, + 371 + ], + "score": 1.0, + "content": "is the energy ratio between target source and noise, i.e.,", + "type": "text" + }, + { + "bbox": [ + 366, + 354, + 468, + 369 + ], + "score": 0.91, + "content": "\\left\\| y \\right\\| ^ { 2 } / ( \\left\\| y \\right\\| ^ { 2 } + \\left\\| x - y \\right\\| ^ { 2 } )", + "type": "inline_equation" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 353, + 468, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 504, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 171, + 384 + ], + "score": 1.0, + "content": "Proposition 1.", + "type": "text" + }, + { + "bbox": [ + 171, + 371, + 246, + 383 + ], + "score": 0.89, + "content": "l o s s _ { w S D R } ( x , y , \\hat { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 370, + 305, + 384 + ], + "score": 1.0, + "content": "is bounded on", + "type": "text" + }, + { + "bbox": [ + 306, + 371, + 331, + 383 + ], + "score": 0.42, + "content": "\\left. l - I , I \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 370, + 415, + 384 + ], + "score": 1.0, + "content": ". Moreover, for fixed", + "type": "text" + }, + { + "bbox": [ + 415, + 371, + 440, + 383 + ], + "score": 0.89, + "content": "y \\ne 0", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 370, + 459, + 384 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 459, + 371, + 501, + 383 + ], + "score": 0.9, + "content": "x - y \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 370, + 505, + 384 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 382, + 420, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 188, + 395 + ], + "score": 1.0, + "content": "the minimum value -", + "type": "text" + }, + { + "bbox": [ + 188, + 384, + 194, + 392 + ], + "score": 0.27, + "content": "\\mathbf { \\nabla } . \\mathit { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 382, + 302, + 395 + ], + "score": 1.0, + "content": "can only be attained when", + "type": "text" + }, + { + "bbox": [ + 303, + 383, + 327, + 394 + ], + "score": 0.9, + "content": "{ \\hat { y } } = y", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 382, + 339, + 395 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 339, + 383, + 369, + 394 + ], + "score": 0.47, + "content": "x \\neq c y", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 382, + 384, + 395 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 384, + 383, + 415, + 393 + ], + "score": 0.79, + "content": "\\forall c \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 382, + 420, + 395 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 370, + 505, + 395 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 506, + 495 + ], + "lines": [ + { + "bbox": [ + 104, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 404, + 308, + 419 + ], + "score": 1.0, + "content": "Proof. Cauchy-Schwarz inequality states that for", + "type": "text" + }, + { + "bbox": [ + 309, + 405, + 343, + 416 + ], + "score": 0.93, + "content": "a \\in \\mathbb { R } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 404, + 363, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 363, + 405, + 397, + 417 + ], + "score": 0.87, + "content": "b \\in \\mathbb { R } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 404, + 399, + 419 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 399, + 405, + 505, + 418 + ], + "score": 0.91, + "content": "- \\| a \\| \\| b \\| \\leq < a , b > \\leq", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 135, + 429 + ], + "score": 0.9, + "content": "\\| a \\| \\| b \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 417, + 334, + 430 + ], + "score": 1.0, + "content": ". By this inequality, [-1,1] becomes the range of", + "type": "text" + }, + { + "bbox": [ + 335, + 418, + 377, + 429 + ], + "score": 0.88, + "content": "l o s s _ { w S D R }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 417, + 506, + 430 + ], + "score": 1.0, + "content": ". To attain the minimum value,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "the equality condition of the Cauchy-Schwarz inequality must be satisfied. This equality condition", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 170, + 452 + ], + "score": 1.0, + "content": "is equivalent to", + "type": "text" + }, + { + "bbox": [ + 171, + 439, + 196, + 450 + ], + "score": 0.9, + "content": "b = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 439, + 208, + 452 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 208, + 439, + 237, + 450 + ], + "score": 0.9, + "content": "a = t b", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 439, + 255, + 452 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 256, + 439, + 286, + 450 + ], + "score": 0.91, + "content": "\\exists t \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 439, + 505, + 452 + ], + "score": 1.0, + "content": ". Applying the equality condition with the assumption", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 108, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 108, + 450, + 181, + 462 + ], + "score": 0.89, + "content": "( y \\neq 0 , x - y \\neq 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 450, + 270, + 463 + ], + "score": 1.0, + "content": "to Equation 9 leads to", + "type": "text" + }, + { + "bbox": [ + 271, + 451, + 304, + 461 + ], + "score": 0.91, + "content": "\\hat { y } = t _ { 1 } y", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 450, + 321, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 450, + 393, + 461 + ], + "score": 0.91, + "content": "x - { \\hat { y } } = t _ { 2 } ( x - y )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 450, + 415, + 463 + ], + "score": 1.0, + "content": ", f or", + "type": "text" + }, + { + "bbox": [ + 415, + 450, + 450, + 461 + ], + "score": 0.91, + "content": "\\exists t _ { 1 } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 450, + 467, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 467, + 450, + 501, + 461 + ], + "score": 0.91, + "content": "\\exists t _ { 2 } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 450, + 506, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 286, + 474 + ], + "score": 1.0, + "content": "By adding these two equations, we can get", + "type": "text" + }, + { + "bbox": [ + 286, + 461, + 385, + 473 + ], + "score": 0.91, + "content": "( 1 - t _ { 1 } ) x = ( t _ { 1 } - t _ { 2 } ) y", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 460, + 468, + 474 + ], + "score": 1.0, + "content": ". By the assumption", + "type": "text" + }, + { + "bbox": [ + 469, + 462, + 501, + 472 + ], + "score": 0.89, + "content": "x \\neq c y", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 460, + 506, + 474 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 253, + 484 + ], + "score": 1.0, + "content": "which is generally satisfied for large", + "type": "text" + }, + { + "bbox": [ + 254, + 473, + 262, + 482 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 471, + 334, + 484 + ], + "score": 1.0, + "content": ", we can conclude", + "type": "text" + }, + { + "bbox": [ + 335, + 473, + 396, + 483 + ], + "score": 0.91, + "content": "t _ { 1 } = 1 , t _ { 1 } = t _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "must be satisfied when the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 217, + 494 + ], + "score": 1.0, + "content": "minimum value is attained.", + "type": "text" + }, + { + "bbox": [ + 494, + 483, + 505, + 494 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 404, + 506, + 494 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "The following property of the weighted-SDR loss shows that the network can also learn from noise-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 335, + 530 + ], + "score": 1.0, + "content": "only training data. In experiments, we add small number", + "type": "text" + }, + { + "bbox": [ + 335, + 520, + 340, + 528 + ], + "score": 0.78, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "to denominators of Equation 9. Thus for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 528, + 264, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 151, + 542 + ], + "score": 1.0, + "content": "the case of", + "type": "text" + }, + { + "bbox": [ + 152, + 529, + 176, + 540 + ], + "score": 0.9, + "content": "y = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 528, + 264, + 542 + ], + "score": 1.0, + "content": ", Equation 9 becomes", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 506, + 505, + 542 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 545, + 388, + 572 + ], + "lines": [ + { + "bbox": [ + 223, + 545, + 388, + 572 + ], + "spans": [ + { + "bbox": [ + 223, + 545, + 388, + 572 + ], + "score": 0.94, + "content": "\\mathit { l o s s } _ { w S D R } ( x , 0 , \\hat { y } ) = - \\frac { < x , x - \\hat { y } > } { \\| x \\| \\| x - \\hat { y } \\| + \\epsilon }", + "type": "interline_equation", + "image_path": "cbf1c5396ad5b5888f7e0fcc8d611367c4b7ddbb3b5890d2d8a141346bfd1c16.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 223, + 545, + 388, + 572 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 577, + 504, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 266, + 591 + ], + "score": 1.0, + "content": "Proposition 2. When we parameterize", + "type": "text" + }, + { + "bbox": [ + 266, + 577, + 309, + 589 + ], + "score": 0.94, + "content": "{ \\hat { y } } = g _ { \\boldsymbol { \\theta } } ( { \\boldsymbol { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 575, + 329, + 591 + ], + "score": 1.0, + "content": ", the", + "type": "text" + }, + { + "bbox": [ + 329, + 577, + 421, + 589 + ], + "score": 0.9, + "content": "\\begin{array} { r } { l o s s _ { w S D R } ( x , y , g _ { \\theta } ( x ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 575, + 505, + 591 + ], + "score": 1.0, + "content": "has a non-zero gra-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 587, + 338, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 189, + 602 + ], + "score": 1.0, + "content": "dient with respect to", + "type": "text" + }, + { + "bbox": [ + 190, + 589, + 196, + 598 + ], + "score": 0.72, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 587, + 338, + 602 + ], + "score": 1.0, + "content": "even if the target source y is empty.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 575, + 505, + 602 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 324, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 325, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 325, + 624 + ], + "score": 1.0, + "content": "Proof. We can calculate partial derivatives as follows:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 612, + 325, + 624 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 628, + 492, + 686 + ], + "lines": [ + { + "bbox": [ + 118, + 628, + 492, + 686 + ], + "spans": [ + { + "bbox": [ + 118, + 628, + 492, + 686 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { { \\frac { \\partial l o s s _ { w S D R } ( x , y , g _ { \\theta } ( x ) ) } { \\partial \\theta } \\Big | _ { ( x = x _ { 0 } , y = 0 , \\theta = \\theta _ { 0 } ) } = \\frac { \\partial l o s s _ { w S D R } ( x _ { 0 } , 0 , g _ { \\theta } ( x _ { 0 } ) ) } { \\partial \\theta } \\Big | _ { \\theta = \\theta _ { 0 } } } } \\\\ & { = < \\frac { x _ { 0 } } { \\| x _ { 0 } \\| } , \\frac { \\partial } { \\partial \\theta } \\{ \\frac { x _ { 0 } - g _ { \\theta } ( x _ { 0 } ) } { \\| x _ { 0 } - g _ { \\theta } ( x _ { 0 } ) \\| + \\epsilon / \\| x _ { 0 } \\| } \\} \\Big | _ { \\theta = \\theta _ { 0 } } > } \\end{array}", + "type": "interline_equation", + "image_path": "607484b75470ad679cd55203bb56e2c358067e2b7035ef5a7c8acbaf96d89f4c.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 118, + 628, + 492, + 647.3333333333334 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 118, + 647.3333333333334, + 492, + 666.6666666666667 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 118, + 666.6666666666667, + 492, + 686.0000000000001 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 700, + 390, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 391, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 284, + 715 + ], + "score": 1.0, + "content": "Thus, the non-zero gradients with respect to", + "type": "text" + }, + { + "bbox": [ + 284, + 702, + 290, + 711 + ], + "score": 0.84, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 699, + 391, + 715 + ], + "score": 1.0, + "content": "can be back-propagated.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 699, + 391, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 222, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 223, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 223, + 94 + ], + "score": 1.0, + "content": "D IRREDUCIBLE ERRORS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 158 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "score": 1.0, + "content": "In this section, we illustrate two possible irreducible errors. Figure 12 (a) shows the irreducible", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "phase error due to lack of phase estimation. Figure 12 (b) shows the irreducible error induced when", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 137 + ], + "score": 1.0, + "content": "bounding the range of mask. Not bounding the range of the mask makes the problem well-posed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "but it may suffer from the wide range of optimization search space because of the lack of prior", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 266, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 266, + 159 + ], + "score": 1.0, + "content": "knowledge on the distribution of cIRM.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 139, + 172, + 472, + 314 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 139, + 172, + 472, + 314 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 139, + 172, + 472, + 314 + ], + "spans": [ + { + "bbox": [ + 139, + 172, + 472, + 314 + ], + "score": 0.97, + "type": "image", + "image_path": "fa32b64835c251e121e7b604cd359d7881d74ae27955ab25b6ba847ad36fa868.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 139, + 172, + 472, + 219.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 139, + 219.33333333333334, + 472, + 266.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 139, + 266.6666666666667, + 472, + 314.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 326, + 505, + 392 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "Figure 12: Two cases of possible irreducible error types in a TF bin in spectrogram. (a) The case", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "where phase is not estimated and phase of mixture is reused. Even if the magnitude estimation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "is assumed to be perfect, there is still an irreducible phase error between the true source and the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "estimation. (b) The case where the magnitude of an estimated mask is bounded to 1. This again", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "induces an irreducible error since there can be sources which have a higher magnitude than the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 381, + 205, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 205, + 394 + ], + "score": 1.0, + "content": "magnitude of a mixture.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + } + ], + "index": 9.25 + }, + { + "type": "title", + "bbox": [ + 107, + 413, + 236, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 237, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 237, + 426 + ], + "score": 1.0, + "content": "E SCATTER PLOTS OF CIRM", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "The scatter plots of cIRM from training set is shown in Figure 13. We show four different scatter", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "plots according to their SNR values of mixture (0, 5, 10, and 15 (dB)). Each scattered point of cIRM,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 456, + 219, + 469 + ], + "spans": [ + { + "bbox": [ + 107, + 457, + 128, + 469 + ], + "score": 0.89, + "content": "M _ { t , f }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 456, + 219, + 468 + ], + "score": 1.0, + "content": ", is defined as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 466, + 365, + 497 + ], + "lines": [ + { + "bbox": [ + 246, + 466, + 365, + 497 + ], + "spans": [ + { + "bbox": [ + 246, + 466, + 365, + 497 + ], + "score": 0.95, + "content": "M _ { t , f } = \\frac { \\left| Y _ { t , f } \\right| } { \\left| X _ { t , f } \\right| } e ^ { i ( \\theta _ { Y _ { t , f } } - \\theta _ { X _ { t , f } } ) }", + "type": "interline_equation", + "image_path": "13f43a390d8b84ff18ffdb9fa906c9401c7d40aadc4f8fe940c5e7d8e660f56b.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 466, + 365, + 481.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 246, + 481.5, + 365, + 497.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 394, + 519 + ], + "score": 1.0, + "content": "The scattered points near origin indicate the TF bins where the value of", + "type": "text" + }, + { + "bbox": [ + 394, + 505, + 418, + 519 + ], + "score": 0.92, + "content": "\\left| Y _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "is significantly small", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 158, + 532 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + }, + { + "bbox": [ + 159, + 518, + 185, + 532 + ], + "score": 0.92, + "content": "\\left| X _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 518, + 506, + 532 + ], + "score": 1.0, + "content": ". Therefore, those TF bins can be interpreted as the bins dominated with noise", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "rather than source. On the other hand, the scattered points near (1,0) indicates the TF bins where the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 140, + 554 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 140, + 540, + 164, + 554 + ], + "score": 0.92, + "content": "\\left| Y _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 540, + 250, + 554 + ], + "score": 1.0, + "content": "is almost the same as", + "type": "text" + }, + { + "bbox": [ + 250, + 541, + 276, + 554 + ], + "score": 0.9, + "content": "\\left| X _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 540, + 506, + 554 + ], + "score": 1.0, + "content": ". In this case, those TF bins can be interpreted as the bins", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "dominated with source rather than noise. Therefore, as SNR becomes higher, the amount of TF bins", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "dominated with clean source becomes larger compared to the lower SNR cases, and consequently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 573, + 367, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 367, + 586 + ], + "score": 1.0, + "content": "the portion of real part close to 1 becomes larger as in Figure 13.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 599, + 286, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 289, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 289, + 611 + ], + "score": 1.0, + "content": "F VISUALIZATION OF ESTIMATED PHASE", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 505, + 730 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "In this section, we show a supplementary visualization of phase of estimated speech. Although the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "raw phase information itself does not show a distinctive pattern, the hidden structure can be revealed", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 641, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 506, + 654 + ], + "score": 1.0, + "content": "with group delay, which is the negative derivative of the phase along frequency axis (Yegnanarayana", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "& Murthy, 1992). With this technique, the phase information can be explicitly shown as in Figure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 662, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 506, + 676 + ], + "score": 1.0, + "content": "14. Figure 14 (d) shows the group delay of clean speech and the corresponding magnitude is shown", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 675, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 686 + ], + "score": 1.0, + "content": "in Figure 14 (a). The two representations shows that the group delay of phase has a similar structure", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "to that of magnitude spectrogram. The estimated phase by our model is shown in in Figure 14 (c).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "score": 1.0, + "content": "While the group delay of noisy speech (Figure 14 (b)) does not show a distinctive harmonic pattern,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 706, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 506, + 720 + ], + "score": 1.0, + "content": "our estimation show the harmonic pattern similar to the group delay of clean speech, as shown in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 718, + 275, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 718, + 275, + 731 + ], + "score": 1.0, + "content": "the yellow boxes in Figure 14 (c) and (d).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 222, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 223, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 223, + 94 + ], + "score": 1.0, + "content": "D IRREDUCIBLE ERRORS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 158 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "score": 1.0, + "content": "In this section, we illustrate two possible irreducible errors. Figure 12 (a) shows the irreducible", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "phase error due to lack of phase estimation. Figure 12 (b) shows the irreducible error induced when", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 137 + ], + "score": 1.0, + "content": "bounding the range of mask. Not bounding the range of the mask makes the problem well-posed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "but it may suffer from the wide range of optimization search space because of the lack of prior", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 266, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 266, + 159 + ], + "score": 1.0, + "content": "knowledge on the distribution of cIRM.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 103, + 505, + 159 + ] + }, + { + "type": "image", + "bbox": [ + 139, + 172, + 472, + 314 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 139, + 172, + 472, + 314 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 139, + 172, + 472, + 314 + ], + "spans": [ + { + "bbox": [ + 139, + 172, + 472, + 314 + ], + "score": 0.97, + "type": "image", + "image_path": "fa32b64835c251e121e7b604cd359d7881d74ae27955ab25b6ba847ad36fa868.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 139, + 172, + 472, + 219.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 139, + 219.33333333333334, + 472, + 266.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 139, + 266.6666666666667, + 472, + 314.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 326, + 505, + 392 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "Figure 12: Two cases of possible irreducible error types in a TF bin in spectrogram. (a) The case", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "where phase is not estimated and phase of mixture is reused. Even if the magnitude estimation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "is assumed to be perfect, there is still an irreducible phase error between the true source and the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "estimation. (b) The case where the magnitude of an estimated mask is bounded to 1. This again", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "induces an irreducible error since there can be sources which have a higher magnitude than the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 381, + 205, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 205, + 394 + ], + "score": 1.0, + "content": "magnitude of a mixture.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + } + ], + "index": 9.25 + }, + { + "type": "title", + "bbox": [ + 107, + 413, + 236, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 237, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 237, + 426 + ], + "score": 1.0, + "content": "E SCATTER PLOTS OF CIRM", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "The scatter plots of cIRM from training set is shown in Figure 13. We show four different scatter", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "plots according to their SNR values of mixture (0, 5, 10, and 15 (dB)). Each scattered point of cIRM,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 456, + 219, + 469 + ], + "spans": [ + { + "bbox": [ + 107, + 457, + 128, + 469 + ], + "score": 0.89, + "content": "M _ { t , f }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 456, + 219, + 468 + ], + "score": 1.0, + "content": ", is defined as follows:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 434, + 505, + 469 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 466, + 365, + 497 + ], + "lines": [ + { + "bbox": [ + 246, + 466, + 365, + 497 + ], + "spans": [ + { + "bbox": [ + 246, + 466, + 365, + 497 + ], + "score": 0.95, + "content": "M _ { t , f } = \\frac { \\left| Y _ { t , f } \\right| } { \\left| X _ { t , f } \\right| } e ^ { i ( \\theta _ { Y _ { t , f } } - \\theta _ { X _ { t , f } } ) }", + "type": "interline_equation", + "image_path": "13f43a390d8b84ff18ffdb9fa906c9401c7d40aadc4f8fe940c5e7d8e660f56b.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 246, + 466, + 365, + 481.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 246, + 481.5, + 365, + 497.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 394, + 519 + ], + "score": 1.0, + "content": "The scattered points near origin indicate the TF bins where the value of", + "type": "text" + }, + { + "bbox": [ + 394, + 505, + 418, + 519 + ], + "score": 0.92, + "content": "\\left| Y _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "is significantly small", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 158, + 532 + ], + "score": 1.0, + "content": "compared to", + "type": "text" + }, + { + "bbox": [ + 159, + 518, + 185, + 532 + ], + "score": 0.92, + "content": "\\left| X _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 518, + 506, + 532 + ], + "score": 1.0, + "content": ". Therefore, those TF bins can be interpreted as the bins dominated with noise", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "rather than source. On the other hand, the scattered points near (1,0) indicates the TF bins where the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 140, + 554 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 140, + 540, + 164, + 554 + ], + "score": 0.92, + "content": "\\left| Y _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 540, + 250, + 554 + ], + "score": 1.0, + "content": "is almost the same as", + "type": "text" + }, + { + "bbox": [ + 250, + 541, + 276, + 554 + ], + "score": 0.9, + "content": "\\left| X _ { t , f } \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 540, + 506, + 554 + ], + "score": 1.0, + "content": ". In this case, those TF bins can be interpreted as the bins", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "dominated with source rather than noise. Therefore, as SNR becomes higher, the amount of TF bins", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "dominated with clean source becomes larger compared to the lower SNR cases, and consequently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 573, + 367, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 367, + 586 + ], + "score": 1.0, + "content": "the portion of real part close to 1 becomes larger as in Figure 13.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 505, + 506, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 599, + 286, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 289, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 289, + 611 + ], + "score": 1.0, + "content": "F VISUALIZATION OF ESTIMATED PHASE", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 505, + 730 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "In this section, we show a supplementary visualization of phase of estimated speech. Although the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "raw phase information itself does not show a distinctive pattern, the hidden structure can be revealed", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 641, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 506, + 654 + ], + "score": 1.0, + "content": "with group delay, which is the negative derivative of the phase along frequency axis (Yegnanarayana", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 665 + ], + "score": 1.0, + "content": "& Murthy, 1992). With this technique, the phase information can be explicitly shown as in Figure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 662, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 506, + 676 + ], + "score": 1.0, + "content": "14. Figure 14 (d) shows the group delay of clean speech and the corresponding magnitude is shown", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 675, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 686 + ], + "score": 1.0, + "content": "in Figure 14 (a). The two representations shows that the group delay of phase has a similar structure", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "to that of magnitude spectrogram. The estimated phase by our model is shown in in Figure 14 (c).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "score": 1.0, + "content": "While the group delay of noisy speech (Figure 14 (b)) does not show a distinctive harmonic pattern,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 706, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 506, + 720 + ], + "score": 1.0, + "content": "our estimation show the harmonic pattern similar to the group delay of clean speech, as shown in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 718, + 275, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 718, + 275, + 731 + ], + "score": 1.0, + "content": "the yellow boxes in Figure 14 (c) and (d).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 619, + 506, + 731 + ] + } + ] + }, + { + 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The group delay of the estimated phase from our model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "(c) shows the similar pattern to that of clean speech (d). 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This can also be verified in Table 5", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "columns PhaseDist(C, N) and Phase Improvement where the values of both columns increase as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 195, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 195, + 149 + ], + "score": 1.0, + "content": "SNR becomes higher.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 81, + 506, + 149 + ] + }, + { + "type": "image", + "bbox": [ + 141, + 172, + 471, + 315 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 141, + 172, + 471, + 315 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 172, + 471, + 315 + ], + "spans": [ + { + "bbox": [ + 141, + 172, + 471, + 315 + ], + "score": 0.97, + "type": "image", + "image_path": "12b9d9801754fe5cc7bd817b128fda482a6f9e543085ebc1dfc0d1cc6c2c528f.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 141, + 172, + 471, + 219.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 141, + 219.66666666666666, + 471, + 267.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 141, + 267.3333333333333, + 471, + 315.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 322, + 505, + 389 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "Figure 15: (a) The case where SNR of given mixture is high. 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For fair comparison, both conventional method", + "type": "text" + }, + { + "bbox": [ + 452, + 445, + 486, + 456 + ], + "score": 0.71, + "content": "( \\mathbf { R M R } \\mathbf { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "proposed method (cRMCn) were set to have the same number of parameters. Also, both models", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "were trained with weighted-SDR loss. 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2.56.714°0.060.440.140.215.32
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HardMedium EasyTotal
DCUnet > Deep Feature Loss0.90 (***）0.82 (***)0.69 (***)0.82 (***)
DCUnet > Wavenet0.98 (***)0.95 (***)0.75 (***)0.93 (***)
DCUnet > SEGAN0.99 (***)0.93 (***)0.71 (***)0.92 (***)
Deep Feature Loss > Wavenet0.67 (***)0.83 (***)0.48 (n.s.)0.74 (***)
Deep Feature Loss > SEGAN0.72 (***)0.81 (***)0.40 (**)0.73 (***)
SEGAN > Wavenet0.53 (n.s.)0.56(**)
0.58 (**)0.50 (n.s.)
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HardMedium EasyTotal
DCUnet > Deep Feature Loss0.90 (***）0.82 (***)0.69 (***)0.82 (***)
DCUnet > Wavenet0.98 (***)0.95 (***)0.75 (***)0.93 (***)
DCUnet > SEGAN0.99 (***)0.93 (***)0.71 (***)0.92 (***)
Deep Feature Loss > Wavenet0.67 (***)0.83 (***)0.48 (n.s.)0.74 (***)
Deep Feature Loss > SEGAN0.72 (***)0.81 (***)0.40 (**)0.73 (***)
SEGAN > Wavenet0.53 (n.s.)0.56(**)
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CSIGCBAKCOVLPESQSSNR
DCU-10|Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR||Spc Wav wSDR|SpcWav wSDR
UBD|3.51 3.713.72 |3.50 3.523.56|3.13 3.153.19|2.78 2.61 2.67[11.86 13.1813.37
BDSS2.94 3.193.21 3.43 3.353.392.81 2.852.892.79 2.61 2.6711.61 12.30 12.30
BDT3.30 3.743.723.52 3.57 3.603.02 3.193.222.79 2.66 2.7212.30 13.5413.60
DCU-16| |SpcWav wSDR||Spc Wav wSDR||SpcWav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD3.97 4.004.07 13.66 3.723.753.46 3.433.482.96 2.85 2.9312.75 14.2514.44
BDSS3.63 3.573.72 3.59 3.513.563.28 3.153.272.98 2.82 2.8712.05 12.8912.89
BDT3.97 4.034.10 3.70 3.743.773.49 3.453.523.01 2.88 2.9312.96 14.3914.44
DCU-20| |Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR||Spc Wav wSDR|Spc WavwSDR
UBD|4.02 4.154.25 3.88 3.934.01|3.57 3.593.70[3.11 3.01 3.1314.81 16.0316.36
BDSS3.77 3.743.93 3.68 3.653.723.40 3.333.473.06 2.98 3.0512.80 13.6613.68
BDT4.02 4.184.24 3.87 3.954.003.58 3.643.693.13 3.08 3.1314.59 15.85 15.95
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CSIGCBAKCOVLPESQSSNR
Wiener (Scalart et al., 1996)3.232.682.672.225.07
SEGAN (Pascual et al., 2017)3.482.942.802.167.73
Wavenet (Rethage et al., 2018)3.623.232.9811
MMSE-GAN (Soni et al., 2018)3.803.123.142.53-
Deep Feature Loss (Germain et al., 2018)3.863.333.2211
DCUnet-20 (ours)4.244.003.693.1315.95
Large-DCUnet-20 (ours)4.344.103.813.2416.85
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DCUnet > Wavenet0.980.950.750.93
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SNR (dB)Phase ImprovementPerformance Difference
PESQCSIGCBAKCOVLSSNR
2.56.714°0.060.440.140.215.32
7.54.408°0.030.420.110.195.21
12.52.982°0.050.390.110.184.96
17.51.702°0.050.300.090.133.93
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sha256:c66868f4f5c60a51cd01553913ef471f850699370890f54395b3affafc61620f +size 1479151 diff --git a/parse/train/TmkN9JmDJx1/TmkN9JmDJx1.md b/parse/train/TmkN9JmDJx1/TmkN9JmDJx1.md new file mode 100644 index 0000000000000000000000000000000000000000..fd6ddc0899b3b1f57556ea14eb5128a0e6124bcf --- /dev/null +++ b/parse/train/TmkN9JmDJx1/TmkN9JmDJx1.md @@ -0,0 +1,502 @@ +# THINKING LIKE TRANSFORMERS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +What is the computational model behind a transformer? Where recurrent neural networks have direct parallels in finite state machines, allowing clear discussion and thought around architecture variants or trained models, transformers have no such familiar parallel. In this paper we aim to change that, proposing a computational model for the transformer-encoder in the form of a programming language. We map the basic components of a transformer-encoder – attention and feed-forward computation – into the simple primitives of select, aggregate and zipmap, around which we form a programming language: the Restricted Access Sequence Processing Language (RASP). We show how RASP can be used to program solutions to tasks that could conceivably be learned by a transformer, augmenting it with tools we discover in our work. In particular, we provide RASP programs for histograms, sorting, and even logical inference similar to that of Clark et al. (2020). We further use our model to relate their difficulty in terms of the number of required layers and attention heads. Finally, we see how insights gained from our abstraction might be used to explain phenomena seen in recent works. + +# 1 INTRODUCTION + +While Yun et al. (2019) show that sufficiently large transformers can approximate any constantlength sequence-to-sequence function, and Hahn (2019) provides theoretical limitations on their ability to compute functions on unbounded input length, neither of these provide insight on how a transformer may achieve a specific task. Orthogonally, Bhattamishra et al. (2020) provide transformer constructions for several counting languages, but this also does not direct us towards a general model. + +This is in stark contrast to other neural network architectures, which do have clear computational models. For example, convolution networks are seen as as a sequence of filters (Zhang et al., 2018), and finite-state automata and their variants have been extensively used both for extraction from and theoretical analysis of recurrent neural networks (RNNs) (Omlin & Giles, 1996; Weiss et al., 2018; Rabusseau et al., 2018; Merrill et al., 2020), even inspiring new RNN variants (Joulin & Mikolov, 2015). + +In this work we propose a computational model for the transformer-encoder, in the form of a simple sequence-processing language which we dub RASP(Restricted Access Sequence Processing Language). Much like how automata describe the token-by-token processing behavior of an RNN, our language captures the unique information flow constraints under which a transformer (Vaswani et al., 2017) operates as it processes input sequences. + +Considering computation problems and their implementation in the RASP language allows us to “think like a transformer” while abstracting away the technical details of a neural network in favor of symbolic programs. A RASP program operates on sequences of values from uniform atomic types, and transforms them by composing a restricted set of sequence processors. One pair of processors is used to select inputs for aggregation, and then aggregate the selected items. Another processor performs arbitrary but local computation over its (localized) input. However, access to the complete sequence is available only through aggregate operations that reduce a stream of numbers to a scalar. The key to performing complex global computations under this model is to compose the aggregations such that they gather the correct information, that can then be locally processed for a final output. + +Given a RASP program, we can analyze it to infer the minimal number of layers and maximum number of heads that is required to implement it as a transformer. We show several examples of expressive programs written in the RASP language, showing how complex operations can be implemented by a transformer. Thinking in terms of the RASP model also allows us to shed light on recent empirical observation of transformer variants (Press et al., 2020) and find concrete limitations of “efficient transformers” with restricted-attention (Tay et al., 2020). + +![](images/6b01396729fafa621584abcc339227175ba7f60fe33baa5593f15be6e16a6598.jpg) +Figure 1: RASP program taking two sequences vals,keys and returning a sequence y sorting the elements of vals according to keys, e.g.: if vals $( \boldsymbol { x } ) \boldsymbol { = } [ \mathsf { a } , \mathsf { b } , \mathsf { c } ]$ and $\mathtt { k e y s ( x ) = [ } 0 , 4 , 2 \mathtt { l }$ , then $y ( x ) = [ a , \mathsf { c } , \mathsf { b } ]$ . + +# 2 THE RESTRICTED ACCESS SEQUENCE PROCESSING LANGUAGE + +In this section, we present the the Restricted Access Sequence Processing Language (RASP). RASP assumes a machine composed of several Turing-complete processors, each of which can only run functions taking and returning a fixed number of primitive arguments, and a simple memory accessor that is controlled by these processors. The select, aggregate, and zipmap operations which we present will define and constrain how the processors work together to process an input sequence. + +We will focus here only on the language itself, leaving the discussion of its exact relation to transformers to Section 3. + +Overview A RASP program works by manipulating sequences, occasionally with the help of selectors. Sequences contain values of uniform atomic type, such as booleans, integers, floats, or strings. They are functions used for selecting elements from sequences, and are used (together with the appropriate operations) only in the process of creating new sequences. All sequences in RASP are lazily evaluated, meaning that their length and contents are not populated until passed an input. + +The Base Sequences Every program in RASP begins from the same set of base sequences, and then creates new ones using a small number of core operations. These base sequences are indices, length, and tokens, evaluated on input $x _ { 1 } , x _ { 2 } , . . . , x _ { n }$ as their names suggest: $( 0 , 1 , . . . , n - 1 )$ , $( n , n , . . . , n )$ (of length $n$ ), and $( x _ { 1 } , x _ { 2 } , . . . , x _ { n } )$ , respectively. + +Combining Sequences Sequences can be combined in an ‘elementwise’ manner, such that the value of the resulting sequence at each position $i$ is a function of the values in the combined sequences at position $i$ (similar to a map operation), or have positions ‘mixed’ in more complicated ways using selectors, which are functions $f : \mathbb { N } \times \mathbb { N } \to \{ \mathrm { T r u e } , \mathrm { F a l s e } \}$ whose sole purpose is to guide the combination of existing sequences into new ones. + +We present the basic ingredients of RASP using an example. Figure 1 shows a simple RASP function for sorting a sequence of values according to a sequence of keys. It accepts an input sequence vals and uses the base sequence indices, that is available to any RASP program, to compute its output in three operations as follows: + +1. count_conditioned of Line 2 creates a new sequence that counts for each element of keys the number of “previous items” it has in keys, where the “previous items” are defined to be all items that have a lesser value, or equal value and lower index. Thus, num_prevs creates a sequence of numbers, representing the target sorted position of each item. 2. select of line 7 creates a new selector which will focus each position $i$ on the corresponding position $j$ for which indices[i] is equal to num_prevs ${ \cdot } j ]$ . Effectively, it will direct the elements in each position $j$ towards their target location $i$ . + +3. Finally, aggregate of line 8 applies select_sorted_val to vals, moving each $i$ -th element of vals to its calculated sorted position num_prevs[i]. + +We now describe the base operations of RASP in-depth, occasionally presenting an example on the hypothetical input sequence $x$ of length $n$ . + +• zipmap The zipmap operation takes a tuple of sequences and an element-processing function f, and applies f per-index to the values in those sequences to create a new sequence. For a simple example, $y 1 = z :$ ipmap((indices,indices), lambda ${ \mathrm { i } } , { \mathrm { j } } : { \mathrm { i } } + { \mathrm { j } } )$ creates a sequence that always evaluates to $( 0 , 2 , . . . , 2 n - 2 )$ . +• aggregate The aggregate operation takes a selector s, a sequence $\mathsf { x }$ , and an optional parameter default, and averages subsets of the values of $\mathsf { x }$ into a new sequence y as follows: for every index $i , y [ i ]$ is the average of $\times [ j _ { 0 } ] , \times [ j _ { 1 } ] , \ldots \times [ j _ { k } ]$ , where $j _ { 0 } , j _ { 1 } , . . . , j _ { k }$ are the indices $j \in [ n ]$ for which $\mathsf { s } ( i , j )$ is True. We say $k$ is the focus width of $s$ at $i$ . If $k = 0$ , then $\mathsf { y } [ j _ { 0 } ]$ is assigned the value in default. For example: if $\mathsf { \Omega } _ { \mathsf { S } } ( i , j )$ returns True iff $i$ is odd and $j { = } 0$ , and the value in default is d, then y will evaluate to $( \mathsf { d } , \mathsf { x } [ 0 ] , \mathsf { d } , \mathsf { x } [ 0 ] , \hdots , \mathsf { y } [ n - 1 ] )$ where $\mathsf { y } [ n - 1 ]$ is either d or $\times [ 0 ]$ depending on the parity of $n$ . +• select The select operation takes two sequences-tuples of lengths $k$ and $l$ , $\mathfrak { m e } { = } ( \mathfrak { m } _ { 1 } , \mathfrak { m } _ { 2 } , \dots { \cdot } , \mathfrak { m } _ { k } )$ and $\mathsf { o t h e r } = ( \mathsf { o t } _ { 1 } , \mathsf { o t } _ { 2 } , \ldots , \mathsf { o t } _ { l } )$ , and a function $\mathsf { f }$ expecting $k + 1$ atomic values and giving boolean output. It composes these to create a selector s as follows: for every two indices $i , j , s ( i , j )$ is the output of $\mathsf { f }$ on the $i$ -th and $j$ -th slice of me and other respectively, i.e., $s ( i , j ) \mathrm { = } \mathsf { f } ( \mathsf { m } _ { 1 } [ i ] , . . . , \mathsf { m } _ { k } [ i ] , \mathsf { o t } _ { 1 } [ j ] . . . \mathsf { o t } _ { l } [ j ] )$ . For a simple example, in $\mathtt { s } =$ select((indices,),(indices,),lambda mi,oti: $m i \% 2 = = 1$ and $\cot \mathtt { i } = = 0 \dot { }$ ), then $\mathfrak { m } _ { 1 } = \mathrm { i }$ ndices, $\mathsf { o t } _ { 1 } \mathsf { = i }$ ndices, and s is the same selector we used for our example in aggregate above. +count_conditioned This operation takes the same parameters me,other and $\mathsf { f }$ as select, but this time returns a sequence y describing the number of selected influencing positions $j$ for each output position $i$ that $\mathtt { S } \mathtt { = }$ select(me,other,f) would have created. In other words, for each $i$ , $y [ i ] = k$ where $j _ { 1 } , . . . , j _ { k }$ is the set of positions $j$ for which $s ( i , j ) =$ True. For example, $h = 0$ count_conditioned((tokens,),(tokens,),lambda a,b: $\mathsf { a } \mathsf { = } \mathsf { = } \mathsf { b } .$ ) returns an in-place histogram for the tokens in the input sequence: h(“abaa”) $= ( 3 , 1 , 3 , 3 )$ . + +This concludes the base operations of RASP – all other operations are shortcuts for combinations of the above 4, occasionally with the base sequences. + +Sugar We implement RASP with a variety of syntactic sugar, presented fully in appendix E. Briefly: + +1. When applying zipmap to a single sequence, it may be passed directly without using a tuple, e.g.: zipmap(indices,f) is equivalent to zipmap((indices,),f). +2. zipmap has sugar for most of the binary operators, e.g.: for two sequences $\mathsf { x } , \mathsf { y } .$ , then $x { + } y$ is sugar for zipmap( $( \mathsf { x } , \mathsf { y } )$ ,lambda $1 , b : a + b )$ . +3. Whenever the focus width of s at some index is $\leq ~ 1$ (“up-to-one selection”), aggregate(s,x,default=d) does not explicitly compute the division. In this case the values of $\mathsf { x }$ do not have to be numbers. +4. aggregate accepts one additional optional parameter elementwise_function. The full order of parameters is $s , x$ ,elementwise_function,default, and the use of elementwise_function is as follows: aggregate $( \mathsf { s } , \mathsf { x } , \mathsf { f } , \mathsf { d } )$ is equivalent to aggregate(s,zipmap $( \times , \mathsf { f } )$ ,defaul $\therefore d$ ). + +# 2.1 EXAMPLES + +We now present some more example RASP programs, by increasing order of complexity. + +Simple Examples The first and simplest example is to compute an in-place histogram for some sequence vals. This is achieved with a single application of count_conditioned: histogram=count_conditioned(vals,vals,lambda a, $b : a = = b :$ ). + +1 def by_frequency (vals , default ) : +2 hist $=$ count_conditioned (vals ,vals , lambda a,b: $a = = b$ ) +3 num_earlier $=$ count_conditioned ( +4 ( indices , vals ) ,( indices , vals ) , +5 lambda iq ,vq ,ik ,vk :( $\mathsf { v q } = = \mathsf { \cdot }$ vk) and ( $\mathbf { i k } < \mathbf { i q } )$ ) ) +6 has_earlier $=$ num_earlier > 0 +7 unique_vals $=$ zipmap (( vals , has_earlier ) , +8 lambda t,h_e: t if not h_e else default ) +9 masked_hist $=$ hist - ( length $\star$ has_earlier ) +10 return sort ( unique_vals , key $=$ - masked_hist ) + +Figure 2: RASP program sorting the unique elements of a sequence vals by decreasing frequency. Lines 6 and 7 show syntactic sugar for multiple simple zipmap operations, e.g., line 6 can be written has_earlier $=$ zipmap(num_earlier,lambda $\mathsf { n } : \mathsf { n } > 0$ ). This program requires a default atom da of the same type as vals, to put in place of all otherwise-duplicated values in its result. For example, by_frequency(tokens_str,“-”)(“abacca”) will return “acb--”. + +From Length to Parity While length is provided as a primitive in the language, it can actually be achieved as a composition of the other base operations and sequences. This is done by computing full_ $\hat { \mathsf { s } } =$ select((),(),lambda :True) followed by 1/aggregate(full_s,indices,lambda i:int $\overset { \cdot } { 1 } = = \varnothing )$ ) (the fraction of elements equal to 0 in indices, inverted). From length and that same full_s we can then define count(vals,v), a function taking any sequence vals and value v and returning a new sequence counting the number of appearances of v in vals. The implementation of count is simply length\*aggregate(full_s,vals,lambda $\mathsf { e : e : } = \mathsf { = v }$ ). count in turn enables us to write programs like parity simply as count(tokens,1) $\% 2 = = 0 ^ { 1 }$ . + +Reverse We can reverse a sequence seq with the help of an up-to-one selector mapping each position to its opposite: flip_s $=$ select(indices, length-1-indices, lambda m,oth: $\scriptstyle { \mathfrak { m } } = = 0$ th). We use flip_s to re-order the tokens of seq: reverse $=$ aggregate(flip_s,seq). + +Balanced Parentheses For balanced parentheses we use count_conditioned twice, storing in prev_opens and prev_closes the number of previous “(” or “)” (respectively) tokens each position has, including itself. The sequence is balanced if prev_opens-prev_closes has no negative values, and is 0 at position length-1. These two qualities can be easily computed using two final select-aggregate pairs, and then combined with a zipmap. + +Most Frequent Tokens In fig. 2 we show how RASP can be used to arrange for any input sequence s the most frequent tokens in s, without repetition. The solution has 2 parts: first, we compute the histogram for all the tokens, and mask it such that all but the first of each token is given a negative value. Then, we sort the tokens according to the masked histogram. The solution uses the sort function from Figure 1. + +1 def count_conditioned (me ,other ,f) : +2 other $=$ other $^ +$ ( indices ,) # tuple concatenation , indices at end +3 s_with $\_ 0 =$ select (me ,other , lambda \*a:f ( $\star$ ( a[: -1]) ) or $( \mathsf { a } [ - 1 ] \mathsf { = } = \mathsf { 0 } )$ ) +4 s_just_0 $=$ select (me ,other , lambda \*a:f ( $\star$ ( a[: -1]) ) and $( \mathsf { a } \left[ - 1 \right] \mathsf { = } \mathsf { = } \mathsf { 0 } )$ ) +5 frac $=$ aggregate ( s_with_0 , indices , lambda i: int ( $\scriptstyle { \dot { 1 } } = = 0$ ) ) +6 count_outside_0 $=$ (1/ frac ) -1 +7 count_in_ $\begin{array} { r l } { \mathbf { \nabla } _ { \mathbf { \nabla } } \theta } & { { } = } \end{array}$ aggregate ( s_with_0 ,() ,lambda :1 , defaul $\mathtt { \iota } = \mathtt { 0 }$ ) +8 return count_outside_0 $^ +$ count_in_0 + +Count Conditioned The operation count_conditioned is a powerful part of RASP, appearing in many other programs. Surprisingly, it is realisable as a composition of the other operations (and base sequence indices). Understanding its implementation is interesting for learning how to “truly” think like a transformer, and we present the code in Figure 3. The intuition is as follows: we compute the select whose width we want to calculate twice, once such that it also selects the position 0, and once such that it only selects this position. We then aggregate both these values, broadcasting 1 from position 0 and 0 from everywhere else, and using default value 0. The first aggregate computes for each position the inverse of the number of selected positions (excluding 0) plus one, and the second computes whether that position would also focus on 0. A straightforward zipmaps then gives us the result. To further help the intuition, we present in fig. 4 the computation flow for a histogram calculation, when count_conditioned is implemented as described here. + +![](images/e3c8af607bbb6d9e300e42e1bf75feb288a546c2db7be6501d8640714ee45873.jpg) +Figure 4: The computation flow for the sequence histogram(tokens_str) applied to input sequence “hello”, when implemented in terms of select, aggregate, and zipmap. + +Note Many of the functions provided with RASP can be expressed in terms of count_conditioned – such as count (which counts how many elements in a sequence are equal to a given value) and contains (which checks if count is greater than 0) – but this is not necessarily an optimal implementation with respect to number of heads it uses. The RASPlibrary provides optimal implementations. + +# 2.2 IMPLEMENTING SYMBOLIC REASONING A LA ROVER + +How might a transformer implement reasoning, as in Clark et al. (2020)? RASP empowers us to clearly think about such problems, and we sketch a solution direction here. We begin by reformulating the task of Clark et al. to a form that focuses on the core problem, moving away from natural language into more ‘concrete’ domain, and by limiting the type of relations we will consider. + +In the work of Clark et al., a transformer is presented a sequence of statements and a query Q, and must recognise whether Q is implied by the previous relations or not. For example: $\mathsf { a } 1 \in \mathsf { A } 1$ , ${ \mathsf { b } } \in { \mathsf { A } } 1 \Longrightarrow { \mathsf { b } } \in { \mathsf { A } } 2 .$ , a $1 \in { \mathsf { A } } 2 ?$ evaluates to True, whereas a1 $\not \in \mathsf { A } \mathsf { 1 }$ , a1 $\not \in { \mathsf { A } } 2 ?$ evaluates to False. Different inputs for this task can have different depth: the number of ‘logical hops’ needed to correctly identify whether the query statement is true. + +Note. The original work accepts this input in natural language, e.g., “Alan is young. [...] If someone is young then...”, and allows statements with more complicated logical form, such as ${ \mathsf { a } } 1 \in { \mathsf { A } } 1 \wedge { \mathsf { a } } 1 \in { \mathsf { A } } 2 \implies { \mathsf { a } } 1 \in { \mathsf { B } }$ . In this section we consider only a simplified and symbolic version, in which the statements are limited to the form of the previous paragraph. We assume the statements are separated by a special token |. + +We sketch this task in RASP as follows: first, mark each statement ‘block’ in the sequence by its index (as a block), by counting for each token the number of separators before it in the input. Set aside the final block (the query), it is the set of tokens with no separators after them. For each remaining block mark whether it is a relation $\in$ or $\notin )$ or inference $( \mapsto )$ statement. Then, for each relation block, note at the position of the set token the element and whether it is inside or outside, and similarly over the element token note the set. Next, for as many repetitions as the logical depth that the program should cover: share set information between all elements and element information between all sets (including from inference blocks, which initially have none initially empty), and then apply one logical step ‘locally’ at each inference block. Finally, for the query, seek any occurrence of the set token in the sequence, and return whether the element token is listed there in its contents. + +Generalisation on Inference Depth A similar solution to the one we have proposed would be to make all of these logical inferences backwards from the query, i.e. by propagating backwards the requirements that would be sufficient to answer the query. If a trained transformer implements both of these solutions in parallel (for instance, to increase its robustness), this may explain the generalisation to greater query depth observed by Clark et al.. + +# 3 RELATION TO TRANSFORMERS, AND ANALYSIS + +We discuss how RASP relates to the real transformer architecture, and how it may also be used to compare transformer variants, or analyse the ‘difficulty’ of a task for transformers. + +Connection of RASP to Transformers The select and aggregate operations of RASP correspond to the attention-scoring and then pooling of transformer self-attention heads, the zipmap operations correspond to the feed-forward sublayers, and the computed sequences correspond to head or feedforward inputs and outputs. indices and tokens represent the initial input, while length and conditioned_contains are in fact combinations of the other primitives. The persistence of sequences – such that they may be accessed multiple times over the course of a RASP program – is encouraged by the existence of skip connections in the transformer architecture. In appendix A we consolidate these connections, giving a full description of transformers, and showing how any given transformer can be represented in the RASP language (provided a slight generalization of select2). + +# .1 PREDICTING TRANSFORMER COMPLEXITY WITH RASP + +The purpose of RASP is to help us reason about the computation process enabled by a transformer. We find that RASP programs lend themselves easily to ‘width’ and ‘depth’ analysis, enabling us to predict the number of heads and layers that a transformer will need to implement the same solution. We discuss this analysis now, and evaluate the predictions it provides in Appendix B. + +For any given RASP program, we can compute the minimal number of layers required to implement it in a transformer, and upper bound the number of heads this implementation requires.3 This is provided its internal dimensions are wide enough to replicate the given processing functions. This analysis can give us intuition regarding the relative difficulty of different tasks for the transformer architecture, where each algorithm we find for a task gives us an upper bound on the number of layers and heads a transformer needs to solve it. + +We implement such an analysis and provide a draw_comp_flow function, which automatically positions each attention head using a greedy scheduler, and displays the computation flow accordingly (see fig. 4, and others in the supplementary material). + +A similar analysis can be done for different "primitive" computations in isolation, giving us intuition on the ‘cost’ of various common computations: how many additional heads and layers each computation adds when applied to a previously computed sequence. For example, our implementation of sort takes 2 layers, and so whenever we apply it to a computed sequence we know it will increase our program’s depth by 2 from that sequence. + +Algorithm for computing program depth and width As noted, the first 3 base operations of RASP – select, aggregate, and zipmap – have direct parallels in the transformer architecture, and so we may easily analyse the result of any RASP program to see how many layers and heads it would take to realise in an actual transformer. (For length and count_conditioned, we analyse them in terms of their deconstruction to the other operations and sequences.) + +The first part of the analysis is simple: every sequence and selector is initiated with a “minimum depth” $d$ , reflecting the earliest layer at which it can be computed, and so the minimum number of layers needed to create any given sequence is $d , d$ is computed for each new sequence as follows: + +1. The base sequences indices and tokens have $d = 0$ , as they are input to the transformer rather than part of its computation. +2. Any sequence created from a zipmap is given $d$ equal to the maximum $d$ of the inputs to that zipmap, as it can be created immediately after the last of them in the same feed forward computation that concludes it.4 +3. Every selector gets $d$ equal to the maximum $d$ of its creating sequences X plus 1, to reflect the fact that all of them must be calculated before it can even begin (as multi-headed attention happens only once, at the beginning of each layer). +4. Any selector created from an aggregate has $d$ equal to at least that of its creating selector, and at least one plus those of the input sequences to the aggregate operation (as they must be passed through the attention to create the new sequences). + +RASP makes it easy to access all sequence and selectors on which an unfinished value is dependent, and allowing us to analyse not only the depth but also the width of a given program. The width of the computation is a reflection of how many (unique) selectors are being used at every layer of the transformer: the number of attention heads needed to mimick that layer in a transformer. We say that a selector $s$ is being used at some layer $l$ if a sequence that is aggregated from $s$ is calculated at $l$ This is because it is possible a selector $s$ may have minimum depth $d$ , but is only or also used at later layers: for instance if the sequences needed for the aggregation operation using $s$ are not yet ready. + +Similarly, a sequence does not need to be computed at its minimum possible depth, as it is possible it will only be needed much later. Hence there is no one analysis for a given program, and there is room for creating a scheduling algorithm that minimises the maximum width of the transformer, i.e., the maximum of the widths of all layers (useful, as transformers tend to be created with uniform width). + +# 4 IMPLICATIONS FOR TRANSFORMER VARIANTS + +# 4.1 RESTRICTED-ATTENTION TRANSFORMERS + +Multiple works propose restricting the attention mechanism of transformers in order to create more efficient transformers, reducing the time complexity of each layer from ${ \mathrm { O } } ( n ^ { 2 } )$ to $\mathrm { { O } } ( n l o g ( n ) )$ or even ${ \bf O } ( n )$ with respect to the input sequence length $n$ (see Tay et al. (2020) for a survey of such approaches and their complexity). Several of these do so using sparse attention, in which the attention is masked using different patterns to reduce the number of locations that can interact (see for instance (Child et al., 2019; Beltagy et al., 2020; Ainslie et al., 2020; Zaheer et al., 2020; Roy et al., 2020)). + +Considering these variants of transformers in terms of the RASP language, allows us to reason about the computations they can and cannot perform. In terms of RASP, these variants of transformers all impose restrictions on the selectors, forcing some of the $n ^ { 2 }$ index pairs $( i , j )$ to False. + +Figure 1 showed how to implement sorting of an input sequence with arbitrary alphabet size, comparison function, and length5. We now prove that RASP variants where the selector is restricted to ${ \bf O } ( n )$ pairs (i.e., transformer variants with sufficiently restricted attention), cannot sort. + +The computation model of RASP and indeed of all transformer variants allows comparison of values between more than one sequence position only during the select operation, i.e., only while computing the attention distribution Hence, all comparisons necessary for sorting must be applied in select. It follows that whenever select is restricted such that it compares at most ${ \bf O } ( n )$ index pairs per head, no constant number of heads and layers will be sufficient for the model to perform sorting on arbitrary length – as sorting is known to require at least $\mathrm { { O } } ( n l o g ( n ) )$ comparisons. + +Thus, variants of transformers in which the attention is masked to impose ${ \bf O } ( n )$ complexity require at least $\mathbf { O } ( l o g ( n ) )$ layers to sort. It also follows that they require $\mathbf { O } ( l o g ( n ) )$ layers to implement count_conditioned, as we see in fig. 1 that count_conditioned can be applied to create a sequence (num_prevs) which is sufficient to complete a sorting operation with only ${ \bf O } ( n )$ further operations. + +# 4.2 SANDWICH TRANSFORMERS + +Recently, Press et al. (2020) showed that reordering the attention and feed-forward sublayers of a transformer affects its ability to train on language modeling tasks. In particular, they showed that 1. pushing feed-forward sublayers towards the bottom of a transformer weakened it, and 2. pushing attention sublayers to the bottom and feed-forward sublayers to the top strengthened it, provided there was still some interleaving in the middle (making a sandwich transformer). + +Considering the base operations of RASP helps us understand the observations of Press et al.. In RASP, the feed-forward and attention sublayers are the zipmap and select-aggregate (or gather for short) operations. Any arrangement of the sublayers into a set architecture, from the ‘vanilla’ transformer to the variations considered in (Press et al., 2020), imposes a restriction on the number and order of RASP operations that can be chained in a RASP program. For example, an architecture in which all feed-forward sublayers appear before the attention sublayers imposes that no zipmap operation may be applied to the results of any gather operation. + +In RASP, there is no value to repeated applications of zipmap before the first gather, as no further information can be generated beyond that already described by indices and tokens. This immediately explains the first observation of Press et al. (2020). Conversely, an architecture beginning with several attention sublayers – i.e., multiple gather operations – will be able to gather a large amount of information into each position early in the computation, if only by simple rules. More complicated gathering rules can be realised by applying zipmaps to the gathered information before generating new selectors6, explaining the interleaved attention/feed-forward middle section present in the discovered architecture. + +# 5 EXPERIMENTS + +To evaluate the relevance of the RASP language to transformers in practice, we train transformers on a small set of synthetic tasks and compare their results to the head- and layer- bounds and attention patterns predicted by RASP programs for the same tasks. + +While no RASP program promises to be a unique solution to the task it solves, several of the trained networks find solutions similar to those predicted by RASP. Among the most striking of these is the transformer trained to compute an in-place histogram, e.g. $\ S \mathrm { a b b d } \mapsto ( 1 , 1 , 2 , 2 , 1 )$ . We considered this task when the input sequences are presented with a beginning-of-sequence (BOS) token §, writing a single-head RASP program for it and training a single-head transformer on it. Visualizing the selection/attention patterns of these two heads (one RASP and one transformer) showed an identical pattern – see Figure 5. + +We present the single-head RASP program for this task in Figure 6. Its operation is as follows: first, the selector same_and_0 focuses each position $i$ on all positions $j$ containing the same token as $i$ , and also on the position 0. Hence the width of this selector at each position $i \neq 0$ is exactly one plus the value $v _ { i }$ that should be output at $i$ . Aggregating the sequence $( 1 , 0 , 0 , . . . , 0 )$ with this selector (which always includes focus on 0) gives us the value $\begin{array} { r } { \dot { a } _ { i } = \frac { 1 } { v _ { i } + 1 } } \end{array}$ at each location $i$ , from which $v _ { i }$ can then be recovered with a simple zipmap. + +![](images/51a057ef1f3bf7ee24295b282a28a69d0d43e0750cdb71f345b40dcdb7b23fac.jpg) +Figure 5: The attention (left) and selection (right) patterns of a single-head transformer and singlehead RASP program both trained/written for the task of in-place histograms. We visualise both on the input sequence §jibbiejig. On the y axis, the sequence acts as the query, describing the output locations for which new values must be computed, and on the $\mathbf { X }$ axis it acts as the key, describing the input locations being selected to compute these new values. Specifically, each row in the attention pattern describes the self-attention distribution of this head over the sequence §jibbiejig. The transformer has clearly learned the same pattern as that used by the RASP program. + +![](images/6ce7b3a2e50844c8e74777ee7640d0860fdb2650f2ce8c55f83bbc33211e99ff.jpg) +Figure 6: Building the RASP sequence histogram_given_BOS that computes an in-place histogram on the input tokens, under the assumption that the first token is always a special beginning-of-sequence character §. The selector same_and_0 builds on the same intuition as the s_with_0 selector from the implementation of count_conditioned. + +1 same_and_0 $=$ select ( tokens ,( tokens , indices ) , +2 lambda t1 ,t2 ,i:( $\displaystyle \ t 1 = = \ t 2$ ) or ( $\mathrm { i } = = 0 )$ ) +3 inverted_count $=$ aggregate ( same_and_0 , indices , +4 lambda i: int( $\mathrm { i } = = 0 \mathrm { i }$ ) ) +5 histogram_given_BOS $=$ (1/ inverted_count ) -1 + +The direct parallel between our program’s only selector and our trained transformer’s attention pattern (Figure 5) suggests that this RASP program describes the exact mechanism that our transformer has discovered. + +We present further experiments on additional tasks in Appendix B. + +# 6 CONCLUSIONS + +We abstract the computation model of the Transformer-encoder into a simple sequence processing language that captures the constraints on information flow in a Transformer. Considering computation problems and their implementation in the RASP language allows us to “think like a transformer” while abstracting away the technical details of a neural network in favor of symbolic programs. Moreover, provided it uses reasonable element-processing functions, we can analyze any RASP program to infer the minimum number of layers and maximum number of heads required to implement it in a transformer. We show several examples of expressive programs written in the RASP language, showing how complex operations can be implemented by a transformer. We train several transformers on these tasks, and find that RASP helps us predict both the correct number of heads and layers to use for these tasks and also the attention patterns that the transformers realise to solve them. Additionally, we use RASP to shed light on an empirical observation over transformer variants made by Press et al. 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Efficient content-based sparse attention with routing transformers, 2020. + +Yi Tay, Mostafa Dehghani, Dara Bahri, and Donald Metzler. Efficient transformers: A survey, 2020. + +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. CoRR, abs/1706.03762, 2017. URL http://arxiv.org/abs/1706.03762. + +Jesse Vig and Yonatan Belinkov. Analyzing the structure of attention in a transformer language model. In Proceedings of the 2019 ACL Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP, pp. 63–76, Florence, Italy, August 2019. Association for Computational Linguistics. doi: 10.18653/v1/W19-4808. URL https://www.aclweb.org/anthology/W19-4808. + +Gail Weiss, Yoav Goldberg, and Eran Yahav. Extracting automata from recurrent neural networks using queries and counterexamples. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, pp. 5244–5253, 2018. URL http://proceedings.mlr.press/v80/weiss18a.html. + +Chulhee Yun, Srinadh Bhojanapalli, Ankit Singh Rawat, Sashank J. Reddi, and Sanjiv Kumar. Are transformers universal approximators of sequence-to-sequence functions?, 2019. + +Manzil Zaheer, Guru Guruganesh, Avinava Dubey, Joshua Ainslie, Chris Alberti, Santiago Ontañón, Philip Pham, Anirudh Ravula, Qifan Wang, Li Yang, and Amr Ahmed. Big bird: Transformers for longer sequences. CoRR, abs/2007.14062, 2020. URL https://arxiv.org/abs/2007.14062. + +Quanshi Zhang, Ying Nian Wu, and Song-Chun Zhu. Interpretable convolutional neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018. + +# A TRANSFORMERS IN RASP + +RASP is almost – but not quite – a strict over-approximation of transformer-encoders. In this section, we show how the addition of score, a generalisation of select that may assign non-boolean values to index pairs, makes RASP a strict over-approximation. In particular, we give an explicit translation from any given transformer-encoder to a RASP program, provided the augmentation with score. If the reader prefers to start there, a full description of transformers is given in section D (with notations in the preceding section). + +We now introduce the score operation, and expand aggregate to receive a scorer: + +• score Similarly to select, the score operation takes two tuples of sequences $\mathfrak { m } \mathbf { e } { = } ( \mathfrak { m } \mathbb { 1 } , \ldots , \mathfrak { m } \mathbf { k } )$ and other $=$ (ot1,...,otl), and a function f expecting $\mathsf { k } { + } \mathsf { 1 }$ atomic values. This time however, f may return any non-negative float value. It creates from these a scorer similarly to how select creates a selector from its inputs. + +• aggregate The aggregate operation is expanded such that it may receive either a scorer or a selector where it previously accepted only a selector. When it receives a scorer, each y[i] is assigned the weighted average of all the values of $\mathsf { x }$ , according to the values in the scorer: + +$$ +\mathsf { y } [ \mathrm { i } ] = \frac { \sum _ { \mathsf { j } \in [ n ] } \mathsf { s } ( \mathrm { i } , \mathrm { j } ) { \cdot } \mathsf { x } [ \mathrm { j } ] } { \sum _ { \mathsf { j } \in [ n ] } \mathsf { s } ( \mathrm { i } , \mathrm { j } ) } +$$ + +Intuitively, the select operation, and the application of aggregate to a selector, can be seen as the special case of a score-select pair in which the scorer has only given scores of 0 and 1. It does have one difference in that it also allows for a default value which may be used when the total score for some index i is 0. However, this can be seen as sugar: it is not difficult to create a mechanism similar to that of count_conditioned in order to recognize when a selector has width 0, and so avoid the direct use of default in aggregate7. Moreover, if the data is always given with a special beginning-of-sequence (BOS) token, then the default value can simply be loaded from that location whenever no other focus is found. + +1 def elementwise_part (x,att_y , el_funcs ) : +2 lA , n2 , ff $=$ el_funcs +3 def apply_elementwise_parts (xy) : +4 $\mathrm { ~ \tt ~ { ~ x ~ y ~ } ~ } =$ tovec (xy) +5 x, att_y $=$ first_half (xy) , second_half (xy) +6 x1 = x + lA( att_y ) # vector addition +7 res = x1 $^ +$ ff(n2(x1) ) +8 return totuple ( res ) +9 return zipmap (x+att_y , apply_elementwise_parts ) +10 +11 +12 def layer (x, head_weights ,n1 , el_funcs ) : +13 multihead_y $=$ ( ) +14 for lq ,lk ,lv in head_weights : +15 s $=$ score (x,x, lambda xixj : +16 exp ( scalarprod ( +17 lq(n1( first_half ( xixj ) ) ) , +18 lk(n1( second_half ( xixj ) ) ) ) ) +19 head_y $=$ aggregate (s,x,lv) +20 multihead_y $=$ multihead_y $^ +$ head_y +21 # $^ +$ here is concatenating tuples of sequences +22 return elementwise_part (x, multihead_y , el_funcs ) +23 +24 def T_y0 ( layer_funcs ,w,p) : +25 $\qquad \times \quad =$ zipmap (( indices , tokens ) ,lambda i, $\mathsf { t } : \mathsf { p } \left( \mathrm { i } \right) + \mathsf { w } \left( \mathrm { t } \right) \big )$ +26 for head_weights ,n1 , el_funcs in layer_funcs : +27 x = layer (x, head_weights ,n1 , el_funcs ) +28 return x # tuple of d float sequences + +Theorem A.1. Let $\mathcal { T } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { d } ) ^ { * }$ be a transformer and $y _ { 0 } : \Sigma \to ( \mathbb { R } ^ { d } ) ^ { * }$ be an input embedding computed as the sum of a token and positional embedding, $y _ { 0 } ( x ) _ { i } = w ( x _ { i } ) + p ( i )$ . Then the computation of $\mathcal { T } _ { y _ { 0 } } \triangleq y _ { 0 } \circ \mathcal { T }$ can be mimicked in a RASP program that writes the output to $d$ float-sequences and uses exactly $L H$ score and aggregate calls and $( H + 1 ) L + 1$ zipmap calls, where $L$ and $H$ are the number of layers and attention heads in $\tau$ , respectively. + +Proof Sketch In figure 7 we present code that, given the token and positional embedding $\boldsymbol { w } : \Sigma \to \mathbb { R } ^ { d }$ and $\dot { p } : \mathbb { N } \to \mathbb { R } ^ { d }$ and all the weights of a transformer, recreates that same transformer in RASP. Our code relies on the helper functions tup2vec, vec2tup, first_half and second_half which help convert between the tuples of values given to the processing functions by zipmap and aggregate and the vectors they represent. For simplicity in the presentation, we assume here that the transformer is given as a collection of linear transformations, layer-norms, and feed-forward functions which can be applied to its internal vectors directly. + +The main routine is $\mathsf { T } \_ \mathsf { y } \boldsymbol { \theta }$ , which applies the initial embedding and stores it in a tuple of $d$ sequences, $\mathsf { x }$ , and then applies each layer to it in turn. This takes $L$ calls to the layer function, each of which computes score-aggregate (with additional zipmap before the aggregate) pair $H$ times to mimic each of the heads, and then calls a zipmap on the concatenation of their result to complete the remaining (elementwise) computations of the layer. + +Note. Recall that, as noted in $\mathrm { E }$ , when a processing function passed to zipmap returns multiple values, zipmap simply generates that same number of sequences. In particular, at all iterations of the loop in $\mathsf { T } _ { - } \mathsf { y } \mathsf { 0 } , \mathsf { x }$ is a tuple of $d$ sequences where $d$ is the embedding dimension of the given transformer. + +We see that, when augmented with score, the RASP language naturally composes the components of a given transformer to reconstruct it exactly. + +Why not have score? The motivation for the omission of score from RASP is cleanliness: it is far easier to think in terms of select as opposed to score, and we have not yet encountered a problem where we used scorers whose values where outside of 0 and 1. In time, as we use the language more and encounter the limitations this choice poses, we may return to score and see what other kind of special cases of it we will benefit from including in RASP. + +Power of RASP As seen in this section, RASP can (provided this slight generalization of select) represent any transformer. Additionally, we see that it is not arbitrarily overpowered. For example, it does not allow iterating over a sequence of arbitrary length one-by-one to perform some gradual computation (as might be done in RNN or DFA), and in general does not allow arbitrary repetition of operations as other languages might allow. This is because the number of operations in a RASP program is predetermined: RASP programs do not admit loops. This distinction between transformers and RNNs is known, and there is interest in bridging it. For example, the Universal Transformer attempts to introduce loops into transformers, by allowing them to also have a control mechanism that decides whether to repeat a layer during computation (Dehghani et al., 2018). + +# B EXPERIMENTS + +For RASP to be useful, it is important to see that RASP programs relate well to actual transformer behavior in practice. In this section, we train transformers on a small set of synthetic tasks for which we can write RASP programs, and see how the these programs relate to our empirical results. + +We consider the following tasks: + +1. Count-a: Return the number of $\prime _ { a } \prime$ tokens in a sequence, e.g., aba $\mapsto 2$ . +2. Histogram: For each token, its number of repetitions in a sequence, presented in-place. For example, aababc $\mapsto ( 3 , 3 , 2 , 3 , 2 , 1 )$ . +3. Reverse: Reversing a sequence, e.g.: abc $\mapsto$ cba. + +For Histogram, we also consider a variant with a special beginning-of-sequence (BOS) token §, appearing exactly once at the beginning of each sequence (and nowhere else). For example, §aabc $\mapsto$ $( 1 , 2 , 2 , 1 , 1 )$ . + +We train transformers for each of these tasks, and test whether our RASP programs accurately predict the minimum number of heads and layers needed to perform them. We also visualise their attention distributions8, and see if they match the selectors used by our programs. + +We find that several of the RASP programs presented in this paper show similar attention (selector) patterns to those of the trained transformers in practice, suggesting that programming in RASP helps us provide reasonable predictions of transformer behavior. Moreover, we often find that reducing the number of heads and layers in a transformer beyond the number needed in our RASP program for the same task significantly degrades its accuracy. This suggests that the specific programs we have presented in for these tasks are also optimal solutions. + +Data and Evaluation Unless stated otherwise: for all of the languages, we use the alphabet $\{ \mathsf { a } , \mathsf { b } , \mathsf { c } , \mathsf { d } , \mathsf { e } \}$ with sequences of sizes 1 through 100. These are generated by first choosing the length uniformly from 1 to 100, and then choosing each token uniformly from the alphabet. We use 50, 000 train samples, $1 , 0 0 0$ test samples, and 1, 000 validation samples. + +For tasks giving a ‘single’ output value – such as Count-a (which gives 1 number) as opposed to Reverse which gives a new sequence – we train the network to return that value in all positions, e.g., ${ \mathsf { a b a } } \mapsto ( 2 , 2 , 2 )$ for Count-a. This makes the visualisation of the attention distributions clearer (as all locations are trying to do something meaningful, as opposed to one), and is also more clearly aligned with the tasks we have described in this work. + +We measure the accuracy of a transformer on a batch of sequences as the fraction of total predictions it made for those sequences that were correct, e.g. $\textstyle { \frac { x } { 5 } }$ for a batch with total sequence length 5. For the train, set, and validation sets, we report accuracy as the average batch accuracy. + +Architecture We use the transformer architecture provided with PyTorch 1.7.0, with an additional single linear transformation and softmax at the end to convert to the output classes prediction. Unless stated otherwise, we use small embedding and feed-forward dimensions: 20 and 40, respectively. We vary the number of heads and layers per task. + +Training Method We train with the ADAM optimizer, sin-cosine positional embedding, and dropout 0.1 in the transformer weights. We use learning rate 0.0003, batch size 50, and PyTorch’s ExponentialLR learning rate scheduler with gamma 0.98 (updated after every epoch). Excluding confirmation of a negative result, we train each network for 20 epochs. If a network hits $1 0 0 \%$ accuracy before then, we stop. + +Helpful RASP functions In this section we will make frequent use of the full-selector full_s $=$ select((),(),lambda :True) and the function frac_condition(sequences,f), which computes aggregate(full_s,sequences,lambda a:int $( \mathsf { f } ( \star \mathsf { a } ) ) ^ { 9 }$ – the fraction of input positions for which the f is satisfied on the values of sequences. Note that frac_condition requires only 1 layer (after sequences have been computed) with 1 head, and that that head is full_s. Recall also that length is computed: length $^ { = 1 }$ /frac_condition(indices,lambda $\mathbf { i } : \mathbf { i } = = \mathbf { \boldsymbol { \theta } } ,$ ). + +# B.1 COUNT-a + +To avoid gradient problems from trying to obtain large numerical values from the transformer (e.g., 8), we encode Count-a as a categorical task. In particular, we create 21 output tokens $\{ \emptyset , 1 , 2 , \dots , 2 \emptyset \}$ , and if there are more than $2 0 \mathsf { a }$ tokens in the sequence we just report 20. + +![](images/4a3ade32726301aa70e8068b57ccfc230295de5eb8730001e338fadf242b80a8.jpg) +Figure 8: The attention distribution in the only head of a transformer trained on Count-a. The sequence is presented on the y-axis as the queries and on the $\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. As predicted by our RASP program (which solves this task using full_s), this distribution is relatively uniform. + +RASP permits a 1-layer, 1-head program for this task: count_a $=$ lengthfrac_condition( tokens, lambda t: $\tan \tt { a } ^ { \prime \prime } \tt { a } ^ { \prime \prime } .$ ). (length and frac_condition share the selector full_s.) Accordingly, training a transformer with 1 layer and 1 head on Count-a succeeded, reaching test accuracy $9 8 . 9 \%$ on the $2 0 ^ { \mathrm { t h } }$ epoch. + +The attention pattern of this transformer on the input sequence abbaabcde is shown in Figure 8. While the distribution is not perfectly uniform, it does seem to focus on the entire sequence, as the use of full_s in our RASP program suggests (contrast for example with the attention distribution for Reverse, in Figure 12)10. + +We stress that this is a different distribution to that which we might intuitively expect for this task – namely, an attention pattern focused solely on instances of a in the sequence – and that RASP has successfully pushed us to predict it! + +# B.2 HISTOGRAM + +As with Count-a, we encode histograms as a categorical task. This time we limit the maximum count to 10, i.e., we only use the output tokens $1 , 2 , \ldots , 1 0$ . (0 is irrelevant as it will not appear in an in-place histogram). Unlike most other tasks, for Histograms we use an alphabet of size 10: $\{ \mathsf { a } , \mathsf { b } , \mathsf { \ldots } , \mathsf { j } \}$ . For histograms, we use an input alphabet of size 10: $\{ \mathsf { a } , \mathsf { b } , \mathsf { \ldots } , \mathsf { j } \}$ . + +![](images/129bcdeb0656cd1545705ffef4321dd806c71604c95e0d41796cc649612a5c55.jpg) +Figure 9: The attention distribution in the single head transformer trained on Histogram. The input sequence abcdeba is presented on the y-axis as the queries and on the $\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. This transformer does not have enough heads: despite training for 50 epochs, it has only reached test accuracy $5 5 \%$ , and generates the incorrect output $( 1 , 2 , 1 , 1 , 1 , 2 , 1 )$ for this sequence. Similarly, it does not manage to recreate the selection pattern predicted by our RASP program for this task, which requires 2 heads. + +![](images/fcbd4bf3c46c659bc9e85192f4cf71c3c99e1d806d1278c586753ee1f80c2e50.jpg) +Figure 10: The attention distribution in the 2-head, single-layer transformer trained on Histogram. The input sequence aabbccacde is presented on the y-axis as the queries and on the $\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. The distribution is similar, but not identical, to our count_conditioned implementation. In particular, it seems that the tokens d and e are to some extent taking the role of the index 0 in our implementation: the first head has most tokens focus on themselves, d, and e, and the second head has most tokens focus slightly more on d and e than others. d and e themselves seem to behave inversely to the other tokens. + +We trained one transformer with 1 layer and 2 heads, and another with only 1 layer and 1 head. After 20 epochs, the transformer with 2 heads reached test accuracy $8 9 . 3 \%$ . In contrast, after 50 epochs, the transformer with only 1 head was still at accuracy $5 5 \% !$ ! Increasing its embedding and feed-forward dimensions to 50 and 100 respectively also did not work: a 1-layer 1-head transformer with these dimensions still only reached $7 9 . 3 \%$ test accuracy after 50 epochs, and this after being past $7 7 + \%$ validation accuracy since the $2 7 ^ { \mathrm { t h } }$ epoch. + +Drawing the attention map for the single-head transformer (Figure 9) did not seem to relate to the selection pattern of count_conditioned at all, unsurprisingly considering that it did not have enough heads. See for example the apparent focus on b by several query positions not containing b, as opposed to our expectations of the count_conditioned focus pattern as shown in Figure 4. + +For the 2-heads transformer, we draw its attention maps in Figure 10. The solution has some clear parallels with our RASP prediction of histogram selection patterns of count_conditioned, such as most tokens focusing on themselves in the one head and sharing distribution patterns in the other. But we can also easily find differences: in particular, the d and e tokens seem to avoid rather than focus on themselves, and the shared focus of most tokens in the similar-attentions head is not on 0 but on d and e. There is a possibility that in this transformer, d and e are playing a role similar to that which we gave to 0 in our implementation of count_conditioned. We leave a deeper exploration of this to future work. + +# B.3 HISTOGRAM WITH BOS + +Recall the implementation of count_conditioned (Figure 3): it calculates the “width" (number of selected locations) of a hypothetical selector s by simulating it along two actual selectors, s_with_0 and s_just_0. s_with_0 is used to calculate for every index the fraction $1 \big / c _ { i } ^ { \prime } { + 1 }$ where $c _ { i } ^ { \prime }$ is the width of s on everything except index 0, and s_just_0 is used to make a final adjustment from $c _ { i } ^ { \prime }$ to the actual width, depending on the focus on 0. + +If, then, the contents at position 0 are constant across all inputs, then the second selector s_just_0 becomes unnecessary: any information it conveys can be hard-coded into the RASP program (practically, the transformer). It follows that for setups where all input sequences are prepended with a special beginning-of-sequence (BOS) token, count_conditioned can be implemented with only one head, using just the s_with_0 selection pattern. + +![](images/88fb3fb4b9a69ff2144db012cb6e75a7e4501fd35a6b36fc57865a9a528636a0.jpg) +Figure 11: The attention distribution for a 1-head, 1-layer transformer trained on Histogram with BOS tokens. The input sequence jibbiejig is presented on the y-axis as the queries and on the $\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. This transformer appears to have implemented Histogram (with BOS) using exactly the same s_with_0 selection pattern as suggested in our count_conditioned implementation! (The second selection pattern in the count_conditioned implementation, s_just_0, is unnecessary when the value at the 0 index is constant.)) + +We prepend all of the original Histogram inputs with a special BOS token § (and their outputs with 1), and train a new 1-layer, 1-head transformer on the resulting data set. For gamma $= 0 . 9 9$ , the results satisfy our predictions perfectly: the transformer reaches ${ \bar { 9 9 } } . 7 \%$ test accuracy in 20 epochs, and drawing its attention distribution (Figure 11) shows that it follows exactly the pattern of the selector s_with_0! (For gamma $= 0 . 9 8$ the attention distribution was also very similar to that of s_with_0, but the model reached only $8 6 . 4 \%$ test accuracy after 20 epochs.) + +Discussion of BOS The significance of such ‘non-input’ tokens in transformers has been previously discussed, with different interpretations. For example, Vig & Belinkov (2019) refer to the attention focused on the initial token of a sequence – seemingly when there is nothing else to focus on – as null attention. They report that the null token gathered as much as $9 7 \%$ of the attention of some of their heads, and suggest that this is consistent with these heads being unimportant to the transformer’s overall performance. Conversely, this new result suggests that the null token at the beginning of a sequence is playing an important role in the transformer calculations, and in particular is directly assisting in counting! + +# B.4 REVERSE + +In RASP, Reverse is implemented using flip_ $; =$ select(indices,length-1-indices,lambda ${ \bf i } , { \bf j } : { \bf i } = = { \bf j } ;$ ) followed by reverse $\boldsymbol { \underline { { \underline { { \mathbf { \delta \pi } } } } } } ,$ aggregate(flip_s,tokens). This takes two layers of one head each (recall that length itself requires one layer to compute). + +![](images/e1827b36e47fc786db62d84a2cbddcfae906f728573391e9202ff89c943181d5.jpg) +Figure 12: The attention distributions in the 2-layer, 1-head (per layer) transformer trained on Reverse. The first layer is on the left and the second on the right. The input sequence abcdeabcde is presented on the y-axis as the queries and on the $\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. The attention distribution in the second layer is exactly as predicted by our RASP program (i.e., a hard, ‘reverse-match’ attention). In the first layer however, it seems the transformer has learned a behavior other than uniform attention to compute the length of the sequence (which it needs for the second layer). + +We train two transformers on Reverse: one with 2 layers and 1 head, and the other with 1 layer and 2 heads, to verify that the separation into 2 layers is indeed necessary. To give room for the index-based selection pattern (i.e., a scoring method that involves comparison of indices and not just tokens), we give the transformers per-head width at least as large as our maximum length. In particular, we use embedding dimension 100 for the 2-layer transformer and 200 for the 1-layer transformer11. We also give them each feed-forward dimension twice their embedding dimension. + +The 2-layer transformer reaches test accuracy $9 9 . 6 \%$ after 20 epochs. In contrast, and as expected, the single-layer transformer remains trapped at $3 9 . 6 \%$ test accuracy even after 50 epochs. Plotting the attention for the 2-layer transformer (Figure 12) matches some of the predictions of our RASP program: the reverse-matching attention is only computed in the $2 ^ { \mathrm { n d } }$ layer, and computed perfectly at that point. We are inclined to believe the length is being computed at the first layer (as we predict). + +![](images/e4eb31866cecec1caca09b4c79cf27b5372473e299dd18ee8c8c8ae47a1d6b8c.jpg) +Figure 13: The attention distributions in the 1-layer, 1-head transformer trained on Reverse with fixed input length 50, on input sequences of length 45, 50, and 55, in order. The y-axes represent output locations (queries) and the $\mathbf { X }$ -axes input locations (keys), i.e., each row is a distribution over the input positions. As expected, the transformer has learned a constant relation between pairs of input locations, and maintains all of the pairs it can find in each input sequence it gets. For missing pairs, such as $( 0 , 4 9 )$ through (4, 45) in the input of length 45 (left), it struggles to focus its attention. + +But there are also devitions from our prediction: the attention pattern suggests that the transformer is computing the length using a different mechanism from the one that we have suggested. + +We now strengthen the claim that the initial layer of the Reverse transformer is computing the sequence length, by showing that when the sequence length is fixed, then a single-layer (and single-head) transformer does succeed on Reverse. We fix the sequence length to 50 and train a 1-layer 1-head transformer on Reverse. This simpler task can be presented in one layer and one head in RASP using the single selector flip50_s=select(indices,49-indices,lambda ${ \mathrm { i } } , { \mathrm { j } } : { \mathrm { i } } = = { \mathrm { j } } { \mathrm { ) } }$ , from which the result is computed reverse $5 0 =$ aggregate(flip50_s,tokens). + +As expected, the transformer succeeds in its task, reaching $1 0 0 \%$ test accuracy in only 3 epochs. In Figure 13 we illustrate that it has indeed learned a constant $( i , 4 9 - i )$ location pairing, by visualising its attention on sequences slightly longer or shorter than it has been trained on. + +For completeness, in Figure 14 we show also the attention patterns of the single-layer transformer trained on variable-length Reverse. In keeping with our predictions, it has not managed to learn the reverse-matching attention pattern at all. This is because it needs an additional layer to compute the length before it can create the correct attention pattern. + +![](images/d6f9bcf3871f784bd1bbff0bb71308e6ccf295939533d58fcde72b3abc37d3aa.jpg) +Figure 14: The attention distributions of the 2 heads in the 1-layer transformer trained on Reverse, as applied to the sequence abcdeabcde. As expected from the RASP program, the transformer is unable to learn the reverse-match attention on any head of its first layer. + +# C NOTATIONS + +Basic Notations For every $n \in \mathbb { N }$ , we denote $[ n ] = \{ 1 , . . . , n \}$ + +Matrices For a matrix $\ b { X } \in \mathbb { R } ^ { n \times d }$ , we refer to its $i$ -th row as $X _ { i } \in \mathbb { R } ^ { d }$ , and for a vector $v \in \mathbb { R } ^ { d }$ we refer to its $i$ -th value as $v _ { i } \in \mathbb { R }$ . We additionally refer to $n$ as the length of $X$ and abusively denote $| X | = n$ . + +For $\ b { X } \in \mathbb { R } ^ { n \times d }$ and $b \in \mathbb { R } ^ { d }$ , we use the shorthand $X + b$ to describe the addition of $b$ to each of the rows of $X$ , i.e.: $( X + b ) _ { i } = X _ { i } + b$ for every $i \in [ n ]$ . For a scalar $\alpha \in \mathbb { R }$ , any operation $X \odot \alpha , \odot \in \{ + , - , \times , \div \}$ is applied elementwise to all the values of $X$ . Matrix multiplication between two matrices $A , B$ is denoted simply $A B$ , and the transpose of a matrix $A$ is denoted $A ^ { T }$ . + +We occasionally treat input or output sequences $x _ { 1 } , . . . , x _ { n } \in \mathbb { R } ^ { d }$ as matrices $\ b { X } \in \mathbb { R } ^ { n \times d }$ whose rows are the individual input vectors: $X _ { i } = x _ { i }$ . When $n$ may be arbitrary, we will say that $X \in ( \mathbb { R } ^ { d } ) ^ { * }$ . + +Definition C.1. $A$ linear transformation with input dimension $d$ and output dimension m is a function $l _ { A , b } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { m } ) ^ { * }$ parameterised by a matrix $A \in \mathbb { R } ^ { d \times m }$ and vector $b \in \mathbb { R } ^ { m }$ as follows: $l _ { A , b } ( X ) = X A + b$ for every $X \in ( \mathbb { R } ^ { d } ) ^ { * }$ . $A$ and $b$ are the weights of the transformation. + +Definition C.2. The softmax function s : $\mathbb { R } ^ { * } \to \mathbb { R } ^ { * }$ is defined: for every $d \in \mathbb { N }$ and $x \in \mathbb { R } ^ { d }$ , $\mathbf { s } ( x ) \in \mathbb { R } ^ { d }$ such that $\begin{array} { r } { \mathrm { s } ( x ) _ { i } = \frac { \bar { e } ^ { x _ { i } } } { \sum _ { j \in [ d ] } e ^ { x _ { j } } } } \end{array}$ = e P xij∈[d] exj for every i ∈ [d]. We also denote by S the row-wise softmax function: for every $n , d \in \mathbb { N }$ and $\dot { X } \in \mathbb { R } ^ { n \times d }$ , $S ( X ) \in \mathbb { R } ^ { n \times d }$ , and ${ \cal { S } } ( X ) _ { i } = \mathrm { s } ( X _ { i } )$ for every $i \in [ n ]$ . + +We denote by ${ \mathcal { R } } ( X )$ the elementwise application of the ReLU function, $r : x \mapsto \operatorname* { m a x } ( 0 , x )$ , to $X$ . + +Function Qualities A function $f : A ^ { * } \to B ^ { * }$ is length preserving if it satisfies $| f ( x ) | = | x |$ for any $x \in ( \mathbb { R } ^ { d } ) ^ { * }$ (i.e., for any input sequence $x _ { 1 } , . . . , x _ { n } \in A$ , $f$ returns a sequence $y _ { 1 } , . . . , y _ { n } \in B$ ). If there also exists a function $g : A B$ such that $f ( x ) _ { i } = g ( x _ { i } )$ for any $i \leq | x |$ , then $f$ is elementwise, and we say that $f$ is an elementwise application of $g$ . Note that linear transformations are elementwise. + +# D TRANSFORMER-ENCODERS + +At the highest level, a transformer-encoder $T$ (henceforth, a transformer) is a parameterised lengthpreserving function $T : ( \mathbb { R } ^ { d } ) ^ { * } ( \mathbb { R } ^ { d } ) ^ { * }$ composed of multiple layers of length-preserving functions $\dot { \ell } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { d } ) ^ { * }$ , i.e. $f = \ell _ { 1 } \circ \ell _ { 2 } \dots \circ \ell _ { L }$ which we will describe in this section. + +Generally speaking, a transformer’s layers are not elementwise, and indeed the transformer would not be interesting if they were. However, when we come to look at their components, we see that this quality rests entirely only on their use of attention12. + +Attention Attention is a function devised to enable ‘recollection’ of previously processed data from a history of arbitrary length (Bahdanau et al., 2015; Luong et al., 2015). Transformers use a variant called scaled dot-product attention to collect data from multiple locations in an input sequence. + +Definition D.1. Scaled Dot-Product Attention is a function a $: ( \mathbb { R } ^ { d } ) ^ { * } ( \mathbb { R } ^ { m } ) ^ { * }$ parameterised by 3 linear transformations, $l _ { Q } , l _ { K } , l _ { V } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { m } ) ^ { * }$ and defined for every $X \in \mathsf { ( } \mathbb { R } ^ { d } ) ^ { * }$ as follows: + +$$ +\mathrm { a } ( X ) = \mathcal { S } \bigg ( \frac { l _ { Q } ( X ) l _ { K } ( X ) ^ { T } } { \sqrt { m } } \bigg ) l _ { V } ( X ) +$$ + +Note. The original definition of scaled dot-product attention allows $l _ { Q } , l _ { K }$ , and $l _ { V }$ to have different output dimensions $m$ . In this case, the denominator is $\sqrt { m _ { k } }$ . + +For convenience, from here on we refer to scaled dot-product attention simply as attention. + +The attention computation can be broken into 3 stages. First, a pairwise score is calculated for each pair of locations, showing how much the input in location $j$ should influence the output in location $i$ : this is the value $\mathbb { S } _ { i , j }$ in the matrix $\begin{array} { r } { \mathbb { S } = \frac { l _ { Q } ( X ) l _ { K } ( X ) ^ { T } } { \sqrt { m } } } \end{array}$ . Then, each input is processed in-place $( l _ { V } ( X ) )$ to create candidate outputs, and finally the candidate outputs are averaged for each output location $i$ , according to the softmaxed scores $S ( \mathbb { S } ) _ { i }$ for that location. In this sense, attention can be seen as a request to gather into each output information from various locations, where $l _ { Q }$ and $l _ { K }$ work together to select information sources, and $l _ { V }$ encodes the transferred information. + +Transformer layers often gather information with multiple attention functions, referred to as attention heads, whose results are concatenated back into a single output at the end: + +Definition D.2. Let $d , H , m \in \mathbb { N }$ be such that $d = H m .$ . $A$ multi-headed attention function with input dimension $d$ and $H$ heads is a function $\mathcal { A } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { d } ) ^ { * }$ parameterised by the weights of $H$ scaled dot-product attention functions $\mathbf { a } _ { 1 } , . . . , \mathbf { a } _ { H }$ as follows: 13 for every $X \in ( \mathbb { R } ^ { \dot { d } } ) ^ { * }$ , + +$$ +\begin{array} { r } { \boldsymbol { \mathcal { A } } ( \boldsymbol { X } ) = \mathrm { a _ { 1 } } ( \boldsymbol { X } ) { \cdot } \mathrm { a _ { 2 } } ( \boldsymbol { X } ) { \cdot } . . . { \cdot } \mathrm { a } _ { H } ( \boldsymbol { X } ) } \end{array} +$$ + +where $\cdot$ denotes row-wise concatenation, i.e. for every $i \in [ | X | ]$ , ${ \mathcal { A } } ( X ) _ { i }$ is the concatenation of $\operatorname { a } _ { 1 } ( X ) _ { i }$ through $\mathrm { a } _ { H } ( X ) _ { i }$ . The functions $\mathbf { a } _ { 1 } , . . . , \mathbf { a } _ { H }$ are referred to as the heads of $\mathcal { A }$ . + +In addition to attention, transformers use layer-norm and feed-forward components, as follows: Definition D.3. $A$ single-row layer-norm over dimension $d$ is a function $g : \mathbb { R } ^ { d } \mathbb { R } ^ { d }$ parameterised by vectors $a , b \in \mathbb { R } ^ { d }$ and constant $\varepsilon \in \mathbb { R }$ , and defined for every $x \in \mathbb { R } ^ { d }$ and $i \leq d$ as follows: + +$$ +g ( x ) _ { i } = { \frac { a _ { i } ( x _ { i } - { \bar { x } } ) } { \mathrm { s t d } ( x ) + \varepsilon } } + b _ { i } +$$ + +where $\begin{array} { r } { \bar { x } = \frac { \sum _ { j \in [ d ] } x _ { j } } { d } } \end{array}$ is the mean of x and $\begin{array} { r } { \mathrm { s t d } ( x ) = \sqrt { \frac { 1 } { d - 1 } \sum _ { i \leq d } ( x _ { i } - \bar { x } ) ^ { 2 } } } \end{array}$ is its standard deviation. + +A layer-norm over dimension $d$ $, \mathbf { n } : ( \mathbb { R } ^ { d } ) ^ { * } ( \mathbb { R } ^ { d } ) ^ { * }$ , is an elementwise application of a single-row layer-norm of dimension $d$ . + +The layer-norm’s function is more a reguliser, and indeed, it will not play a part in our abstraction. + +Definition D.4. $A$ feed-forward function with input dimension d and internal dimension m is an elementwise function $\mathcal { F } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { d } ) ^ { * }$ obtained by composing two linear transformations $L _ { 1 } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { m } ) ^ { * } , L _ { 2 } : ( \mathbb { R } ^ { m } ) ^ { * } \to ( \mathbb { R } ^ { d } ) ^ { * }$ and ReLU, as follows:14 ${ \mathcal { F } } ( X ) \triangleq L _ { 2 } ( { \mathcal { R } } ( L _ { 1 } ( X ) ) )$ . + +The feed-forward component is elementwise, and the combination of two linear transformations with nonlinear activation provides strong expressive capacity (Hornik et al., 1989). + +$$ +\begin{array} { r } { X _ { 1 } = X + l _ { A } ( \mathcal { A } ( \mathbf { n } _ { 1 } ( X ) ) ) } \\ { \ell ( X ) = X _ { 1 } + \mathcal { F } ( \mathbf { n } _ { 2 } ( X _ { 1 } ) ) } \end{array} +$$ + +We refer to the additions in both equations as a ‘skip connection’. Note that the layernorm, feedforward, and skip connection components of the layer are elementwise: were it not for the attention, the entire layer would be elementwise. + +Finally, we present the full encoder architecture: + +Definition D.6 (Transformer-Encoder Vaswani et al. (2017)). A transformer-encoder with $L$ layers, $H$ heads, and input and internal dimensions $d , m$ is a length-preserving function $\mathcal { T } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \dot { \mathbb { R } } ^ { d } ) ^ { * }$ parameterised by the weights of $L$ transformer-encoder layers $\ell _ { 1 } , . . . , \ell _ { L }$ , each with $H$ heads and input and internal dimensions $d , m$ , and defined for every $X \in ( \mathbb { R } ^ { d } ) ^ { * }$ as follows: + +$$ +\mathcal { T } ( X ) = \ell _ { L } ( . . . \ell _ { 2 } ( \ell _ { 1 } ( X ) ) ) +$$ + +Permutation Invariance of Transformers An interesting trait of the transformer architecture is that it has no inherent positional awareness. Specifically: for any transformer $\mathcal { T } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { d } ) ^ { * }$ , input sequence $\mathbf { x } = x _ { 1 } , x _ { 2 } , . . . , x _ { n } \in \mathbb { R } ^ { d }$ , and permutation $\pi$ , we have ${ \mathcal T } ( \pi ( \mathbf { x } ) ) = \pi ( { \mathcal T } ( \mathbf { x } ) ) ^ { \mathrm { ~ l ~ } }$ 15 . + +Discrete Input Transformers $\mathcal { T } : ( \mathbb { R } ^ { d } ) ^ { * } \to ( \mathbb { R } ^ { d } ) ^ { * }$ are used to process non-empty sequences over a finite alphabet $\Sigma$ by composing them with a simple length-preserving function, $\mathsf { \bar { y } _ { 0 } : \dot { Z } ^ { + } ( \mathbb { R } ^ { d } ) ^ { * } }$ : ${ \mathcal T } _ { y _ { 0 } } ( x ) \triangleq { \mathcal T } ( y _ { 0 } ( x ) )$ . This $y _ { 0 }$ is in turn composed from a token embedding $\boldsymbol { w } : \Sigma \to \mathbb { R } ^ { d }$ and position embedding $p : \mathbb { N } \to \mathbb { R } ^ { d }$ , which are normally combined using addition: for every $x = x _ { 1 } . . . x _ { n } \in \Sigma ^ { * }$ , $y _ { 0 } ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } ) _ { i } = w ( x _ { i } ) + p ( i ) ^ { 1 6 }$ . + +From here, whenever we refer to a ‘transformer over (some finite alphabet) $\Sigma ^ { \prime }$ , we mean a transformer paired with an initial embedding $y _ { 0 }$ as described above. + +# E ADDITIONAL DETAILS ABOUT RASP + +A note about types in RASP Technically, there is no one ‘tokens’, rather 5 options: tokens_str, tokens_int, tokens_float and tokens_bool cast the input sequence to the corresponding atomic types, and tokens_asis takes the input sequence as-is. For brevity, we refer to all of these as tokens here. + +RASP Sequences RASP operates exclusively on sequence- and selector-generating functions, which we refer to as sequences and selectors respectively and RASP-functions together. All RASP-functions take as input exactly one non-empty sequence, and when describing them and how RASP manipulates them to create new RASP-functions we will do so in terms of the sequences and selectors that they and their manipulations generate from each input sequence. When it is clear from context, we will simply refer to them as sequences and selectors, and describe them directly in terms of their outputs. For example, if we say that a RASP operation applies $+ 1$ elementwise to each value in a sequence u, we actually mean that it returns a new sequence v such that for every input $x$ and position $0 \leq i < | x |$ , $\mathsf { v } ( x ) [ i ] { = } \mathsf { u } ( x ) [ i ] { + } 1$ . + +Note In this section we will refer to the $i$ -th value in a sequence s as $\mathsf { s } [ i ]$ , such that $\mathsf { s } [ i ]$ is in one of the atomic types. We stress however that this is only for the discussion, and not a part of the language. + +Additional operations For brevity in code, the RASP also comes with the following syntactic sugar: + +• Anywhere that a tuple of sequences is passed into an operation, a single sequence may be passed in as-is as well. For example, $y = z .$ ipmap((indices,),(indices,),lambda a $, b : a + b ) ,$ ) can also be written $y = z \mathrm { i }$ pmap(indices,indices,lambda a, $b : a + b ) ,$ ). +• The zipmap operation is accessible through a large range of operators, covering its application to all of the base binary and unary operations on the atomic types. For example, the above y can equivalently be defined as y=indices+indices. These operators can also be mixed with constants from the atomic primitives, such that an equal y be obtained using $y = 2 +$ indices. +• aggregate may receive a tuple of sequences xx instead of a single sequence x. In this case, +it returns a new tuple (of the same length) of sequences, the result of applying aggregate to each of the sequences in xx. For example, the line a,b $=$ aggregate(s,(x,y)) is equivalent to a,b $=$ aggregate $( s , \mathsf { x } )$ , aggregate(s,y). +• The processing function passed to zipmap may return more than one value (provided the number of values it returns is constant). In this case, the operation will arrange the output values into the same number of output sequences. For example, y1, $y 2 = 2$ ipmap(x,lambda $\mathsf { v } : \mathsf { v } ^ { + 1 } , \mathsf { v } ^ { + 2 } )$ is equivalent to writing $y 1 = x + 1$ and then $y 2 = x + 2$ . +• Anywhere that a processing function is expected, if one is not provided, the identity function +is used17. + +$l _ { K } ( X ) , l _ { V } ( X )$ , and $l _ { Q } ( X ) _ { i }$ , where the order of the rows of $l _ { K } ( X )$ and $l _ { V } ( X )$ does not matter as long as they remain aligned. + +16Note that without the position embedding, $y _ { 0 }$ would be elementwise, and so its combination with $\tau$ would be permutation invariant – an undesirable trait for sequence processing. + +17This is consistent with aggregate allowing you to choose whether to pass a single sequence, or a tuple of sequences and a processing function. + +• The function select1 which takes only a single tuple of sequences xx and an indexcomputing function $\mathsf { f i }$ , and is syntactic sugar for select(xx,(indices,),lambda $\star a : f \mathrm { i } ( \mathsf { a } [ : - 1 ] ) = = a [ - 1 ] \gamma$ . select1 is guaranteed to return a true select satisfying the "up-to-one" property, i.e., that can be successfully paired in aggregates with an $\mathsf { x }$ that does not contain numbers. \ No newline at end of file diff --git a/parse/train/TmkN9JmDJx1/TmkN9JmDJx1_content_list.json b/parse/train/TmkN9JmDJx1/TmkN9JmDJx1_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..bbf0a5b198a0becf4502725cad01bd5c3bdce3af --- /dev/null +++ b/parse/train/TmkN9JmDJx1/TmkN9JmDJx1_content_list.json @@ -0,0 +1,2391 @@ +[ + { + "type": "text", + "text": "THINKING LIKE TRANSFORMERS ", + "text_level": 1, + "bbox": [ + 174, + 99, + 573, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 145, + 398, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 210, + 544, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "What is the computational model behind a transformer? Where recurrent neural networks have direct parallels in finite state machines, allowing clear discussion and thought around architecture variants or trained models, transformers have no such familiar parallel. In this paper we aim to change that, proposing a computational model for the transformer-encoder in the form of a programming language. We map the basic components of a transformer-encoder – attention and feed-forward computation – into the simple primitives of select, aggregate and zipmap, around which we form a programming language: the Restricted Access Sequence Processing Language (RASP). We show how RASP can be used to program solutions to tasks that could conceivably be learned by a transformer, augmenting it with tools we discover in our work. In particular, we provide RASP programs for histograms, sorting, and even logical inference similar to that of Clark et al. (2020). We further use our model to relate their difficulty in terms of the number of required layers and attention heads. Finally, we see how insights gained from our abstraction might be used to explain phenomena seen in recent works. ", + "bbox": [ + 233, + 239, + 766, + 448 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 474, + 336, + 489 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While Yun et al. (2019) show that sufficiently large transformers can approximate any constantlength sequence-to-sequence function, and Hahn (2019) provides theoretical limitations on their ability to compute functions on unbounded input length, neither of these provide insight on how a transformer may achieve a specific task. Orthogonally, Bhattamishra et al. (2020) provide transformer constructions for several counting languages, but this also does not direct us towards a general model. ", + "bbox": [ + 174, + 505, + 825, + 575 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This is in stark contrast to other neural network architectures, which do have clear computational models. For example, convolution networks are seen as as a sequence of filters (Zhang et al., 2018), and finite-state automata and their variants have been extensively used both for extraction from and theoretical analysis of recurrent neural networks (RNNs) (Omlin & Giles, 1996; Weiss et al., 2018; Rabusseau et al., 2018; Merrill et al., 2020), even inspiring new RNN variants (Joulin & Mikolov, 2015). ", + "bbox": [ + 174, + 582, + 825, + 665 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work we propose a computational model for the transformer-encoder, in the form of a simple sequence-processing language which we dub RASP(Restricted Access Sequence Processing Language). Much like how automata describe the token-by-token processing behavior of an RNN, our language captures the unique information flow constraints under which a transformer (Vaswani et al., 2017) operates as it processes input sequences. ", + "bbox": [ + 174, + 672, + 825, + 742 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Considering computation problems and their implementation in the RASP language allows us to “think like a transformer” while abstracting away the technical details of a neural network in favor of symbolic programs. A RASP program operates on sequences of values from uniform atomic types, and transforms them by composing a restricted set of sequence processors. One pair of processors is used to select inputs for aggregation, and then aggregate the selected items. Another processor performs arbitrary but local computation over its (localized) input. However, access to the complete sequence is available only through aggregate operations that reduce a stream of numbers to a scalar. The key to performing complex global computations under this model is to compose the aggregations such that they gather the correct information, that can then be locally processed for a final output. ", + "bbox": [ + 173, + 750, + 825, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Given a RASP program, we can analyze it to infer the minimal number of layers and maximum number of heads that is required to implement it as a transformer. We show several examples of expressive programs written in the RASP language, showing how complex operations can be implemented by a transformer. Thinking in terms of the RASP model also allows us to shed light on recent empirical observation of transformer variants (Press et al., 2020) and find concrete limitations of “efficient transformers” with restricted-attention (Tay et al., 2020). ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/6b01396729fafa621584abcc339227175ba7f60fe33baa5593f15be6e16a6598.jpg", + "image_caption": [ + "Figure 1: RASP program taking two sequences vals,keys and returning a sequence y sorting the elements of vals according to keys, e.g.: if vals $( \\boldsymbol { x } ) \\boldsymbol { = } [ \\mathsf { a } , \\mathsf { b } , \\mathsf { c } ]$ and $\\mathtt { k e y s ( x ) = [ } 0 , 4 , 2 \\mathtt { l }$ , then $y ( x ) = [ a , \\mathsf { c } , \\mathsf { b } ]$ . " + ], + "image_footnote": [], + "bbox": [ + 150, + 108, + 753, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 303, + 823, + 345 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 THE RESTRICTED ACCESS SEQUENCE PROCESSING LANGUAGE ", + "text_level": 1, + "bbox": [ + 174, + 366, + 735, + 382 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we present the the Restricted Access Sequence Processing Language (RASP). RASP assumes a machine composed of several Turing-complete processors, each of which can only run functions taking and returning a fixed number of primitive arguments, and a simple memory accessor that is controlled by these processors. The select, aggregate, and zipmap operations which we present will define and constrain how the processors work together to process an input sequence. ", + "bbox": [ + 174, + 397, + 825, + 468 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We will focus here only on the language itself, leaving the discussion of its exact relation to transformers to Section 3. ", + "bbox": [ + 174, + 474, + 823, + 502 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Overview A RASP program works by manipulating sequences, occasionally with the help of selectors. Sequences contain values of uniform atomic type, such as booleans, integers, floats, or strings. They are functions used for selecting elements from sequences, and are used (together with the appropriate operations) only in the process of creating new sequences. All sequences in RASP are lazily evaluated, meaning that their length and contents are not populated until passed an input. ", + "bbox": [ + 174, + 518, + 825, + 588 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The Base Sequences Every program in RASP begins from the same set of base sequences, and then creates new ones using a small number of core operations. These base sequences are indices, length, and tokens, evaluated on input $x _ { 1 } , x _ { 2 } , . . . , x _ { n }$ as their names suggest: $( 0 , 1 , . . . , n - 1 )$ , $( n , n , . . . , n )$ (of length $n$ ), and $( x _ { 1 } , x _ { 2 } , . . . , x _ { n } )$ , respectively. ", + "bbox": [ + 174, + 604, + 825, + 661 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Combining Sequences Sequences can be combined in an ‘elementwise’ manner, such that the value of the resulting sequence at each position $i$ is a function of the values in the combined sequences at position $i$ (similar to a map operation), or have positions ‘mixed’ in more complicated ways using selectors, which are functions $f : \\mathbb { N } \\times \\mathbb { N } \\to \\{ \\mathrm { T r u e } , \\mathrm { F a l s e } \\}$ whose sole purpose is to guide the combination of existing sequences into new ones. ", + "bbox": [ + 174, + 676, + 825, + 746 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We present the basic ingredients of RASP using an example. Figure 1 shows a simple RASP function for sorting a sequence of values according to a sequence of keys. It accepts an input sequence vals and uses the base sequence indices, that is available to any RASP program, to compute its output in three operations as follows: ", + "bbox": [ + 176, + 752, + 825, + 809 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. count_conditioned of Line 2 creates a new sequence that counts for each element of keys the number of “previous items” it has in keys, where the “previous items” are defined to be all items that have a lesser value, or equal value and lower index. Thus, num_prevs creates a sequence of numbers, representing the target sorted position of each item. 2. select of line 7 creates a new selector which will focus each position $i$ on the corresponding position $j$ for which indices[i] is equal to num_prevs ${ \\cdot } j ]$ . Effectively, it will direct the elements in each position $j$ towards their target location $i$ . ", + "bbox": [ + 212, + 820, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3. Finally, aggregate of line 8 applies select_sorted_val to vals, moving each $i$ -th element of vals to its calculated sorted position num_prevs[i]. ", + "bbox": [ + 207, + 103, + 823, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We now describe the base operations of RASP in-depth, occasionally presenting an example on the hypothetical input sequence $x$ of length $n$ . ", + "bbox": [ + 174, + 145, + 821, + 172 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• zipmap The zipmap operation takes a tuple of sequences and an element-processing function f, and applies f per-index to the values in those sequences to create a new sequence. For a simple example, $y 1 = z :$ ipmap((indices,indices), lambda ${ \\mathrm { i } } , { \\mathrm { j } } : { \\mathrm { i } } + { \\mathrm { j } } )$ creates a sequence that always evaluates to $( 0 , 2 , . . . , 2 n - 2 )$ . \n• aggregate The aggregate operation takes a selector s, a sequence $\\mathsf { x }$ , and an optional parameter default, and averages subsets of the values of $\\mathsf { x }$ into a new sequence y as follows: for every index $i , y [ i ]$ is the average of $\\times [ j _ { 0 } ] , \\times [ j _ { 1 } ] , \\ldots \\times [ j _ { k } ]$ , where $j _ { 0 } , j _ { 1 } , . . . , j _ { k }$ are the indices $j \\in [ n ]$ for which $\\mathsf { s } ( i , j )$ is True. We say $k$ is the focus width of $s$ at $i$ . If $k = 0$ , then $\\mathsf { y } [ j _ { 0 } ]$ is assigned the value in default. For example: if $\\mathsf { \\Omega } _ { \\mathsf { S } } ( i , j )$ returns True iff $i$ is odd and $j { = } 0$ , and the value in default is d, then y will evaluate to $( \\mathsf { d } , \\mathsf { x } [ 0 ] , \\mathsf { d } , \\mathsf { x } [ 0 ] , \\hdots , \\mathsf { y } [ n - 1 ] )$ where $\\mathsf { y } [ n - 1 ]$ is either d or $\\times [ 0 ]$ depending on the parity of $n$ . \n• select The select operation takes two sequences-tuples of lengths $k$ and $l$ , $\\mathfrak { m e } { = } ( \\mathfrak { m } _ { 1 } , \\mathfrak { m } _ { 2 } , \\dots { \\cdot } , \\mathfrak { m } _ { k } )$ and $\\mathsf { o t h e r } = ( \\mathsf { o t } _ { 1 } , \\mathsf { o t } _ { 2 } , \\ldots , \\mathsf { o t } _ { l } )$ , and a function $\\mathsf { f }$ expecting $k + 1$ atomic values and giving boolean output. It composes these to create a selector s as follows: for every two indices $i , j , s ( i , j )$ is the output of $\\mathsf { f }$ on the $i$ -th and $j$ -th slice of me and other respectively, i.e., $s ( i , j ) \\mathrm { = } \\mathsf { f } ( \\mathsf { m } _ { 1 } [ i ] , . . . , \\mathsf { m } _ { k } [ i ] , \\mathsf { o t } _ { 1 } [ j ] . . . \\mathsf { o t } _ { l } [ j ] )$ . For a simple example, in $\\mathtt { s } =$ select((indices,),(indices,),lambda mi,oti: $m i \\% 2 = = 1$ and $\\cot \\mathtt { i } = = 0 \\dot { }$ ), then $\\mathfrak { m } _ { 1 } = \\mathrm { i }$ ndices, $\\mathsf { o t } _ { 1 } \\mathsf { = i }$ ndices, and s is the same selector we used for our example in aggregate above. \ncount_conditioned This operation takes the same parameters me,other and $\\mathsf { f }$ as select, but this time returns a sequence y describing the number of selected influencing positions $j$ for each output position $i$ that $\\mathtt { S } \\mathtt { = }$ select(me,other,f) would have created. In other words, for each $i$ , $y [ i ] = k$ where $j _ { 1 } , . . . , j _ { k }$ is the set of positions $j$ for which $s ( i , j ) =$ True. For example, $h = 0$ count_conditioned((tokens,),(tokens,),lambda a,b: $\\mathsf { a } \\mathsf { = } \\mathsf { = } \\mathsf { b } .$ ) returns an in-place histogram for the tokens in the input sequence: h(“abaa”) $= ( 3 , 1 , 3 , 3 )$ . ", + "bbox": [ + 215, + 185, + 828, + 554 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This concludes the base operations of RASP – all other operations are shortcuts for combinations of the above 4, occasionally with the base sequences. ", + "bbox": [ + 176, + 565, + 823, + 593 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Sugar We implement RASP with a variety of syntactic sugar, presented fully in appendix E. Briefly: ", + "bbox": [ + 176, + 611, + 823, + 626 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. When applying zipmap to a single sequence, it may be passed directly without using a tuple, e.g.: zipmap(indices,f) is equivalent to zipmap((indices,),f). \n2. zipmap has sugar for most of the binary operators, e.g.: for two sequences $\\mathsf { x } , \\mathsf { y } .$ , then $x { + } y$ is sugar for zipmap( $( \\mathsf { x } , \\mathsf { y } )$ ,lambda $1 , b : a + b )$ . \n3. Whenever the focus width of s at some index is $\\leq ~ 1$ (“up-to-one selection”), aggregate(s,x,default=d) does not explicitly compute the division. In this case the values of $\\mathsf { x }$ do not have to be numbers. \n4. aggregate accepts one additional optional parameter elementwise_function. The full order of parameters is $s , x$ ,elementwise_function,default, and the use of elementwise_function is as follows: aggregate $( \\mathsf { s } , \\mathsf { x } , \\mathsf { f } , \\mathsf { d } )$ is equivalent to aggregate(s,zipmap $( \\times , \\mathsf { f } )$ ,defaul $\\therefore d$ ). ", + "bbox": [ + 210, + 638, + 825, + 811 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 EXAMPLES ", + "text_level": 1, + "bbox": [ + 174, + 829, + 292, + 844 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We now present some more example RASP programs, by increasing order of complexity. ", + "bbox": [ + 173, + 856, + 754, + 871 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Simple Examples The first and simplest example is to compute an in-place histogram for some sequence vals. This is achieved with a single application of count_conditioned: histogram=count_conditioned(vals,vals,lambda a, $b : a = = b :$ ). ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1 def by_frequency (vals , default ) : \n2 hist $=$ count_conditioned (vals ,vals , lambda a,b: $a = = b$ ) \n3 num_earlier $=$ count_conditioned ( \n4 ( indices , vals ) ,( indices , vals ) , \n5 lambda iq ,vq ,ik ,vk :( $\\mathsf { v q } = = \\mathsf { \\cdot }$ vk) and ( $\\mathbf { i k } < \\mathbf { i q } )$ ) ) \n6 has_earlier $=$ num_earlier > 0 \n7 unique_vals $=$ zipmap (( vals , has_earlier ) , \n8 lambda t,h_e: t if not h_e else default ) \n9 masked_hist $=$ hist - ( length $\\star$ has_earlier ) \n10 return sort ( unique_vals , key $=$ - masked_hist ) ", + "bbox": [ + 150, + 108, + 756, + 236 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 2: RASP program sorting the unique elements of a sequence vals by decreasing frequency. Lines 6 and 7 show syntactic sugar for multiple simple zipmap operations, e.g., line 6 can be written has_earlier $=$ zipmap(num_earlier,lambda $\\mathsf { n } : \\mathsf { n } > 0$ ). This program requires a default atom da of the same type as vals, to put in place of all otherwise-duplicated values in its result. For example, by_frequency(tokens_str,“-”)(“abacca”) will return “acb--”. ", + "bbox": [ + 173, + 253, + 826, + 324 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "From Length to Parity While length is provided as a primitive in the language, it can actually be achieved as a composition of the other base operations and sequences. This is done by computing full_ $\\hat { \\mathsf { s } } =$ select((),(),lambda :True) followed by 1/aggregate(full_s,indices,lambda i:int $\\overset { \\cdot } { 1 } = = \\varnothing )$ ) (the fraction of elements equal to 0 in indices, inverted). From length and that same full_s we can then define count(vals,v), a function taking any sequence vals and value v and returning a new sequence counting the number of appearances of v in vals. The implementation of count is simply length\\*aggregate(full_s,vals,lambda $\\mathsf { e : e : } = \\mathsf { = v }$ ). count in turn enables us to write programs like parity simply as count(tokens,1) $\\% 2 = = 0 ^ { 1 }$ . ", + "bbox": [ + 173, + 361, + 825, + 473 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Reverse We can reverse a sequence seq with the help of an up-to-one selector mapping each position to its opposite: flip_s $=$ select(indices, length-1-indices, lambda m,oth: $\\scriptstyle { \\mathfrak { m } } = = 0$ th). We use flip_s to re-order the tokens of seq: reverse $=$ aggregate(flip_s,seq). ", + "bbox": [ + 174, + 483, + 825, + 525 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Balanced Parentheses For balanced parentheses we use count_conditioned twice, storing in prev_opens and prev_closes the number of previous “(” or “)” (respectively) tokens each position has, including itself. The sequence is balanced if prev_opens-prev_closes has no negative values, and is 0 at position length-1. These two qualities can be easily computed using two final select-aggregate pairs, and then combined with a zipmap. ", + "bbox": [ + 174, + 536, + 825, + 606 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Most Frequent Tokens In fig. 2 we show how RASP can be used to arrange for any input sequence s the most frequent tokens in s, without repetition. The solution has 2 parts: first, we compute the histogram for all the tokens, and mask it such that all but the first of each token is given a negative value. Then, we sort the tokens according to the masked histogram. The solution uses the sort function from Figure 1. ", + "bbox": [ + 173, + 617, + 825, + 686 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1 def count_conditioned (me ,other ,f) : \n2 other $=$ other $^ +$ ( indices ,) # tuple concatenation , indices at end \n3 s_with $\\_ 0 =$ select (me ,other , lambda \\*a:f ( $\\star$ ( a[: -1]) ) or $( \\mathsf { a } [ - 1 ] \\mathsf { = } = \\mathsf { 0 } )$ ) \n4 s_just_0 $=$ select (me ,other , lambda \\*a:f ( $\\star$ ( a[: -1]) ) and $( \\mathsf { a } \\left[ - 1 \\right] \\mathsf { = } \\mathsf { = } \\mathsf { 0 } )$ ) \n5 frac $=$ aggregate ( s_with_0 , indices , lambda i: int ( $\\scriptstyle { \\dot { 1 } } = = 0$ ) ) \n6 count_outside_0 $=$ (1/ frac ) -1 \n7 count_in_ $\\begin{array} { r l } { \\mathbf { \\nabla } _ { \\mathbf { \\nabla } } \\theta } & { { } = } \\end{array}$ aggregate ( s_with_0 ,() ,lambda :1 , defaul $\\mathtt { \\iota } = \\mathtt { 0 }$ ) \n8 return count_outside_0 $^ +$ count_in_0 ", + "bbox": [ + 150, + 717, + 782, + 819 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Count Conditioned The operation count_conditioned is a powerful part of RASP, appearing in many other programs. Surprisingly, it is realisable as a composition of the other operations (and base sequence indices). Understanding its implementation is interesting for learning how to “truly” think like a transformer, and we present the code in Figure 3. The intuition is as follows: we compute the select whose width we want to calculate twice, once such that it also selects the position 0, and once such that it only selects this position. We then aggregate both these values, broadcasting 1 from position 0 and 0 from everywhere else, and using default value 0. The first aggregate computes for each position the inverse of the number of selected positions (excluding 0) plus one, and the second computes whether that position would also focus on 0. A straightforward zipmaps then gives us the result. To further help the intuition, we present in fig. 4 the computation flow for a histogram calculation, when count_conditioned is implemented as described here. ", + "bbox": [ + 173, + 103, + 825, + 256 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/e3c8af607bbb6d9e300e42e1bf75feb288a546c2db7be6501d8640714ee45873.jpg", + "image_caption": [ + "Figure 4: The computation flow for the sequence histogram(tokens_str) applied to input sequence “hello”, when implemented in terms of select, aggregate, and zipmap. " + ], + "image_footnote": [], + "bbox": [ + 284, + 270, + 712, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note Many of the functions provided with RASP can be expressed in terms of count_conditioned – such as count (which counts how many elements in a sequence are equal to a given value) and contains (which checks if count is greater than 0) – but this is not necessarily an optimal implementation with respect to number of heads it uses. The RASPlibrary provides optimal implementations. ", + "bbox": [ + 173, + 580, + 825, + 637 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.2 IMPLEMENTING SYMBOLIC REASONING A LA ROVER ", + "text_level": 1, + "bbox": [ + 173, + 652, + 583, + 667 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "How might a transformer implement reasoning, as in Clark et al. (2020)? RASP empowers us to clearly think about such problems, and we sketch a solution direction here. We begin by reformulating the task of Clark et al. to a form that focuses on the core problem, moving away from natural language into more ‘concrete’ domain, and by limiting the type of relations we will consider. ", + "bbox": [ + 174, + 679, + 825, + 734 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the work of Clark et al., a transformer is presented a sequence of statements and a query Q, and must recognise whether Q is implied by the previous relations or not. For example: $\\mathsf { a } 1 \\in \\mathsf { A } 1$ , ${ \\mathsf { b } } \\in { \\mathsf { A } } 1 \\Longrightarrow { \\mathsf { b } } \\in { \\mathsf { A } } 2 .$ , a $1 \\in { \\mathsf { A } } 2 ?$ evaluates to True, whereas a1 $\\not \\in \\mathsf { A } \\mathsf { 1 }$ , a1 $\\not \\in { \\mathsf { A } } 2 ?$ evaluates to False. Different inputs for this task can have different depth: the number of ‘logical hops’ needed to correctly identify whether the query statement is true. ", + "bbox": [ + 174, + 742, + 825, + 811 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note. The original work accepts this input in natural language, e.g., “Alan is young. [...] If someone is young then...”, and allows statements with more complicated logical form, such as ${ \\mathsf { a } } 1 \\in { \\mathsf { A } } 1 \\wedge { \\mathsf { a } } 1 \\in { \\mathsf { A } } 2 \\implies { \\mathsf { a } } 1 \\in { \\mathsf { B } }$ . In this section we consider only a simplified and symbolic version, in which the statements are limited to the form of the previous paragraph. We assume the statements are separated by a special token |. ", + "bbox": [ + 174, + 819, + 825, + 888 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We sketch this task in RASP as follows: first, mark each statement ‘block’ in the sequence by its index (as a block), by counting for each token the number of separators before it in the input. Set aside the final block (the query), it is the set of tokens with no separators after them. For each remaining block mark whether it is a relation $\\in$ or $\\notin )$ or inference $( \\mapsto )$ statement. Then, for each relation block, note at the position of the set token the element and whether it is inside or outside, and similarly over the element token note the set. Next, for as many repetitions as the logical depth that the program should cover: share set information between all elements and element information between all sets (including from inference blocks, which initially have none initially empty), and then apply one logical step ‘locally’ at each inference block. Finally, for the query, seek any occurrence of the set token in the sequence, and return whether the element token is listed there in its contents. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 214 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Generalisation on Inference Depth A similar solution to the one we have proposed would be to make all of these logical inferences backwards from the query, i.e. by propagating backwards the requirements that would be sufficient to answer the query. If a trained transformer implements both of these solutions in parallel (for instance, to increase its robustness), this may explain the generalisation to greater query depth observed by Clark et al.. ", + "bbox": [ + 174, + 234, + 825, + 304 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 RELATION TO TRANSFORMERS, AND ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 328, + 601, + 343 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We discuss how RASP relates to the real transformer architecture, and how it may also be used to compare transformer variants, or analyse the ‘difficulty’ of a task for transformers. ", + "bbox": [ + 176, + 361, + 823, + 390 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Connection of RASP to Transformers The select and aggregate operations of RASP correspond to the attention-scoring and then pooling of transformer self-attention heads, the zipmap operations correspond to the feed-forward sublayers, and the computed sequences correspond to head or feedforward inputs and outputs. indices and tokens represent the initial input, while length and conditioned_contains are in fact combinations of the other primitives. The persistence of sequences – such that they may be accessed multiple times over the course of a RASP program – is encouraged by the existence of skip connections in the transformer architecture. In appendix A we consolidate these connections, giving a full description of transformers, and showing how any given transformer can be represented in the RASP language (provided a slight generalization of select2). ", + "bbox": [ + 174, + 409, + 825, + 534 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": ".1 PREDICTING TRANSFORMER COMPLEXITY WITH RASP ", + "text_level": 1, + "bbox": [ + 186, + 554, + 591, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The purpose of RASP is to help us reason about the computation process enabled by a transformer. We find that RASP programs lend themselves easily to ‘width’ and ‘depth’ analysis, enabling us to predict the number of heads and layers that a transformer will need to implement the same solution. We discuss this analysis now, and evaluate the predictions it provides in Appendix B. ", + "bbox": [ + 174, + 582, + 825, + 637 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For any given RASP program, we can compute the minimal number of layers required to implement it in a transformer, and upper bound the number of heads this implementation requires.3 This is provided its internal dimensions are wide enough to replicate the given processing functions. This analysis can give us intuition regarding the relative difficulty of different tasks for the transformer architecture, where each algorithm we find for a task gives us an upper bound on the number of layers and heads a transformer needs to solve it. ", + "bbox": [ + 174, + 645, + 825, + 728 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We implement such an analysis and provide a draw_comp_flow function, which automatically positions each attention head using a greedy scheduler, and displays the computation flow accordingly (see fig. 4, and others in the supplementary material). ", + "bbox": [ + 174, + 736, + 825, + 777 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A similar analysis can be done for different \"primitive\" computations in isolation, giving us intuition on the ‘cost’ of various common computations: how many additional heads and layers each computation adds when applied to a previously computed sequence. For example, our implementation of sort takes 2 layers, and so whenever we apply it to a computed sequence we know it will increase our program’s depth by 2 from that sequence. ", + "bbox": [ + 174, + 785, + 825, + 854 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm for computing program depth and width As noted, the first 3 base operations of RASP – select, aggregate, and zipmap – have direct parallels in the transformer architecture, and so we may easily analyse the result of any RASP program to see how many layers and heads it would take to realise in an actual transformer. (For length and count_conditioned, we analyse them in terms of their deconstruction to the other operations and sequences.) ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The first part of the analysis is simple: every sequence and selector is initiated with a “minimum depth” $d$ , reflecting the earliest layer at which it can be computed, and so the minimum number of layers needed to create any given sequence is $d , d$ is computed for each new sequence as follows: ", + "bbox": [ + 176, + 180, + 825, + 223 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. The base sequences indices and tokens have $d = 0$ , as they are input to the transformer rather than part of its computation. \n2. Any sequence created from a zipmap is given $d$ equal to the maximum $d$ of the inputs to that zipmap, as it can be created immediately after the last of them in the same feed forward computation that concludes it.4 \n3. Every selector gets $d$ equal to the maximum $d$ of its creating sequences X plus 1, to reflect the fact that all of them must be calculated before it can even begin (as multi-headed attention happens only once, at the beginning of each layer). \n4. Any selector created from an aggregate has $d$ equal to at least that of its creating selector, and at least one plus those of the input sequences to the aggregate operation (as they must be passed through the attention to create the new sequences). ", + "bbox": [ + 210, + 234, + 825, + 402 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "RASP makes it easy to access all sequence and selectors on which an unfinished value is dependent, and allowing us to analyse not only the depth but also the width of a given program. The width of the computation is a reflection of how many (unique) selectors are being used at every layer of the transformer: the number of attention heads needed to mimick that layer in a transformer. We say that a selector $s$ is being used at some layer $l$ if a sequence that is aggregated from $s$ is calculated at $l$ This is because it is possible a selector $s$ may have minimum depth $d$ , but is only or also used at later layers: for instance if the sequences needed for the aggregation operation using $s$ are not yet ready. ", + "bbox": [ + 174, + 415, + 825, + 512 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Similarly, a sequence does not need to be computed at its minimum possible depth, as it is possible it will only be needed much later. Hence there is no one analysis for a given program, and there is room for creating a scheduling algorithm that minimises the maximum width of the transformer, i.e., the maximum of the widths of all layers (useful, as transformers tend to be created with uniform width). ", + "bbox": [ + 174, + 518, + 825, + 575 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 IMPLICATIONS FOR TRANSFORMER VARIANTS ", + "text_level": 1, + "bbox": [ + 176, + 595, + 591, + 611 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 RESTRICTED-ATTENTION TRANSFORMERS ", + "text_level": 1, + "bbox": [ + 174, + 627, + 511, + 641 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Multiple works propose restricting the attention mechanism of transformers in order to create more efficient transformers, reducing the time complexity of each layer from ${ \\mathrm { O } } ( n ^ { 2 } )$ to $\\mathrm { { O } } ( n l o g ( n ) )$ or even ${ \\bf O } ( n )$ with respect to the input sequence length $n$ (see Tay et al. (2020) for a survey of such approaches and their complexity). Several of these do so using sparse attention, in which the attention is masked using different patterns to reduce the number of locations that can interact (see for instance (Child et al., 2019; Beltagy et al., 2020; Ainslie et al., 2020; Zaheer et al., 2020; Roy et al., 2020)). ", + "bbox": [ + 174, + 652, + 825, + 738 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Considering these variants of transformers in terms of the RASP language, allows us to reason about the computations they can and cannot perform. In terms of RASP, these variants of transformers all impose restrictions on the selectors, forcing some of the $n ^ { 2 }$ index pairs $( i , j )$ to False. ", + "bbox": [ + 174, + 744, + 825, + 786 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 1 showed how to implement sorting of an input sequence with arbitrary alphabet size, comparison function, and length5. We now prove that RASP variants where the selector is restricted to ${ \\bf O } ( n )$ pairs (i.e., transformer variants with sufficiently restricted attention), cannot sort. ", + "bbox": [ + 176, + 792, + 823, + 835 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The computation model of RASP and indeed of all transformer variants allows comparison of values between more than one sequence position only during the select operation, i.e., only while computing the attention distribution Hence, all comparisons necessary for sorting must be applied in select. It follows that whenever select is restricted such that it compares at most ${ \\bf O } ( n )$ index pairs per head, no constant number of heads and layers will be sufficient for the model to perform sorting on arbitrary length – as sorting is known to require at least $\\mathrm { { O } } ( n l o g ( n ) )$ comparisons. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Thus, variants of transformers in which the attention is masked to impose ${ \\bf O } ( n )$ complexity require at least $\\mathbf { O } ( l o g ( n ) )$ layers to sort. It also follows that they require $\\mathbf { O } ( l o g ( n ) )$ layers to implement count_conditioned, as we see in fig. 1 that count_conditioned can be applied to create a sequence (num_prevs) which is sufficient to complete a sorting operation with only ${ \\bf O } ( n )$ further operations. ", + "bbox": [ + 174, + 194, + 825, + 251 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 SANDWICH TRANSFORMERS ", + "text_level": 1, + "bbox": [ + 176, + 270, + 411, + 284 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recently, Press et al. (2020) showed that reordering the attention and feed-forward sublayers of a transformer affects its ability to train on language modeling tasks. In particular, they showed that 1. pushing feed-forward sublayers towards the bottom of a transformer weakened it, and 2. pushing attention sublayers to the bottom and feed-forward sublayers to the top strengthened it, provided there was still some interleaving in the middle (making a sandwich transformer). ", + "bbox": [ + 174, + 296, + 825, + 366 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Considering the base operations of RASP helps us understand the observations of Press et al.. In RASP, the feed-forward and attention sublayers are the zipmap and select-aggregate (or gather for short) operations. Any arrangement of the sublayers into a set architecture, from the ‘vanilla’ transformer to the variations considered in (Press et al., 2020), imposes a restriction on the number and order of RASP operations that can be chained in a RASP program. For example, an architecture in which all feed-forward sublayers appear before the attention sublayers imposes that no zipmap operation may be applied to the results of any gather operation. ", + "bbox": [ + 174, + 372, + 825, + 470 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In RASP, there is no value to repeated applications of zipmap before the first gather, as no further information can be generated beyond that already described by indices and tokens. This immediately explains the first observation of Press et al. (2020). Conversely, an architecture beginning with several attention sublayers – i.e., multiple gather operations – will be able to gather a large amount of information into each position early in the computation, if only by simple rules. More complicated gathering rules can be realised by applying zipmaps to the gathered information before generating new selectors6, explaining the interleaved attention/feed-forward middle section present in the discovered architecture. ", + "bbox": [ + 174, + 478, + 825, + 588 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 612, + 326, + 627 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To evaluate the relevance of the RASP language to transformers in practice, we train transformers on a small set of synthetic tasks and compare their results to the head- and layer- bounds and attention patterns predicted by RASP programs for the same tasks. ", + "bbox": [ + 174, + 645, + 825, + 686 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "While no RASP program promises to be a unique solution to the task it solves, several of the trained networks find solutions similar to those predicted by RASP. Among the most striking of these is the transformer trained to compute an in-place histogram, e.g. $\\ S \\mathrm { a b b d } \\mapsto ( 1 , 1 , 2 , 2 , 1 )$ . We considered this task when the input sequences are presented with a beginning-of-sequence (BOS) token §, writing a single-head RASP program for it and training a single-head transformer on it. Visualizing the selection/attention patterns of these two heads (one RASP and one transformer) showed an identical pattern – see Figure 5. ", + "bbox": [ + 174, + 694, + 825, + 791 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We present the single-head RASP program for this task in Figure 6. Its operation is as follows: first, the selector same_and_0 focuses each position $i$ on all positions $j$ containing the same token as $i$ , and also on the position 0. Hence the width of this selector at each position $i \\neq 0$ is exactly one plus the value $v _ { i }$ that should be output at $i$ . Aggregating the sequence $( 1 , 0 , 0 , . . . , 0 )$ with this selector (which always includes focus on 0) gives us the value $\\begin{array} { r } { \\dot { a } _ { i } = \\frac { 1 } { v _ { i } + 1 } } \\end{array}$ at each location $i$ , from which $v _ { i }$ can then be recovered with a simple zipmap. ", + "bbox": [ + 174, + 797, + 825, + 883 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/51a057ef1f3bf7ee24295b282a28a69d0d43e0750cdb71f345b40dcdb7b23fac.jpg", + "image_caption": [ + "Figure 5: The attention (left) and selection (right) patterns of a single-head transformer and singlehead RASP program both trained/written for the task of in-place histograms. We visualise both on the input sequence §jibbiejig. On the y axis, the sequence acts as the query, describing the output locations for which new values must be computed, and on the $\\mathbf { X }$ axis it acts as the key, describing the input locations being selected to compute these new values. Specifically, each row in the attention pattern describes the self-attention distribution of this head over the sequence §jibbiejig. The transformer has clearly learned the same pattern as that used by the RASP program. " + ], + "image_footnote": [], + "bbox": [ + 181, + 107, + 457, + 316 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/6ce7b3a2e50844c8e74777ee7640d0860fdb2650f2ce8c55f83bbc33211e99ff.jpg", + "image_caption": [ + "Figure 6: Building the RASP sequence histogram_given_BOS that computes an in-place histogram on the input tokens, under the assumption that the first token is always a special beginning-of-sequence character §. The selector same_and_0 builds on the same intuition as the s_with_0 selector from the implementation of count_conditioned. " + ], + "image_footnote": [], + "bbox": [ + 562, + 104, + 805, + 306 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "1 same_and_0 $=$ select ( tokens ,( tokens , indices ) , \n2 lambda t1 ,t2 ,i:( $\\displaystyle \\ t 1 = = \\ t 2$ ) or ( $\\mathrm { i } = = 0 )$ ) \n3 inverted_count $=$ aggregate ( same_and_0 , indices , \n4 lambda i: int( $\\mathrm { i } = = 0 \\mathrm { i }$ ) ) \n5 histogram_given_BOS $=$ (1/ inverted_count ) -1 ", + "bbox": [ + 151, + 455, + 660, + 521 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The direct parallel between our program’s only selector and our trained transformer’s attention pattern (Figure 5) suggests that this RASP program describes the exact mechanism that our transformer has discovered. ", + "bbox": [ + 174, + 628, + 823, + 671 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We present further experiments on additional tasks in Appendix B. ", + "bbox": [ + 173, + 678, + 609, + 693 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 720, + 328, + 737 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We abstract the computation model of the Transformer-encoder into a simple sequence processing language that captures the constraints on information flow in a Transformer. Considering computation problems and their implementation in the RASP language allows us to “think like a transformer” while abstracting away the technical details of a neural network in favor of symbolic programs. Moreover, provided it uses reasonable element-processing functions, we can analyze any RASP program to infer the minimum number of layers and maximum number of heads required to implement it in a transformer. We show several examples of expressive programs written in the RASP language, showing how complex operations can be implemented by a transformer. 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", + "bbox": [ + 173, + 425, + 826, + 467 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A TRANSFORMERS IN RASP ", + "text_level": 1, + "bbox": [ + 178, + 491, + 429, + 507 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "RASP is almost – but not quite – a strict over-approximation of transformer-encoders. In this section, we show how the addition of score, a generalisation of select that may assign non-boolean values to index pairs, makes RASP a strict over-approximation. In particular, we give an explicit translation from any given transformer-encoder to a RASP program, provided the augmentation with score. If the reader prefers to start there, a full description of transformers is given in section D (with notations in the preceding section). ", + "bbox": [ + 174, + 522, + 826, + 606 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We now introduce the score operation, and expand aggregate to receive a scorer: ", + "bbox": [ + 173, + 613, + 723, + 627 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "• score Similarly to select, the score operation takes two tuples of sequences $\\mathfrak { m } \\mathbf { e } { = } ( \\mathfrak { m } \\mathbb { 1 } , \\ldots , \\mathfrak { m } \\mathbf { k } )$ and other $=$ (ot1,...,otl), and a function f expecting $\\mathsf { k } { + } \\mathsf { 1 }$ atomic values. This time however, f may return any non-negative float value. It creates from these a scorer similarly to how select creates a selector from its inputs. ", + "bbox": [ + 218, + 637, + 826, + 693 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "• aggregate The aggregate operation is expanded such that it may receive either a scorer or a selector where it previously accepted only a selector. When it receives a scorer, each y[i] is assigned the weighted average of all the values of $\\mathsf { x }$ , according to the values in the scorer: ", + "bbox": [ + 220, + 696, + 825, + 752 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/798b34723a08407cac986c1773e7382bbd2c8fdd42c9fff8524a698ed6a6e6b5.jpg", + "text": "$$\n\\mathsf { y } [ \\mathrm { i } ] = \\frac { \\sum _ { \\mathsf { j } \\in [ n ] } \\mathsf { s } ( \\mathrm { i } , \\mathrm { j } ) { \\cdot } \\mathsf { x } [ \\mathrm { j } ] } { \\sum _ { \\mathsf { j } \\in [ n ] } \\mathsf { s } ( \\mathrm { i } , \\mathrm { j } ) }\n$$", + "text_format": "latex", + "bbox": [ + 426, + 747, + 629, + 787 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Intuitively, the select operation, and the application of aggregate to a selector, can be seen as the special case of a score-select pair in which the scorer has only given scores of 0 and 1. It does have one difference in that it also allows for a default value which may be used when the total score for some index i is 0. However, this can be seen as sugar: it is not difficult to create a mechanism similar to that of count_conditioned in order to recognize when a selector has width 0, and so avoid the direct use of default in aggregate7. Moreover, if the data is always given with a special beginning-of-sequence (BOS) token, then the default value can simply be loaded from that location whenever no other focus is found. ", + "bbox": [ + 173, + 792, + 825, + 904 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "1 def elementwise_part (x,att_y , el_funcs ) : \n2 lA , n2 , ff $=$ el_funcs \n3 def apply_elementwise_parts (xy) : \n4 $\\mathrm { ~ \\tt ~ { ~ x ~ y ~ } ~ } =$ tovec (xy) \n5 x, att_y $=$ first_half (xy) , second_half (xy) \n6 x1 = x + lA( att_y ) # vector addition \n7 res = x1 $^ +$ ff(n2(x1) ) \n8 return totuple ( res ) \n9 return zipmap (x+att_y , apply_elementwise_parts ) \n10 \n11 \n12 def layer (x, head_weights ,n1 , el_funcs ) : \n13 multihead_y $=$ ( ) \n14 for lq ,lk ,lv in head_weights : \n15 s $=$ score (x,x, lambda xixj : \n16 exp ( scalarprod ( \n17 lq(n1( first_half ( xixj ) ) ) , \n18 lk(n1( second_half ( xixj ) ) ) ) ) \n19 head_y $=$ aggregate (s,x,lv) \n20 multihead_y $=$ multihead_y $^ +$ head_y \n21 # $^ +$ here is concatenating tuples of sequences \n22 return elementwise_part (x, multihead_y , el_funcs ) \n23 \n24 def T_y0 ( layer_funcs ,w,p) : \n25 $\\qquad \\times \\quad =$ zipmap (( indices , tokens ) ,lambda i, $\\mathsf { t } : \\mathsf { p } \\left( \\mathrm { i } \\right) + \\mathsf { w } \\left( \\mathrm { t } \\right) \\big )$ \n26 for head_weights ,n1 , el_funcs in layer_funcs : \n27 x = layer (x, head_weights ,n1 , el_funcs ) \n28 return x # tuple of d float sequences ", + "bbox": [ + 147, + 108, + 620, + 460 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Theorem A.1. Let $\\mathcal { T } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ be a transformer and $y _ { 0 } : \\Sigma \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ be an input embedding computed as the sum of a token and positional embedding, $y _ { 0 } ( x ) _ { i } = w ( x _ { i } ) + p ( i )$ . Then the computation of $\\mathcal { T } _ { y _ { 0 } } \\triangleq y _ { 0 } \\circ \\mathcal { T }$ can be mimicked in a RASP program that writes the output to $d$ float-sequences and uses exactly $L H$ score and aggregate calls and $( H + 1 ) L + 1$ zipmap calls, where $L$ and $H$ are the number of layers and attention heads in $\\tau$ , respectively. ", + "bbox": [ + 173, + 575, + 825, + 650 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof Sketch In figure 7 we present code that, given the token and positional embedding $\\boldsymbol { w } : \\Sigma \\to \\mathbb { R } ^ { d }$ and $\\dot { p } : \\mathbb { N } \\to \\mathbb { R } ^ { d }$ and all the weights of a transformer, recreates that same transformer in RASP. Our code relies on the helper functions tup2vec, vec2tup, first_half and second_half which help convert between the tuples of values given to the processing functions by zipmap and aggregate and the vectors they represent. For simplicity in the presentation, we assume here that the transformer is given as a collection of linear transformations, layer-norms, and feed-forward functions which can be applied to its internal vectors directly. ", + "bbox": [ + 173, + 665, + 825, + 762 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The main routine is $\\mathsf { T } \\_ \\mathsf { y } \\boldsymbol { \\theta }$ , which applies the initial embedding and stores it in a tuple of $d$ sequences, $\\mathsf { x }$ , and then applies each layer to it in turn. This takes $L$ calls to the layer function, each of which computes score-aggregate (with additional zipmap before the aggregate) pair $H$ times to mimic each of the heads, and then calls a zipmap on the concatenation of their result to complete the remaining (elementwise) computations of the layer. ", + "bbox": [ + 174, + 768, + 825, + 839 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Note. Recall that, as noted in $\\mathrm { E }$ , when a processing function passed to zipmap returns multiple values, zipmap simply generates that same number of sequences. In particular, at all iterations of the loop in $\\mathsf { T } _ { - } \\mathsf { y } \\mathsf { 0 } , \\mathsf { x }$ is a tuple of $d$ sequences where $d$ is the embedding dimension of the given transformer. ", + "bbox": [ + 174, + 847, + 825, + 888 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We see that, when augmented with score, the RASP language naturally composes the components of a given transformer to reconstruct it exactly. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Why not have score? The motivation for the omission of score from RASP is cleanliness: it is far easier to think in terms of select as opposed to score, and we have not yet encountered a problem where we used scorers whose values where outside of 0 and 1. In time, as we use the language more and encounter the limitations this choice poses, we may return to score and see what other kind of special cases of it we will benefit from including in RASP. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Power of RASP As seen in this section, RASP can (provided this slight generalization of select) represent any transformer. Additionally, we see that it is not arbitrarily overpowered. For example, it does not allow iterating over a sequence of arbitrary length one-by-one to perform some gradual computation (as might be done in RNN or DFA), and in general does not allow arbitrary repetition of operations as other languages might allow. This is because the number of operations in a RASP program is predetermined: RASP programs do not admit loops. This distinction between transformers and RNNs is known, and there is interest in bridging it. For example, the Universal Transformer attempts to introduce loops into transformers, by allowing them to also have a control mechanism that decides whether to repeat a layer during computation (Dehghani et al., 2018). ", + "bbox": [ + 174, + 189, + 825, + 314 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 334, + 330, + 351 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For RASP to be useful, it is important to see that RASP programs relate well to actual transformer behavior in practice. In this section, we train transformers on a small set of synthetic tasks for which we can write RASP programs, and see how the these programs relate to our empirical results. ", + "bbox": [ + 176, + 366, + 825, + 409 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We consider the following tasks: ", + "bbox": [ + 174, + 415, + 388, + 430 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. Count-a: Return the number of $\\prime _ { a } \\prime$ tokens in a sequence, e.g., aba $\\mapsto 2$ . \n2. Histogram: For each token, its number of repetitions in a sequence, presented in-place. For example, aababc $\\mapsto ( 3 , 3 , 2 , 3 , 2 , 1 )$ . \n3. Reverse: Reversing a sequence, e.g.: abc $\\mapsto$ cba. ", + "bbox": [ + 210, + 441, + 825, + 508 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For Histogram, we also consider a variant with a special beginning-of-sequence (BOS) token §, appearing exactly once at the beginning of each sequence (and nowhere else). For example, §aabc $\\mapsto$ $( 1 , 2 , 2 , 1 , 1 )$ . ", + "bbox": [ + 178, + 520, + 825, + 563 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We train transformers for each of these tasks, and test whether our RASP programs accurately predict the minimum number of heads and layers needed to perform them. We also visualise their attention distributions8, and see if they match the selectors used by our programs. ", + "bbox": [ + 174, + 569, + 825, + 612 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We find that several of the RASP programs presented in this paper show similar attention (selector) patterns to those of the trained transformers in practice, suggesting that programming in RASP helps us provide reasonable predictions of transformer behavior. Moreover, we often find that reducing the number of heads and layers in a transformer beyond the number needed in our RASP program for the same task significantly degrades its accuracy. This suggests that the specific programs we have presented in for these tasks are also optimal solutions. ", + "bbox": [ + 174, + 618, + 825, + 702 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Data and Evaluation Unless stated otherwise: for all of the languages, we use the alphabet $\\{ \\mathsf { a } , \\mathsf { b } , \\mathsf { c } , \\mathsf { d } , \\mathsf { e } \\}$ with sequences of sizes 1 through 100. These are generated by first choosing the length uniformly from 1 to 100, and then choosing each token uniformly from the alphabet. We use 50, 000 train samples, $1 , 0 0 0$ test samples, and 1, 000 validation samples. ", + "bbox": [ + 174, + 718, + 825, + 773 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For tasks giving a ‘single’ output value – such as Count-a (which gives 1 number) as opposed to Reverse which gives a new sequence – we train the network to return that value in all positions, e.g., ${ \\mathsf { a b a } } \\mapsto ( 2 , 2 , 2 )$ for Count-a. This makes the visualisation of the attention distributions clearer (as all locations are trying to do something meaningful, as opposed to one), and is also more clearly aligned with the tasks we have described in this work. ", + "bbox": [ + 174, + 780, + 825, + 851 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We measure the accuracy of a transformer on a batch of sequences as the fraction of total predictions it made for those sequences that were correct, e.g. $\\textstyle { \\frac { x } { 5 } }$ for a batch with total sequence length 5. For the train, set, and validation sets, we report accuracy as the average batch accuracy. ", + "bbox": [ + 176, + 857, + 823, + 900 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Architecture We use the transformer architecture provided with PyTorch 1.7.0, with an additional single linear transformation and softmax at the end to convert to the output classes prediction. Unless stated otherwise, we use small embedding and feed-forward dimensions: 20 and 40, respectively. We vary the number of heads and layers per task. ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Training Method We train with the ADAM optimizer, sin-cosine positional embedding, and dropout 0.1 in the transformer weights. We use learning rate 0.0003, batch size 50, and PyTorch’s ExponentialLR learning rate scheduler with gamma 0.98 (updated after every epoch). Excluding confirmation of a negative result, we train each network for 20 epochs. If a network hits $1 0 0 \\%$ accuracy before then, we stop. ", + "bbox": [ + 174, + 174, + 825, + 244 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Helpful RASP functions In this section we will make frequent use of the full-selector full_s $=$ select((),(),lambda :True) and the function frac_condition(sequences,f), which computes aggregate(full_s,sequences,lambda a:int $( \\mathsf { f } ( \\star \\mathsf { a } ) ) ^ { 9 }$ – the fraction of input positions for which the f is satisfied on the values of sequences. Note that frac_condition requires only 1 layer (after sequences have been computed) with 1 head, and that that head is full_s. Recall also that length is computed: length $^ { = 1 }$ /frac_condition(indices,lambda $\\mathbf { i } : \\mathbf { i } = = \\mathbf { \\boldsymbol { \\theta } } ,$ ). ", + "bbox": [ + 174, + 260, + 825, + 344 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.1 COUNT-a ", + "text_level": 1, + "bbox": [ + 174, + 361, + 282, + 375 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "To avoid gradient problems from trying to obtain large numerical values from the transformer (e.g., 8), we encode Count-a as a categorical task. In particular, we create 21 output tokens $\\{ \\emptyset , 1 , 2 , \\dots , 2 \\emptyset \\}$ , and if there are more than $2 0 \\mathsf { a }$ tokens in the sequence we just report 20. ", + "bbox": [ + 174, + 387, + 826, + 429 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/4a3ade32726301aa70e8068b57ccfc230295de5eb8730001e338fadf242b80a8.jpg", + "image_caption": [ + "Figure 8: The attention distribution in the only head of a transformer trained on Count-a. The sequence is presented on the y-axis as the queries and on the $\\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. As predicted by our RASP program (which solves this task using full_s), this distribution is relatively uniform. " + ], + "image_footnote": [], + "bbox": [ + 357, + 445, + 635, + 654 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "RASP permits a 1-layer, 1-head program for this task: count_a $=$ lengthfrac_condition( tokens, lambda t: $\\tan \\tt { a } ^ { \\prime \\prime } \\tt { a } ^ { \\prime \\prime } .$ ). (length and frac_condition share the selector full_s.) Accordingly, training a transformer with 1 layer and 1 head on Count-a succeeded, reaching test accuracy $9 8 . 9 \\%$ on the $2 0 ^ { \\mathrm { t h } }$ epoch. ", + "bbox": [ + 174, + 741, + 825, + 797 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The attention pattern of this transformer on the input sequence abbaabcde is shown in Figure 8. While the distribution is not perfectly uniform, it does seem to focus on the entire sequence, as the use of full_s in our RASP program suggests (contrast for example with the attention distribution for Reverse, in Figure 12)10. ", + "bbox": [ + 174, + 805, + 825, + 861 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We stress that this is a different distribution to that which we might intuitively expect for this task – namely, an attention pattern focused solely on instances of a in the sequence – and that RASP has successfully pushed us to predict it! ", + "bbox": [ + 174, + 103, + 826, + 146 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.2 HISTOGRAM ", + "text_level": 1, + "bbox": [ + 174, + 167, + 305, + 181 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "As with Count-a, we encode histograms as a categorical task. This time we limit the maximum count to 10, i.e., we only use the output tokens $1 , 2 , \\ldots , 1 0$ . (0 is irrelevant as it will not appear in an in-place histogram). Unlike most other tasks, for Histograms we use an alphabet of size 10: $\\{ \\mathsf { a } , \\mathsf { b } , \\mathsf { \\ldots } , \\mathsf { j } \\}$ . For histograms, we use an input alphabet of size 10: $\\{ \\mathsf { a } , \\mathsf { b } , \\mathsf { \\ldots } , \\mathsf { j } \\}$ . ", + "bbox": [ + 173, + 195, + 826, + 252 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/129bcdeb0656cd1545705ffef4321dd806c71604c95e0d41796cc649612a5c55.jpg", + "image_caption": [ + "Figure 9: The attention distribution in the single head transformer trained on Histogram. The input sequence abcdeba is presented on the y-axis as the queries and on the $\\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. This transformer does not have enough heads: despite training for 50 epochs, it has only reached test accuracy $5 5 \\%$ , and generates the incorrect output $( 1 , 2 , 1 , 1 , 1 , 2 , 1 )$ for this sequence. Similarly, it does not manage to recreate the selection pattern predicted by our RASP program for this task, which requires 2 heads. " + ], + "image_footnote": [], + "bbox": [ + 325, + 271, + 671, + 472 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/fcbd4bf3c46c659bc9e85192f4cf71c3c99e1d806d1278c586753ee1f80c2e50.jpg", + "image_caption": [ + "Figure 10: The attention distribution in the 2-head, single-layer transformer trained on Histogram. The input sequence aabbccacde is presented on the y-axis as the queries and on the $\\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. The distribution is similar, but not identical, to our count_conditioned implementation. In particular, it seems that the tokens d and e are to some extent taking the role of the index 0 in our implementation: the first head has most tokens focus on themselves, d, and e, and the second head has most tokens focus slightly more on d and e than others. d and e themselves seem to behave inversely to the other tokens. " + ], + "image_footnote": [], + "bbox": [ + 258, + 597, + 785, + 808 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We trained one transformer with 1 layer and 2 heads, and another with only 1 layer and 1 head. After 20 epochs, the transformer with 2 heads reached test accuracy $8 9 . 3 \\%$ . In contrast, after 50 epochs, the transformer with only 1 head was still at accuracy $5 5 \\% !$ ! Increasing its embedding and feed-forward dimensions to 50 and 100 respectively also did not work: a 1-layer 1-head transformer with these dimensions still only reached $7 9 . 3 \\%$ test accuracy after 50 epochs, and this after being past $7 7 + \\%$ validation accuracy since the $2 7 ^ { \\mathrm { t h } }$ epoch. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Drawing the attention map for the single-head transformer (Figure 9) did not seem to relate to the selection pattern of count_conditioned at all, unsurprisingly considering that it did not have enough heads. See for example the apparent focus on b by several query positions not containing b, as opposed to our expectations of the count_conditioned focus pattern as shown in Figure 4. ", + "bbox": [ + 174, + 194, + 825, + 251 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For the 2-heads transformer, we draw its attention maps in Figure 10. The solution has some clear parallels with our RASP prediction of histogram selection patterns of count_conditioned, such as most tokens focusing on themselves in the one head and sharing distribution patterns in the other. But we can also easily find differences: in particular, the d and e tokens seem to avoid rather than focus on themselves, and the shared focus of most tokens in the similar-attentions head is not on 0 but on d and e. There is a possibility that in this transformer, d and e are playing a role similar to that which we gave to 0 in our implementation of count_conditioned. We leave a deeper exploration of this to future work. ", + "bbox": [ + 174, + 257, + 825, + 368 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.3 HISTOGRAM WITH BOS ", + "text_level": 1, + "bbox": [ + 176, + 397, + 383, + 411 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Recall the implementation of count_conditioned (Figure 3): it calculates the “width\" (number of selected locations) of a hypothetical selector s by simulating it along two actual selectors, s_with_0 and s_just_0. s_with_0 is used to calculate for every index the fraction $1 \\big / c _ { i } ^ { \\prime } { + 1 }$ where $c _ { i } ^ { \\prime }$ is the width of s on everything except index 0, and s_just_0 is used to make a final adjustment from $c _ { i } ^ { \\prime }$ to the actual width, depending on the focus on 0. ", + "bbox": [ + 174, + 428, + 825, + 497 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "If, then, the contents at position 0 are constant across all inputs, then the second selector s_just_0 becomes unnecessary: any information it conveys can be hard-coded into the RASP program (practically, the transformer). It follows that for setups where all input sequences are prepended with a special beginning-of-sequence (BOS) token, count_conditioned can be implemented with only one head, using just the s_with_0 selection pattern. ", + "bbox": [ + 173, + 505, + 825, + 574 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/88fb3fb4b9a69ff2144db012cb6e75a7e4501fd35a6b36fc57865a9a528636a0.jpg", + "image_caption": [ + "Figure 11: The attention distribution for a 1-head, 1-layer transformer trained on Histogram with BOS tokens. The input sequence jibbiejig is presented on the y-axis as the queries and on the $\\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. This transformer appears to have implemented Histogram (with BOS) using exactly the same s_with_0 selection pattern as suggested in our count_conditioned implementation! (The second selection pattern in the count_conditioned implementation, s_just_0, is unnecessary when the value at the 0 index is constant.)) " + ], + "image_footnote": [], + "bbox": [ + 361, + 599, + 637, + 810 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We prepend all of the original Histogram inputs with a special BOS token § (and their outputs with 1), and train a new 1-layer, 1-head transformer on the resulting data set. For gamma $= 0 . 9 9$ , the results satisfy our predictions perfectly: the transformer reaches ${ \\bar { 9 9 } } . 7 \\%$ test accuracy in 20 epochs, and drawing its attention distribution (Figure 11) shows that it follows exactly the pattern of the selector s_with_0! (For gamma $= 0 . 9 8$ the attention distribution was also very similar to that of s_with_0, but the model reached only $8 6 . 4 \\%$ test accuracy after 20 epochs.) ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Discussion of BOS The significance of such ‘non-input’ tokens in transformers has been previously discussed, with different interpretations. For example, Vig & Belinkov (2019) refer to the attention focused on the initial token of a sequence – seemingly when there is nothing else to focus on – as null attention. They report that the null token gathered as much as $9 7 \\%$ of the attention of some of their heads, and suggest that this is consistent with these heads being unimportant to the transformer’s overall performance. Conversely, this new result suggests that the null token at the beginning of a sequence is playing an important role in the transformer calculations, and in particular is directly assisting in counting! ", + "bbox": [ + 174, + 203, + 825, + 315 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.4 REVERSE ", + "text_level": 1, + "bbox": [ + 176, + 333, + 284, + 347 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In RASP, Reverse is implemented using flip_ $; =$ select(indices,length-1-indices,lambda ${ \\bf i } , { \\bf j } : { \\bf i } = = { \\bf j } ;$ ) followed by reverse $\\boldsymbol { \\underline { { \\underline { { \\mathbf { \\delta \\pi } } } } } } ,$ aggregate(flip_s,tokens). This takes two layers of one head each (recall that length itself requires one layer to compute). ", + "bbox": [ + 176, + 359, + 823, + 401 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/e1827b36e47fc786db62d84a2cbddcfae906f728573391e9202ff89c943181d5.jpg", + "image_caption": [ + "Figure 12: The attention distributions in the 2-layer, 1-head (per layer) transformer trained on Reverse. The first layer is on the left and the second on the right. The input sequence abcdeabcde is presented on the y-axis as the queries and on the $\\mathbf { X }$ -axis as the keys, i.e., each row is a distribution over the input positions. The attention distribution in the second layer is exactly as predicted by our RASP program (i.e., a hard, ‘reverse-match’ attention). In the first layer however, it seems the transformer has learned a behavior other than uniform attention to compute the length of the sequence (which it needs for the second layer). " + ], + "image_footnote": [], + "bbox": [ + 284, + 416, + 707, + 584 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We train two transformers on Reverse: one with 2 layers and 1 head, and the other with 1 layer and 2 heads, to verify that the separation into 2 layers is indeed necessary. To give room for the index-based selection pattern (i.e., a scoring method that involves comparison of indices and not just tokens), we give the transformers per-head width at least as large as our maximum length. In particular, we use embedding dimension 100 for the 2-layer transformer and 200 for the 1-layer transformer11. We also give them each feed-forward dimension twice their embedding dimension. ", + "bbox": [ + 174, + 712, + 825, + 796 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The 2-layer transformer reaches test accuracy $9 9 . 6 \\%$ after 20 epochs. In contrast, and as expected, the single-layer transformer remains trapped at $3 9 . 6 \\%$ test accuracy even after 50 epochs. Plotting the attention for the 2-layer transformer (Figure 12) matches some of the predictions of our RASP program: the reverse-matching attention is only computed in the $2 ^ { \\mathrm { n d } }$ layer, and computed perfectly at that point. We are inclined to believe the length is being computed at the first layer (as we predict). ", + "bbox": [ + 174, + 803, + 825, + 873 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/e4eb31866cecec1caca09b4c79cf27b5372473e299dd18ee8c8c8ae47a1d6b8c.jpg", + "image_caption": [ + "Figure 13: The attention distributions in the 1-layer, 1-head transformer trained on Reverse with fixed input length 50, on input sequences of length 45, 50, and 55, in order. The y-axes represent output locations (queries) and the $\\mathbf { X }$ -axes input locations (keys), i.e., each row is a distribution over the input positions. As expected, the transformer has learned a constant relation between pairs of input locations, and maintains all of the pairs it can find in each input sequence it gets. For missing pairs, such as $( 0 , 4 9 )$ through (4, 45) in the input of length 45 (left), it struggles to focus its attention. " + ], + "image_footnote": [], + "bbox": [ + 251, + 102, + 746, + 248 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "But there are also devitions from our prediction: the attention pattern suggests that the transformer is computing the length using a different mechanism from the one that we have suggested. ", + "bbox": [ + 171, + 380, + 823, + 409 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We now strengthen the claim that the initial layer of the Reverse transformer is computing the sequence length, by showing that when the sequence length is fixed, then a single-layer (and single-head) transformer does succeed on Reverse. We fix the sequence length to 50 and train a 1-layer 1-head transformer on Reverse. This simpler task can be presented in one layer and one head in RASP using the single selector flip50_s=select(indices,49-indices,lambda ${ \\mathrm { i } } , { \\mathrm { j } } : { \\mathrm { i } } = = { \\mathrm { j } } { \\mathrm { ) } }$ , from which the result is computed reverse $5 0 =$ aggregate(flip50_s,tokens). ", + "bbox": [ + 173, + 415, + 825, + 500 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As expected, the transformer succeeds in its task, reaching $1 0 0 \\%$ test accuracy in only 3 epochs. In Figure 13 we illustrate that it has indeed learned a constant $( i , 4 9 - i )$ location pairing, by visualising its attention on sequences slightly longer or shorter than it has been trained on. ", + "bbox": [ + 174, + 506, + 825, + 547 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For completeness, in Figure 14 we show also the attention patterns of the single-layer transformer trained on variable-length Reverse. In keeping with our predictions, it has not managed to learn the reverse-matching attention pattern at all. This is because it needs an additional layer to compute the length before it can create the correct attention pattern. ", + "bbox": [ + 174, + 554, + 825, + 611 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/d6f9bcf3871f784bd1bbff0bb71308e6ccf295939533d58fcde72b3abc37d3aa.jpg", + "image_caption": [ + "Figure 14: The attention distributions of the 2 heads in the 1-layer transformer trained on Reverse, as applied to the sequence abcdeabcde. As expected from the RASP program, the transformer is unable to learn the reverse-match attention on any head of its first layer. " + ], + "image_footnote": [], + "bbox": [ + 343, + 631, + 671, + 768 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C NOTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 872, + 305, + 888 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Basic Notations For every $n \\in \\mathbb { N }$ , we denote $[ n ] = \\{ 1 , . . . , n \\}$ ", + "bbox": [ + 173, + 909, + 593, + 925 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Matrices For a matrix $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ , we refer to its $i$ -th row as $X _ { i } \\in \\mathbb { R } ^ { d }$ , and for a vector $v \\in \\mathbb { R } ^ { d }$ we refer to its $i$ -th value as $v _ { i } \\in \\mathbb { R }$ . We additionally refer to $n$ as the length of $X$ and abusively denote $| X | = n$ . ", + "bbox": [ + 173, + 102, + 825, + 146 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ and $b \\in \\mathbb { R } ^ { d }$ , we use the shorthand $X + b$ to describe the addition of $b$ to each of the rows of $X$ , i.e.: $( X + b ) _ { i } = X _ { i } + b$ for every $i \\in [ n ]$ . For a scalar $\\alpha \\in \\mathbb { R }$ , any operation $X \\odot \\alpha , \\odot \\in \\{ + , - , \\times , \\div \\}$ is applied elementwise to all the values of $X$ . Matrix multiplication between two matrices $A , B$ is denoted simply $A B$ , and the transpose of a matrix $A$ is denoted $A ^ { T }$ . ", + "bbox": [ + 174, + 151, + 825, + 209 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We occasionally treat input or output sequences $x _ { 1 } , . . . , x _ { n } \\in \\mathbb { R } ^ { d }$ as matrices $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ whose rows are the individual input vectors: $X _ { i } = x _ { i }$ . When $n$ may be arbitrary, we will say that $X \\in ( \\mathbb { R } ^ { d } ) ^ { * }$ . ", + "bbox": [ + 171, + 214, + 823, + 244 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Definition C.1. $A$ linear transformation with input dimension $d$ and output dimension m is a function $l _ { A , b } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { m } ) ^ { * }$ parameterised by a matrix $A \\in \\mathbb { R } ^ { d \\times m }$ and vector $b \\in \\mathbb { R } ^ { m }$ as follows: $l _ { A , b } ( X ) = X A + b$ for every $X \\in ( \\mathbb { R } ^ { d } ) ^ { * }$ . $A$ and $b$ are the weights of the transformation. ", + "bbox": [ + 173, + 248, + 826, + 292 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Definition C.2. The softmax function s : $\\mathbb { R } ^ { * } \\to \\mathbb { R } ^ { * }$ is defined: for every $d \\in \\mathbb { N }$ and $x \\in \\mathbb { R } ^ { d }$ , $\\mathbf { s } ( x ) \\in \\mathbb { R } ^ { d }$ such that $\\begin{array} { r } { \\mathrm { s } ( x ) _ { i } = \\frac { \\bar { e } ^ { x _ { i } } } { \\sum _ { j \\in [ d ] } e ^ { x _ { j } } } } \\end{array}$ = e P xij∈[d] exj for every i ∈ [d]. We also denote by S the row-wise softmax function: for every $n , d \\in \\mathbb { N }$ and $\\dot { X } \\in \\mathbb { R } ^ { n \\times d }$ , $S ( X ) \\in \\mathbb { R } ^ { n \\times d }$ , and ${ \\cal { S } } ( X ) _ { i } = \\mathrm { s } ( X _ { i } )$ for every $i \\in [ n ]$ . ", + "bbox": [ + 173, + 297, + 826, + 351 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We denote by ${ \\mathcal { R } } ( X )$ the elementwise application of the ReLU function, $r : x \\mapsto \\operatorname* { m a x } ( 0 , x )$ , to $X$ . ", + "bbox": [ + 171, + 361, + 816, + 376 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Function Qualities A function $f : A ^ { * } \\to B ^ { * }$ is length preserving if it satisfies $| f ( x ) | = | x |$ for any $x \\in ( \\mathbb { R } ^ { d } ) ^ { * }$ (i.e., for any input sequence $x _ { 1 } , . . . , x _ { n } \\in A$ , $f$ returns a sequence $y _ { 1 } , . . . , y _ { n } \\in B$ ). If there also exists a function $g : A B$ such that $f ( x ) _ { i } = g ( x _ { i } )$ for any $i \\leq | x |$ , then $f$ is elementwise, and we say that $f$ is an elementwise application of $g$ . Note that linear transformations are elementwise. ", + "bbox": [ + 173, + 390, + 825, + 448 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "D TRANSFORMER-ENCODERS ", + "text_level": 1, + "bbox": [ + 176, + 468, + 439, + 484 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "At the highest level, a transformer-encoder $T$ (henceforth, a transformer) is a parameterised lengthpreserving function $T : ( \\mathbb { R } ^ { d } ) ^ { * } ( \\mathbb { R } ^ { d } ) ^ { * }$ composed of multiple layers of length-preserving functions $\\dot { \\ell } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ , i.e. $f = \\ell _ { 1 } \\circ \\ell _ { 2 } \\dots \\circ \\ell _ { L }$ which we will describe in this section. ", + "bbox": [ + 174, + 500, + 826, + 542 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Generally speaking, a transformer’s layers are not elementwise, and indeed the transformer would not be interesting if they were. However, when we come to look at their components, we see that this quality rests entirely only on their use of attention12. ", + "bbox": [ + 174, + 547, + 825, + 592 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Attention Attention is a function devised to enable ‘recollection’ of previously processed data from a history of arbitrary length (Bahdanau et al., 2015; Luong et al., 2015). Transformers use a variant called scaled dot-product attention to collect data from multiple locations in an input sequence. ", + "bbox": [ + 174, + 606, + 825, + 648 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Definition D.1. Scaled Dot-Product Attention is a function a $: ( \\mathbb { R } ^ { d } ) ^ { * } ( \\mathbb { R } ^ { m } ) ^ { * }$ parameterised by 3 linear transformations, $l _ { Q } , l _ { K } , l _ { V } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { m } ) ^ { * }$ and defined for every $X \\in \\mathsf { ( } \\mathbb { R } ^ { d } ) ^ { * }$ as follows: ", + "bbox": [ + 173, + 652, + 823, + 683 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/16fa50a212a0a666f63e1fc19bf48a131296732dd6b96062cee380176701b89c.jpg", + "text": "$$\n\\mathrm { a } ( X ) = \\mathcal { S } \\bigg ( \\frac { l _ { Q } ( X ) l _ { K } ( X ) ^ { T } } { \\sqrt { m } } \\bigg ) l _ { V } ( X )\n$$", + "text_format": "latex", + "bbox": [ + 377, + 689, + 620, + 724 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Note. The original definition of scaled dot-product attention allows $l _ { Q } , l _ { K }$ , and $l _ { V }$ to have different output dimensions $m$ . In this case, the denominator is $\\sqrt { m _ { k } }$ . ", + "bbox": [ + 171, + 738, + 825, + 767 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For convenience, from here on we refer to scaled dot-product attention simply as attention. ", + "bbox": [ + 169, + 773, + 767, + 789 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The attention computation can be broken into 3 stages. First, a pairwise score is calculated for each pair of locations, showing how much the input in location $j$ should influence the output in location $i$ : this is the value $\\mathbb { S } _ { i , j }$ in the matrix $\\begin{array} { r } { \\mathbb { S } = \\frac { l _ { Q } ( X ) l _ { K } ( X ) ^ { T } } { \\sqrt { m } } } \\end{array}$ . Then, each input is processed in-place $( l _ { V } ( X ) )$ to create candidate outputs, and finally the candidate outputs are averaged for each output location $i$ , according to the softmaxed scores $S ( \\mathbb { S } ) _ { i }$ for that location. In this sense, attention can be seen as a request to gather into each output information from various locations, where $l _ { Q }$ and $l _ { K }$ work together to select information sources, and $l _ { V }$ encodes the transferred information. ", + "bbox": [ + 173, + 794, + 826, + 900 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Transformer layers often gather information with multiple attention functions, referred to as attention heads, whose results are concatenated back into a single output at the end: ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Definition D.2. Let $d , H , m \\in \\mathbb { N }$ be such that $d = H m .$ . $A$ multi-headed attention function with input dimension $d$ and $H$ heads is a function $\\mathcal { A } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ parameterised by the weights of $H$ scaled dot-product attention functions $\\mathbf { a } _ { 1 } , . . . , \\mathbf { a } _ { H }$ as follows: 13 for every $X \\in ( \\mathbb { R } ^ { \\dot { d } } ) ^ { * }$ , ", + "bbox": [ + 174, + 135, + 825, + 178 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/fa4348d7defd9335552388d0d24790f1e47886a3e1d916b46caca5c1be57fb53.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { \\mathcal { A } } ( \\boldsymbol { X } ) = \\mathrm { a _ { 1 } } ( \\boldsymbol { X } ) { \\cdot } \\mathrm { a _ { 2 } } ( \\boldsymbol { X } ) { \\cdot } . . . { \\cdot } \\mathrm { a } _ { H } ( \\boldsymbol { X } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 385, + 180, + 612, + 196 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\cdot$ denotes row-wise concatenation, i.e. for every $i \\in [ | X | ]$ , ${ \\mathcal { A } } ( X ) _ { i }$ is the concatenation of $\\operatorname { a } _ { 1 } ( X ) _ { i }$ through $\\mathrm { a } _ { H } ( X ) _ { i }$ . The functions $\\mathbf { a } _ { 1 } , . . . , \\mathbf { a } _ { H }$ are referred to as the heads of $\\mathcal { A }$ . ", + "bbox": [ + 174, + 199, + 825, + 229 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In addition to attention, transformers use layer-norm and feed-forward components, as follows: Definition D.3. $A$ single-row layer-norm over dimension $d$ is a function $g : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ parameterised by vectors $a , b \\in \\mathbb { R } ^ { d }$ and constant $\\varepsilon \\in \\mathbb { R }$ , and defined for every $x \\in \\mathbb { R } ^ { d }$ and $i \\leq d$ as follows: ", + "bbox": [ + 171, + 237, + 826, + 285 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f345055cff1703fef28cdadfd5ba080fed173a0acf3203be53545b7d3fda2c4f.jpg", + "text": "$$\ng ( x ) _ { i } = { \\frac { a _ { i } ( x _ { i } - { \\bar { x } } ) } { \\mathrm { s t d } ( x ) + \\varepsilon } } + b _ { i }\n$$", + "text_format": "latex", + "bbox": [ + 413, + 286, + 583, + 320 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\bar { x } = \\frac { \\sum _ { j \\in [ d ] } x _ { j } } { d } } \\end{array}$ is the mean of x and $\\begin{array} { r } { \\mathrm { s t d } ( x ) = \\sqrt { \\frac { 1 } { d - 1 } \\sum _ { i \\leq d } ( x _ { i } - \\bar { x } ) ^ { 2 } } } \\end{array}$ is its standard deviation. ", + "bbox": [ + 174, + 323, + 825, + 348 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "A layer-norm over dimension $d$ $, \\mathbf { n } : ( \\mathbb { R } ^ { d } ) ^ { * } ( \\mathbb { R } ^ { d } ) ^ { * }$ , is an elementwise application of a single-row layer-norm of dimension $d$ . ", + "bbox": [ + 173, + 354, + 823, + 386 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The layer-norm’s function is more a reguliser, and indeed, it will not play a part in our abstraction. ", + "bbox": [ + 176, + 393, + 818, + 410 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Definition D.4. $A$ feed-forward function with input dimension d and internal dimension m is an elementwise function $\\mathcal { F } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ obtained by composing two linear transformations $L _ { 1 } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { m } ) ^ { * } , L _ { 2 } : ( \\mathbb { R } ^ { m } ) ^ { * } \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ and ReLU, as follows:14 ${ \\mathcal { F } } ( X ) \\triangleq L _ { 2 } ( { \\mathcal { R } } ( L _ { 1 } ( X ) ) )$ . ", + "bbox": [ + 173, + 412, + 825, + 458 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The feed-forward component is elementwise, and the combination of two linear transformations with nonlinear activation provides strong expressive capacity (Hornik et al., 1989). ", + "bbox": [ + 173, + 467, + 823, + 496 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a8eb6ff14a4f20c9949ea6dcddfe31bc73790c363bb4c4e033e3b880eb04ea4c.jpg", + "text": "$$\n\\begin{array} { r } { X _ { 1 } = X + l _ { A } ( \\mathcal { A } ( \\mathbf { n } _ { 1 } ( X ) ) ) } \\\\ { \\ell ( X ) = X _ { 1 } + \\mathcal { F } ( \\mathbf { n } _ { 2 } ( X _ { 1 } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 408, + 570, + 589, + 608 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We refer to the additions in both equations as a ‘skip connection’. Note that the layernorm, feedforward, and skip connection components of the layer are elementwise: were it not for the attention, the entire layer would be elementwise. ", + "bbox": [ + 176, + 616, + 826, + 659 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Finally, we present the full encoder architecture: ", + "bbox": [ + 174, + 665, + 490, + 679 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Definition D.6 (Transformer-Encoder Vaswani et al. (2017)). A transformer-encoder with $L$ layers, $H$ heads, and input and internal dimensions $d , m$ is a length-preserving function $\\mathcal { T } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\dot { \\mathbb { R } } ^ { d } ) ^ { * }$ parameterised by the weights of $L$ transformer-encoder layers $\\ell _ { 1 } , . . . , \\ell _ { L }$ , each with $H$ heads and input and internal dimensions $d , m$ , and defined for every $X \\in ( \\mathbb { R } ^ { d } ) ^ { * }$ as follows: ", + "bbox": [ + 173, + 683, + 826, + 738 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/d71589b3502d58d6f62395c1c9f83c44642d0f57508a51c7038a42b0cb9a379a.jpg", + "text": "$$\n\\mathcal { T } ( X ) = \\ell _ { L } ( . . . \\ell _ { 2 } ( \\ell _ { 1 } ( X ) ) )\n$$", + "text_format": "latex", + "bbox": [ + 410, + 741, + 586, + 758 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Permutation Invariance of Transformers An interesting trait of the transformer architecture is that it has no inherent positional awareness. Specifically: for any transformer $\\mathcal { T } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ , input sequence $\\mathbf { x } = x _ { 1 } , x _ { 2 } , . . . , x _ { n } \\in \\mathbb { R } ^ { d }$ , and permutation $\\pi$ , we have ${ \\mathcal T } ( \\pi ( \\mathbf { x } ) ) = \\pi ( { \\mathcal T } ( \\mathbf { x } ) ) ^ { \\mathrm { ~ l ~ } }$ 15 . ", + "bbox": [ + 173, + 768, + 826, + 811 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Discrete Input Transformers $\\mathcal { T } : ( \\mathbb { R } ^ { d } ) ^ { * } \\to ( \\mathbb { R } ^ { d } ) ^ { * }$ are used to process non-empty sequences over a finite alphabet $\\Sigma$ by composing them with a simple length-preserving function, $\\mathsf { \\bar { y } _ { 0 } : \\dot { Z } ^ { + } ( \\mathbb { R } ^ { d } ) ^ { * } }$ : ${ \\mathcal T } _ { y _ { 0 } } ( x ) \\triangleq { \\mathcal T } ( y _ { 0 } ( x ) )$ . This $y _ { 0 }$ is in turn composed from a token embedding $\\boldsymbol { w } : \\Sigma \\to \\mathbb { R } ^ { d }$ and position embedding $p : \\mathbb { N } \\to \\mathbb { R } ^ { d }$ , which are normally combined using addition: for every $x = x _ { 1 } . . . x _ { n } \\in \\Sigma ^ { * }$ , $y _ { 0 } ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } ) _ { i } = w ( x _ { i } ) + p ( i ) ^ { 1 6 }$ . ", + "bbox": [ + 173, + 102, + 826, + 179 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "From here, whenever we refer to a ‘transformer over (some finite alphabet) $\\Sigma ^ { \\prime }$ , we mean a transformer paired with an initial embedding $y _ { 0 }$ as described above. ", + "bbox": [ + 174, + 184, + 825, + 213 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E ADDITIONAL DETAILS ABOUT RASP ", + "text_level": 1, + "bbox": [ + 173, + 234, + 501, + 251 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "A note about types in RASP Technically, there is no one ‘tokens’, rather 5 options: tokens_str, tokens_int, tokens_float and tokens_bool cast the input sequence to the corresponding atomic types, and tokens_asis takes the input sequence as-is. For brevity, we refer to all of these as tokens here. ", + "bbox": [ + 174, + 266, + 825, + 321 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "RASP Sequences RASP operates exclusively on sequence- and selector-generating functions, which we refer to as sequences and selectors respectively and RASP-functions together. All RASP-functions take as input exactly one non-empty sequence, and when describing them and how RASP manipulates them to create new RASP-functions we will do so in terms of the sequences and selectors that they and their manipulations generate from each input sequence. When it is clear from context, we will simply refer to them as sequences and selectors, and describe them directly in terms of their outputs. For example, if we say that a RASP operation applies $+ 1$ elementwise to each value in a sequence u, we actually mean that it returns a new sequence v such that for every input $x$ and position $0 \\leq i < | x |$ , $\\mathsf { v } ( x ) [ i ] { = } \\mathsf { u } ( x ) [ i ] { + } 1$ . ", + "bbox": [ + 173, + 339, + 826, + 465 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Note In this section we will refer to the $i$ -th value in a sequence s as $\\mathsf { s } [ i ]$ , such that $\\mathsf { s } [ i ]$ is in one of the atomic types. We stress however that this is only for the discussion, and not a part of the language. ", + "bbox": [ + 174, + 476, + 823, + 503 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Additional operations For brevity in code, the RASP also comes with the following syntactic sugar: ", + "bbox": [ + 173, + 520, + 823, + 549 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Anywhere that a tuple of sequences is passed into an operation, a single sequence may be passed in as-is as well. For example, $y = z .$ ipmap((indices,),(indices,),lambda a $, b : a + b ) ,$ ) can also be written $y = z \\mathrm { i }$ pmap(indices,indices,lambda a, $b : a + b ) ,$ ). \n• The zipmap operation is accessible through a large range of operators, covering its application to all of the base binary and unary operations on the atomic types. For example, the above y can equivalently be defined as y=indices+indices. These operators can also be mixed with constants from the atomic primitives, such that an equal y be obtained using $y = 2 +$ indices. \n• aggregate may receive a tuple of sequences xx instead of a single sequence x. In this case, \nit returns a new tuple (of the same length) of sequences, the result of applying aggregate to each of the sequences in xx. For example, the line a,b $=$ aggregate(s,(x,y)) is equivalent to a,b $=$ aggregate $( s , \\mathsf { x } )$ , aggregate(s,y). \n• The processing function passed to zipmap may return more than one value (provided the number of values it returns is constant). In this case, the operation will arrange the output values into the same number of output sequences. For example, y1, $y 2 = 2$ ipmap(x,lambda $\\mathsf { v } : \\mathsf { v } ^ { + 1 } , \\mathsf { v } ^ { + 2 } )$ is equivalent to writing $y 1 = x + 1$ and then $y 2 = x + 2$ . \n• Anywhere that a processing function is expected, if one is not provided, the identity function \nis used17. ", + "bbox": [ + 217, + 560, + 825, + 837 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "$l _ { K } ( X ) , l _ { V } ( X )$ , and $l _ { Q } ( X ) _ { i }$ , where the order of the rows of $l _ { K } ( X )$ and $l _ { V } ( X )$ does not matter as long as they remain aligned. ", + "bbox": [ + 178, + 844, + 823, + 871 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "16Note that without the position embedding, $y _ { 0 }$ would be elementwise, and so its combination with $\\tau$ would be permutation invariant – an undesirable trait for sequence processing. ", + "bbox": [ + 173, + 872, + 821, + 897 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "17This is consistent with aggregate allowing you to choose whether to pass a single sequence, or a tuple of sequences and a processing function. ", + "bbox": [ + 171, + 898, + 826, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• The function select1 which takes only a single tuple of sequences xx and an indexcomputing function $\\mathsf { f i }$ , and is syntactic sugar for select(xx,(indices,),lambda $\\star a : f \\mathrm { i } ( \\mathsf { a } [ : - 1 ] ) = = a [ - 1 ] \\gamma$ . select1 is guaranteed to return a true select satisfying the \"up-to-one\" property, i.e., that can be successfully paired in aggregates with an $\\mathsf { x }$ that does not contain numbers. 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a/parse/train/iFODavhthGZ/iFODavhthGZ_span.pdf b/parse/train/iFODavhthGZ/iFODavhthGZ_span.pdf new file mode 100644 index 0000000000000000000000000000000000000000..716e882edcecad05d1b8aa7ac028961589e8b073 --- /dev/null +++ b/parse/train/iFODavhthGZ/iFODavhthGZ_span.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6b404b48d1d1492bbf4f5f7fdb45f329b0cc98a516e24b90c148dc437067d990 +size 1580755 diff --git a/parse/train/jh-rTtvkGeM/jh-rTtvkGeM.md b/parse/train/jh-rTtvkGeM/jh-rTtvkGeM.md new file mode 100644 index 0000000000000000000000000000000000000000..80d4f84537e13098387bce48faf722fe4bc283c4 --- /dev/null +++ b/parse/train/jh-rTtvkGeM/jh-rTtvkGeM.md @@ -0,0 +1,1118 @@ +# GRADIENT DESCENT ON NEURAL NETWORKS TYPICALLY OCCURS AT THE EDGE OF STABILITY + +Jeremy Cohen Simran Kaur Yuanzhi Li J. Zico Kolter1 and Ameet Talwalkar2 +Carnegie Mellon University and: 1Bosch AI 2 Determined AI +Correspondence to: jeremycohen@cmu.edu + +# ABSTRACT + +We empirically demonstrate that full-batch gradient descent on neural network training objectives typically operates in a regime we call the Edge of Stability. In this regime, the maximum eigenvalue of the training loss Hessian hovers just above the value 2/(step size), and the training loss behaves non-monotonically over short timescales, yet consistently decreases over long timescales. Since this behavior is inconsistent with several widespread presumptions in the field of optimization, our findings raise questions as to whether these presumptions are relevant to neural network training. We hope that our findings will inspire future efforts aimed at rigorously understanding optimization at the Edge of Stability. + +# 1 INTRODUCTION + +Neural networks are almost never trained using (full-batch) gradient descent, even though gradient descent is the conceptual basis for popular optimization algorithms such as SGD. In this paper, we train neural networks using gradient descent, and find two surprises. First, while little is known about the dynamics of neural network training in general, we find that in the special case of gradient descent, there is a simple characterization that holds across a broad range of network architectures and tasks. Second, this characterization is strongly at odds with prevailing beliefs in optimization. + +In more detail, as we train neural networks using gradient descent with step size $\eta$ , we measure the evolution of the sharpness — the maximum eigenvalue of the training loss Hessian. Empirically, the behavior of the sharpness is consistent across architectures and tasks: so long as the sharpness is less than the value $2 / \eta$ , it tends to continually rise (§3.1). We call this phenomenon progressive sharpening. The significance of the value $2 / \eta$ is that gradient descent on quadratic objectives is unstable if the sharpness exceeds this threshold (§2). Indeed, in neural network training, if the sharpness ever crosses $2 / \eta$ , gradient descent quickly becomes destabilized — that is, the iterates start to oscillate with ever-increasing magnitude along the direction of greatest curvature. Yet once this happens, gradient descent does not diverge entirely or stall. Instead, it enters a new regime we call the Edge of Stability1 (§3.2), in which (1) the sharpness hovers right at, or just above, the value $2 / \eta$ ; and (2) the train loss behaves non-monotonically, yet consistently decreases over long timescales. In this regime, gradient descent is constantly “trying” to increase the sharpness, but is constantly restrained from doing so. The net effect is that gradient descent continues to successfully optimize the training objective, but in such a way as to avoid further increasing the sharpness.2 + +![](images/3abb073f42e9b646f68463de4cad2806e2fd057b656681f0cbce75095a7b79b5.jpg) +Figure 1: Gradient descent typically occurs at the Edge of Stability. On three separate architectures, we run gradient descent at a range of step sizes $\eta$ , and plot both the train loss (top row) and the sharpness (bottom row). For each step size $\eta$ , observe that the sharpness rises to $2 / \eta$ (marked by the horizontal dashed line of the appropriate color) and then hovers right at, or just above, this value. + +In principle, it is possible to run gradient descent at step sizes $\eta$ so small that the sharpness never rises to $2 / \eta$ . However, these step sizes are suboptimal from the point of view of training speed, sometimes dramatically so. In particular, for standard architectures on the standard dataset CIFAR-10, such step sizes are so small as to be completely unreasonable — at all reasonable step sizes, gradient descent eventually enters the Edge of Stability (see $\ S 4$ ). Thus, at least for standard networks on CIFAR-10, the Edge of Stability regime should be viewed as the “rule,” not the “exception.” + +As we describe in $\ S 5$ , the Edge of Stability regime is inconsistent with several pieces of conventional wisdom in optimization theory: convergence analyses based on $L$ -smoothness or monotone descent, quadratic Taylor approximations as a model for local progress, and certain heuristics for step size selection. We hope that our empirical findings will both nudge the optimization community away from widespread presumptions that appear to be untrue in the case of neural network training, and also point the way forward by identifying precise empirical phenomena suitable for further study. + +Certain aspects of the Edge of Stability have been observed in previous empirical studies of fullbatch gradient descent (Xing et al., 2018; Wu et al., 2018); our paper provides a unified explanation for these observations. Furthermore, Jastrz˛ebski et al. (2020) proposed a simplified model for the evolution of the sharpness during stochastic gradient descent which matches our empirical observations in the special case of full-batch SGD (i.e. gradient descent). However, outside the full-batch special case, there is no evidence that their model matches experiments with any degree of quantitative precision, although their model does successfully predict the directional trend that large step sizes and/or small batch sizes steer SGD into regions of low sharpness. We discuss SGD at greater length in $\ S 6$ . To summarize, while the sharpness does not obey simple dynamics during SGD (as it does during GD), there are indications that the “Edge of Stability” intuition might generalize somehow to SGD, just in a way that does not center around the sharpness. + +# 2 BACKGROUND: STABILITY OF GRADIENT DESCENT ON QUADRATICS + +In this section, we review the stability properties of gradient descent on quadratic functions. Later, we will see that the stability of gradient descent on neural training objectives is partly well-modeled by the stability of gradient descent on the quadratic Taylor approximation. + +On a quadratic objective function $\begin{array} { r } { f ( \mathbf { x } ) = \frac { 1 } { 2 } \mathbf { x } ^ { T } \mathbf { A } \mathbf { x } + \mathbf { b } ^ { T } \mathbf { x } + c , } \end{array}$ , gradient descent with step size $\eta$ will diverge if3 any eigenvalue of A exceeds the threshold $2 / \eta$ . To see why, consider first the onedimensional quadratic $\textstyle f ( x ) = { \frac { 1 } { 2 } } a x ^ { 2 } + b x + c$ , with $a > 0$ . This function has optimum $x ^ { * } = - b / a$ . Consider running gradient descent with step size $\eta$ starting from $x _ { 0 }$ . The update rule is $x _ { t + 1 } =$ $x _ { t } - \eta ( a x _ { t } + b )$ , which means that the error $x _ { t } - x ^ { * }$ evolves as $( x _ { t + 1 } - x ^ { * } ) = ( 1 - \eta a ) ( x _ { t } - x ^ { * } )$ . Therefore, the error at step $t$ is $( x _ { t } - x ^ { * } ) = ( 1 - \eta a ) ^ { t } ( x _ { 0 } - \dot { x } ^ { * } )$ , and so the iterate at step $t$ is $x _ { t } = ( 1 - \eta a ) ^ { t } ( x _ { 0 } - x ^ { * } ) \bar { + } x ^ { * }$ . If $a > 2 / \eta$ , then $( 1 - \eta a ) < - 1$ , so the sequence $\{ x _ { t } \}$ will oscillate around $x ^ { * }$ with ever-increasing magnitude, and diverge. + +Now consider the general $d$ -dimensional case. Let $\left( a _ { i } , \mathbf { q } _ { i } \right)$ be the $i$ -th largest eigenvalue/eigenvector of $\mathbf { A }$ . As shown in Appendix $\mathbf { B }$ , when the gradient descent iterates $\left\{ \mathbf { x } _ { t } \right\}$ are expressed in the special coordinate system whose axes are the eigenvectors of A, each coordinate evolves separately. In particular, the coordinate for each eigenvector $\mathbf { q } _ { i }$ , namely $\left. \mathbf { q } _ { i } , \mathbf { x } _ { t } \right.$ , evolves according to the dynamics of gradient descent on a one-dimensional quadratic objective with second derivative $a _ { i }$ . + +Therefore, if $a _ { i } > 2 / \eta$ , then the sequence $\left\{ \left. \mathbf { q } _ { i } , \mathbf { x } _ { t } \right. \right\}$ will oscillate with ever-increasing magnitude; in this case, we say that the iterates $\left\{ { \bf x } _ { t } \right\}$ diverge along the direction $\mathbf { q } _ { i }$ . To illustrate, Figure 2 shows a quadratic function with eigenvalues $a _ { 1 } ~ = ~ 2 0$ and $a _ { 2 } ~ = ~ 1$ . In Figure 2(a), we run gradient descent with step size $\eta = 0 . 0 9$ ; since $0 < a _ { 2 } < a _ { 1 } < 2 / \eta$ , gradient descent converges along both ${ \bf q } _ { 1 }$ and $\mathbf { q } _ { 2 }$ . In Figure 2(b), we use step size $\eta ~ = ~ 0 . 1 1$ ; since $0 < a _ { 2 } < 2 / \eta < a _ { 1 }$ , gradient descent converges along $\mathbf { q } _ { 2 }$ yet diverges along ${ \bf q } _ { 1 }$ , so diverges overall. + +![](images/b1e8b3a66fc1d4a03837e613f4987a4ce7a363ef61bd9bcea1e3a8afc30dbe59.jpg) +Figure 2: Gradient descent on a quadratic with eigenvalues $a _ { 1 } = 2 0$ and $a _ { 2 } = 1$ . + +Polyak momentum (Polyak, 1964) and Nesterov momentum (Nesterov, 1983; Sutskever et al., 2013) are notable variants of gradient descent which often improve the convergence speed. On quadratic functions, these two algorithms also diverge if the sharpness exceeds a certain threshold, which we call the “maximum stable sharpness,” or MSS. In particular, we prove in Appendix B that gradient descent with step size $\eta$ and momentum parameter $\beta$ diverges if the sharpness exceeds: + +$$ +\mathbf { M S S } _ { \mathrm { P o l y a k } } ( \eta , \beta ) = \frac { 1 } { \eta } \left( 2 + 2 \beta \right) , \quad \mathbf { M S S } _ { \mathrm { N e s t e r o v } } ( \eta , \beta ) = \frac { 1 } { \eta } \left( \frac { 2 + 2 \beta } { 1 + 2 \beta } \right) . +$$ + +The Polyak result previously appeared in Goh (2017); the Nesterov one seems to be new. Note that this discussion only applies to full-batch gradient descent. As we discuss in $\ S 6$ , several recent papers have proposed stability analyses for SGD (Wu et al., 2018; Jastrz˛ebski et al., 2020). + +Neural network training objectives are not globally quadratic. However, the second-order Taylor approximation around any point $\mathbf { x } _ { \mathrm { 0 } }$ in parameter space is a quadratic function whose “A” matrix is the Hessian at $\mathbf { x } _ { \mathrm { 0 } }$ . If any eigenvalue of this Hessian exceeds $2 / \eta$ , gradient descent with step size $\eta$ would diverge if run on this quadratic function — the iterates would oscillate with ever-increasing magnitude along the corresponding eigenvector. Therefore, at any point $\mathbf { x } _ { \mathrm { 0 } }$ in parameter space where the sharpness exceeds $2 / \eta$ , gradient descent with step size $\eta$ would diverge if run on the quadratic Taylor approximation to the training objective around $\mathbf { x } _ { \mathrm { 0 } }$ . + +# 3 GRADIENT DESCENT ON NEURAL NETWORKS + +In this section, we empirically characterize the behavior of gradient descent on neural network training objectives. Section 4 will show that this characterization holds broadly. + +# 3.1 PROGRESSIVE SHARPENING + +When training neural networks, it seems to be a general rule that so long as the sharpness is small enough for gradient descent to be stable $( < 2 / \eta$ , for vanilla gradient descent), gradient descent has an overwhelming tendency to continually increase the sharpness. We call this phenomenon progressive sharpening. By “overwhelming tendency,” we mean that gradient descent can occasionally decrease the sharpness (especially at the beginning of training), but these brief decreases always seem be followed by a return to continual increase. Jastrz˛ebski et al. (2020) previously hypothesized (in their Assumption 4) that a similar phenomenon may hold for SGD, but the evidence for, and the precise scope of, this effect are both currently far clearer for gradient descent than for SGD. + +![](images/56cc0c19b654465ab0c58a070a13f44506ef5b48cb3127451a26a57a6f217e3d.jpg) +Figure 3: So long as the sharpness is less than $2 / \eta$ , it tends to continually increase during gradient descent. We train a network to completion $9 9 \%$ accuracy) using gradient descent with a very small step size. We consider both MSE loss (left) and cross-entropy loss (right). + +Progressive sharpening is illustrated in Figure 3. Here, we use (full-batch) gradient descent to train a network on a subset of 5,000 examples from CIFAR-10, and we monitor the evolution of the sharpness during training. The network is a fully-connected architecture with two hidden layers of width 200, and tanh activations. In Figure 3(a), we train using the mean squared error loss for classification (Hui & Belkin, 2020), encoding the correct class with 1 and the other classes with 0. We use the small step size of $\eta \ : = \ : 2 / 6 0 0$ , and stop when the training accuracy reaches $9 9 \%$ . We plot both the train loss and the sharpness, with a horizontal dashed line marking the stability threshold $2 / \eta$ . Observe that the sharpness continually rises during training (except for a brief dip at the beginning). This is progressive sharpening. For this experiment, we intentionally chose a step size $\eta$ small enough that the sharpness remained beneath $2 / \bar { \eta }$ for the entire duration of training. + +Cross-entropy. When training with cross-entropy loss, there is an exception to the rule that the sharpness tends to continually increase: with cross-entropy loss, the sharpness typically drops at the end of training. This behavior can be seen in Figure 3(b), where we train the same network using the cross-entropy loss rather than MSE. This drop occurs because once most data points are classified correctly, gradient descent tries to drive the cross-entropy loss to zero by scaling up the margins, as detailed in Soudry et al. (2018). As we explain in Appendix C, this causes the sharpness to drop. + +The effect of width. It is known that when networks parameterized in a certain way (the “NTK parameterization”) are made infinitely wide, the Hessian moves a vanishingly small amount during training (Jacot et al., 2018; Lee et al., 2019; Li & Liang, 2018), which implies that no progressive sharpening occurs. In Appendix D, we experiment with networks of varying width, under both NTK and standard parameterizations. We find that progressive sharpening occurs to a lesser degree as networks become increasingly wide. Nevertheless, our experiments in $\ S 4$ demonstrate that progressive sharpening occurs to a dramatic degree for standard architectures on the standard dataset CIFAR-10. + +We do not know why progressive sharpening occurs, or whether “sharp” solutions differ in any important way from “not sharp” solutions. These are important questions for future work. Note that Mulayoff & Michaeli (2020) studied the latter question in the context of deep linear networks. + +# 3.2 THE EDGE OF STABILITY + +In the preceding section, we ran gradient descent using step sizes $\eta$ so small that the sharpness never reached the stability threshold $2 / \eta$ . In Figure 4(a), we start to train the same network at the larger step size of $\eta = 0 . 0 1$ , and pause training once the sharpness rises to $2 / \eta = 2 0 0$ . Recall from $\ S 2$ that in any region where the sharpness exceeds $2 / \eta$ , gradient descent with step size $\eta$ would be unstable if run on the quadratic Taylor approximation to the training objective — the gradient descent iterates would oscillate with ever-increasing magnitude along the leading Hessian eigenvector. Empirically, we find that gradient descent on the real neural training objective behaves similarly — at first. Namely, let ${ \bf q } _ { 1 }$ be the leading Hessian eigenvector at the iteration where the sharpness reaches $2 / \eta$ . In Figure 4(b), we resume training the network, and we monitor both the train loss and the quantity $\left. \mathbf { q } _ { 1 } , \mathbf { x } _ { t } \right.$ for the next 215 iterations. Observe that $\left. \mathbf { q } _ { 1 } , \mathbf { x } _ { t } \right.$ oscillates with ever-increasing magnitude, similar to the divergent quadratic example in Figure 2(b). At first, these oscillations are too small to affect the objective appreciably, and so the train loss continues to monotonically decrease. But eventually, these oscillations grow big enough that the train loss spikes. + +![](images/d167549c8548e4247c63c0dafe3ea4b11804fc67c00c956313b55282c60c6c65.jpg) +Figure 4: Once the sharpness crosses $2 / \eta$ , gradient descent becomes destabilized. We run gradient descent at $\eta = 0 . 0 1$ . (a) The sharpness eventually reaches $2 / \eta$ . (b) Once the sharpness crosses $2 / \eta$ , the iterates start to oscillate along ${ \bf q } _ { 1 }$ with ever-increasing magnitude. (c) Somehow, GD does not diverge entirely; instead, the train loss continues to decrease, albeit non-monotonically. + +Once gradient descent becomes destabilized in this manner, classical optimization theory gives no clues as to what will happen next. One might imagine that perhaps gradient descent might diverge entirely, or that gradient descent might stall while failing to make progress, or that gradient descent might jump to a flatter region and remain there. In reality, none of these outcomes occurs. In Figure 4(c), we plot both the train loss and $\left. \mathbf { q } _ { 1 } , \mathbf { x } _ { t } \right.$ for 1000 iterations after the sharpness first crossed $2 / \eta$ . Observe that gradient descent somehow avoids diverging entirely. Instead, after initially spiking around iteration 215, the train loss continues to decrease, albeit non-monotonically. + +This numerical example is representative. In general, after the sharpness initially crosses $2 / \eta$ , gradient descent enters a regime we call the Edge of Stability, in which (1) the sharpness hovers right at, or just above, the value $2 / \eta$ ; and (2) the train loss behaves non-monotonically over short timescales, yet decreases consistently over long timescales. Indeed, in Figure 5, we run gradient descent at a range of step sizes using both MSE and cross-entropy loss. The left plane plots the train loss curves, with a vertical dotted line (of the appropriate color) marking the iteration where the sharpness first crosses $2 / \eta$ . Observe that the train loss decreases monotonically before this dotted line, but behaves non-monotonically afterwards. The middle plane plots the evolution of the sharpness, with a horizontal dashed line (of the appropriate color) at the value $2 / \eta$ . Observe that once the sharpness reaches $2 / \eta$ , it ceases to increase further, and instead hovers right at, or just above, the value $2 / \eta$ for the remainder of training. (The precise meaning of “just above” varies: in Figure 5, for MSE loss, the sharpness hovers just a minuscule amount above $2 / \eta$ , while for cross-entropy loss, the gap between the sharpness and $2 / \eta$ is small yet non-miniscule.) + +At the Edge of Stability, gradient descent is “trying” to increase the sharpness further, but is being restrained from doing so. To demonstrate this, in Figure 7, we train at step size $2 / 2 0 0$ until reaching the Edge of Stability, and then at iteration 6,000 (marked by the vertical black line), we drop the step size to $\eta = 2 / 3 0 0$ . Observe that after the learning rate drop, the sharpness immediately starts to increase, and only stops increasing once gradient descent is back at the Edge of Stability. Appendix O repeats this experiment on more architectures. Intuitively, gradient descent with fixed step sizes acts like a constrained optimization algorithm: the use of step size $\eta$ imposes an implicit $2 / \eta$ constraint on the sharpness (Nar & Sastry, 2018), and at the Edge of Stability this constraint is “active.” + +Observe from Figure 5 that there do exist step sizes $\eta$ (in purple) small enough that the sharpness never rises to $2 / \eta$ . We call such a step size stable. However, observe that with cross-entropy loss, it takes 3700 iterations to train at the stable step size in purple, but only 1000 iterations to train at the larger step size in blue. In general, we always observe that stable step sizes are suboptimal in terms of convergence speed. In fact, in $\ S 4$ we will see that for standard networks on CIFAR-10, stable step sizes are so suboptimally small that they are completely unreasonable. + +![](images/1d1a6cde8e4d00e092499a010db67cddcc397824f725bfbf3fd1dc2519962684.jpg) +Figure 5: After the sharpness reaches $2 / \eta$ , gradient descent enters the Edge of Stability. A network is trained with gradient descent at a range of step sizes (see legend), using both MSE loss (top row) and cross-entropy (bottom row). Left: the train loss curves, with a vertical dotted line at the iteration where the sharpness first crosses $2 / \eta$ . Center: the sharpness, with a horizontal dashed line at the value $2 / \eta$ . Right: sharpness plotted by time $\mathit { \Psi } = \mathit { \Psi }$ iteration $\times \eta$ ) rather than iteration. + +![](images/1f252f03c147233621dfa8b75b997a2e37764b54085eddf968800144261a3aa1.jpg) +Figure 6: Momentum. We run GD with step size $\eta = 0 . 0 1$ and Polyak or Nesterov momentum at various $\beta$ . For each algorithm, the horizontal dashed line marks the MSS from Equation 1. + +![](images/46bb801641419ab649df218983ffaa1aa8aacf166e8d5d6ec575bf2ff10b7ccd.jpg) +Figure 7: After a learning rate drop, progressive sharpening resumes. We start training at $\eta = 2 / 2 0 0$ (orange) and then after 6000 iterations (dotted vertical black line), we cut the step size to $\eta = 2 / 3 0 0$ (green). Observe that as soon as the step size is cut, the sharpness starts to rise. + +The “Edge of Stability” effect generalizes to gradient descent with momentum. In Figure 6, we train using gradient descent with step size $\eta = 0 . 0 1$ , and varying amounts of either Polyak or Nesterov momentum. Observe that in each case, the sharpness rises until reaching the MSS given by Equation 1, and then plateaus there. Appendix $\mathbf { N }$ has more momentum experiments. + +In Appendix P, we briefly examine the evolution of the next few Hessian eigenvalues during gradient descent. We find that each of these eigenvalues rises until plateauing near $\bar { 2 } / \eta$ . + +Prior work. Aspects of the Edge of Stability have been observed previously in the literature. Wu et al. (2018) noted that the sharpness at the solution reached by full-batch gradient descent was not just less than $2 / \eta$ , as was expected due to stability considerations, but was mysteriously approximately equal to $2 / \eta$ . In retrospect, we can attribute this observation to progressive sharpening. Xing et al. (2018) observed that full-batch gradient descent eventually enters a regime (the Edge of Stability) in which the training loss behaves non-monotonically, and the iterates oscillate along the direction of largest curvature; however, they did not relate this regime to the sharpness. Lewkowycz et al. (2020) found that in neural network training, if the sharpness at initialization is larger than $2 / \eta$ , then after becoming initially destabilized, gradient descent does not always diverge entirely (as the quadratic Taylor approximation would suggest), but rather sometimes “catapults” into a flatter region that is flat enough to stably accommodate the step size. It seems plausible that whichever properties of neural training objectives permit this so-called “catapult” behavior may also be the same properties that permit successful optimization at the Edge of Stability. Indeed, optimization at the Edge of Stability can conceivably be viewed as a never-ending series of micro-catapults. As we discuss at greater length in $\ S 6$ , several papers (Jastrz˛ebski et al., 2017; 2019) have observed that large step sizes steer stochastic gradient descent into less sharp regions of the loss landscape, and Jastrz˛ebski et al. (2020) attributed this effect to the stability properties of SGD. Finally, our precise characterization of the behavior of the sharpness during full-batch gradient descent adds to a growing body of work that empirically investigates the Hessian spectrum of neural networks (Sagun et al., 2017; Ghorbani et al., 2019; Li et al., 2020a; Papyan, 2018; 2019; 2020). + +# 3.3 THE GRADIENT FLOW TRAJECTORY + +In the right pane of Figure 5, we plot the evolution of the sharpness during gradient descent, with “time” $=$ iteration $\times \eta$ , rather than iteration, on the $\mathbf { X }$ -axis. This allows us to directly compare the sharpness after, say, 100 iterations at $\eta = 0 . 0 1$ to the sharpness after 50 iterations at $\eta = 0 . 0 2$ ; both are time 1. Observe that when plotted by time, the sharpnesses for gradient descent at different step sizes coincide until the time where each reaches $2 / \eta$ . This is because for this network, gradient descent at $\eta = 0 . 0 1$ and gradient descent at $\eta = 0 . 0 2$ initially travel the same path (moving at a speed proportional to $\eta$ ) until each reaches the point on that path where the sharpness hits $2 / \eta$ . This path is the gradient flow trajectory. The gradient flow solution at time $t$ is defined as the limit as $\eta 0$ of the gradient descent iterate at iteration $t / \eta$ (if this limit exists). The empirical finding of interest is that for this particular network, gradient descent does not only track the gradient flow trajectory in the limit of infinitesimally small step sizes, but for any step size that is less than 2/sharpness. + +We can numerically approximate gradient flow trajectories by using the Runge-Kutta RK4 algorithm (Press et al., 1992) to numerically integrate the gradient flow ODE. Empirically, for many but not all networks studied in this paper, we find that gradient descent at any step size $\eta$ closely tracks the Runge-Kutta trajectory until reaching the point on that trajectory where the sharpness hits $2 / \eta$ . (This sometimes occurs even for networks with ReLU activations or max-pooling, which give rise to training objectives that are not continuously differentiable, which means that the gradient flow trajectory is not necessarily guaranteed to exist.) For such networks, the gradient flow trajectory provides a coherent framework for reasoning about which step sizes will eventually enter the Edge of Stability. Let $\lambda _ { 0 }$ be the sharpness at initialization, and let $\lambda _ { \mathrm { m a x } }$ be the maximum sharpness along the gradient flow trajectory. If $\eta < 2 / \lambda _ { \mathrm { m a x } }$ , then gradient descent will stably track the gradient flow trajectory for the entire duration of training, and will never enter the Edge of Stability. On the other hand, if $\eta \in [ 2 / \lambda _ { \operatorname* { m a x } } , 2 / \lambda _ { 0 } ]$ , then gradient descent will stably track the gradient flow trajectory only until reaching the point on that trajectory where the sharpness hits $2 / \eta$ ; shortly afterwards, gradient descent will become destabilized, depart the gradient flow trajectory, and enter the Edge of Stability. + +# 4 FURTHER EXPERIMENTS + +Section 3 focused for exposition on a single architecture and task. In this section, we show that our characterization of gradient descent holds broadly across a wide range of architectures and tasks. We detail several known caveats and qualifications in Appendix A. + +Architectures. In Appendix J, we fix the task of training a $5 \mathrm { k }$ subset of CIFAR-10, and we systematically vary the network architecture. We consider fully-connected networks, as well as convolutional networks with both max-pooling and average pooling. For all of these architectures, we consider tanh, ReLU, and ELU activations, and for fully-connected networks we moreover consider softplus and hardtanh —- eleven networks in total. We train each network with both cross-entropy and MSE loss. In each case, we successfully reproduce Figure 5. + +Since batch normalization (Ioffe & Szegedy, 2015) is known to have unusual optimization properties (Li & Arora, 2019), it is natural to wonder whether our findings still hold with batch normalization. In Appendix K, we confirm that they do, and we reconcile this point with Santurkar et al. (2018). + +Tasks. In Appendix L, we verify our findings on: (1) a Transformer trained on the WikiText2 language modeling task; (2) fully-connected tanh networks with one hidden layer, trained on a synthetic one-dimensional toy regression task; and (3) deep linear networks trained on Gaussian data. In each case, the sharpness rises until hovering right at, or just above, the value $2 / \eta$ . + +Standard networks on CIFAR-10. In Appendix M, we verify our findings on three standard architectures trained on the full CIFAR-10 dataset: a ResNet with BN, a VGG with BN, and a VGG without BN. For all three architectures, we find that progressive sharpening occurs to a dramatic degree, and, relatedly, that stable step sizes are dramatically suboptimal. For example, when we train the VGG-BN to $9 9 \%$ accuracy using gradient flow / Runge-Kutta, we find that the sharpness rises from 6.3 at initialization to a peak sharpness of 2227.6. Since this is an architecture for which gradient descent closely hews to the gradient flow trajectory, we can conclude that any stable step size for gradient descent would need to be less than $2 / 2 2 2 7 . 6 = 0 . 0 0 0 8 9 7 .$ . Training finishes at time 14.91, so gradient descent at any stable step size would require at least $1 4 . 9 1 / 0 . 0 \bar { 0 } 0 8 9 7 = 1 6 , 6 2 2$ iterations. Yet empirically, this network can be trained to completion at the larger, “Edge of Stability” step size of $\eta = 0 . 1 6$ in just 329 iterations. Therefore, training at a stable step size is suboptimal by a factor of at least $1 6 6 2 2 / 3 2 9 = 5 0 . 5 $ . The situation is similar for the other two architectures we consider. In short, for standard architectures on the standard dataset CIFAR-10, stable step sizes are not just suboptimally small, they are so suboptimal as to be completely unreasonable. For these networks, gradient descent at any reasonable step size eventually enters the Edge of Stability regime. + +Tracking gradient flow. Recall that for some (but not all) architectures, gradient descent closely hews to the gradient flow trajectory so long as the sharpness is less than $2 / \eta$ . Among the architectures considered in Appendix J, we found this to be true for the architectures with continuously differentiable components, as well as some, but not all, with ReLU, hardtanh, and max-pooling. Among the architectures in Appendix L, we found this to be true for the tanh network, but not for the deep linear network or the Transformer. Finally, we did find this to be true for the three standard architectures in Appendix M, even though those architectures use ReLU. + +# 5 DISCUSSION + +We now explain why the behavior of gradient descent at the Edge of Stability contradicts several pieces of conventional wisdom in optimization. + +At reasonable step sizes, gradient descent cannot be analyzed using (even local) $L$ -smoothness Many convergence analyses of gradient descent assume a bound on the sharpness — either globally or, at the very least, along the optimization trajectory. This condition, called $L$ -smoothness (Nesterov, 1998), is intended to guarantee that each gradient step will decrease the training objective by a certain amount; the weakest guarantee is that if the local sharpness is less than $2 / \eta$ , then a gradient step with size $\eta$ is guaranteed to decrease (rather than increase) the training objective. At a bare minimum, any convergence analysis of gradient descent based on $L$ -smoothness will require the sharpness along the optimization trajectory to be less than $2 / \eta$ . Yet to the contrary, at the Edge of Stability, the sharpness hovers just above $2 / \eta$ . Therefore, at any step size for which gradient descent enters the Edge of Stability (which, on realistic architectures, includes any reasonable step size), gradient descent cannot be analyzed using $L$ -smoothness. Li et al. (2020b) previously argued that convergence analyses based on $L$ -smoothness do not apply to networks with both batch normalization and weight decay; our paper empirically extends this to neural networks without either. + +$L$ -smoothness may be inappropriate when analyzing other optimization algorithms too It is common for optimization papers seemingly motivated by deep learning to analyze algorithms under the “non-convex but $L$ -smooth”’ setting (Reddi et al., 2016; Agarwal et al., 2017; Zaheer et al., 2018; Zhou et al., 2018; Chen et al., 2019; Li & Orabona, 2019; Ward et al., 2019; You et al., 2020; Vaswani et al., 2019; Sankararaman et al., 2019; Reddi et al., 2021; Défossez et al., 2020; Xie et al., 2020; Liu et al., 2020; Defazio, 2020). Since our experiments focus on gradient descent, it does not necessarily follow that $L$ -smoothness assumptions are unjustified when analyzing other optimization algorithms. However, gradient descent is arguably the simplest optimization algorithm, so we believe that the fact that (even local) $L$ -smoothness fails even there should raise serious questions about the suitability of the $L$ -smoothness assumption in neural network optimization more generally. In particular, the burden of proof should be on authors to empirically justify this assumption. + +At reasonable step sizes, gradient descent does not monotonically decrease the training loss In neural network training, SGD does not monotonically decrease the training objective, in part due to minibatch randomness. However, it is often assumed that full-batch gradient descent would monotonically decrease the training objective, were it used to train neural networks. For example, Zhang et al. (2020) proposed a “relaxed $L$ -smoothness” condition that is less restrictive than standard $L$ -smoothness, and proved a convergence guarantee for gradient descent under this condition which asserted that the training objective will decrease monotonically. Likewise, some neural network analyses such as Allen-Zhu et al. (2019) also assert that the training objective will monotonically decrease. Yet, at the Edge of Stability, the training loss behaves non-monotonically over short timescales even as it consistently decreases over long timescales. Therefore, convergence analyses which assert monotone descent cannot possibly apply to gradient descent at reasonable step sizes. + +The Edge of Stability is inherently non-quadratic It is tempting to try to reason about the behavior of gradient descent on neural network training objectives by analyzing, as a proxy, the behavior of gradient descent on the local quadratic Taylor approximation (LeCun et al., 1998). However, at the Edge of Stability, the behavior of gradient descent on the real neural training objective is irreconcilably different from the behavior of gradient descent on the quadratic Taylor approximation: the former makes consistent (if choppy) progress, whereas the latter would diverge (and this divergence would happen quickly, as we demonstrate in Appendix E). Thus, the behavior of gradient descent at the Edge of Stability is inherently non-quadratic. + +Dogma for step size selection may be unjustified An influential piece of conventional wisdom concerning step size selection has its roots in the quadratic Taylor approximation model of gradient descent. This conventional wisdom (LeCun et al., 1993; 1998; Schaul et al., 2013) holds that if the sharpness at step $t$ is $\lambda _ { t }$ , then the current step size $\eta _ { t }$ must be set no greater than $2 / \lambda _ { t }$ (in order to prevent divergence); and furthermore, barring additional information about the objective function, that $\eta _ { t }$ should optimally be set to $1 / \lambda _ { t }$ . Our findings complicate this conventional wisdom. + +To start, it is nearly impossible to satisfy these prescriptions with a fixed step size: for any fixed (and reasonable) step size $\eta _ { t } = \eta$ , progressive sharpening eventually drives gradient descent into regions where the sharpness is just a bit greater than $2 / \eta$ — which means that the step size $\eta$ is purportedly impermissible. Furthermore, in Appendix F, we try running gradient descent with the purportedly optimal $\eta _ { t } = 1 / \lambda _ { t }$ rule, and find that this algorithm is soundly outperformed by the purportedly impermissible baseline of gradient descent with a fixed $\eta _ { t } = 1 / \lambda _ { 0 }$ step size, where $\lambda _ { 0 }$ is the sharpness at initialization. The $\eta _ { t } = 1 / \lambda _ { t }$ rule continually anneals the step size, and in so doing ensures that the training objective will decrease at each iteration, whereas the fixed $\eta _ { t } = 1 / \lambda _ { 0 }$ step size often increases the training objective. However, this non-monotonicity turns out to be a worthwhile price to pay in return for the ability to take larger steps. + +# 6 STOCHASTIC GRADIENT DESCENT + +Our precise characterization of the behavior of the sharpness only applies to full-batch gradient descent. In contrast, during SGD, the sharpness does not always settle at any fixed value (Appendix G), let alone one that can be numerically predicted from the hyperparameters. Nevertheless, prior works (Jastrz˛ebski et al., 2017; 2019; 2020) have demonstrated that large step sizes do steer SGD into regions of the landscape with lower sharpness; the $2 / \eta$ rule we have identified for full-batch gradient descent is a special case of this observation. Furthermore, small batch sizes also steer SGD into regions with lower sharpness (Keskar et al., 2016; Jastrz˛ebski et al., 2017), as we illustrate in Appendix G. Jastrz˛ebski et al. (2020) attributed this phenomenon to the stability properties of SGD. + +Even though our findings only strictly hold for gradient descent, they may have relevance to SGD as well. First, since gradient descent is a special case of SGD, any general characterization of the dynamics of SGD must reduce to the Edge of Stability in the full-batch special case. Second, there are indications that the Edge of Stability may have some analogue for SGD. One way to interpret our main findings is that gradient descent “acclimates” to the step size in such a way that each update sometimes increases and sometimes decrease the train loss, yet an update with a smaller step size would consistently decrease the training loss. Along similar lines, in Appendix H we demonstrate that SGD “acclimates” to the step size and batch size in such a way that each SGD update sometimes increases and sometimes decreases the training loss in expectation, yet an SGD update with a smaller step size or larger batch size would consistently decrease the training loss in expectation. + +In extending these findings to SGD, the question arises of how to model “stability” of SGD. This is a highly active area of research. Wu et al. (2018) proposed modeling stability in expectation, and gave a sufficient (but not necessary) criterion for the stability of SGD in expectation. Building on this framework, Jastrz˛ebski et al. (2020) argued that if the Hessian is aligned with the second moment matrix of per-example gradients, then SGD is stable so long as a certain expression (involving the sharpness) is below a certain threshold. In the special full-batch case, their criterion reduces to the sharpness being beneath $2 / \eta$ — a constraint which we have shown is “tight” throughout training. However, in the general SGD case, there is no evidence that their stability constraint is tight throughout training. Giladi et al. (2020) showed that the generalization gap in asynchronous SGD can be mostly ameliorated by setting the step size so as to ensure that stability properties in expectation remain identical to those of a well-tuned implementation of synchronous SGD. Finally, a number of papers have attempted to mathematically model the propensity of SGD to “escape from sharp minima” (Hu et al., 2017; Zhu et al., 2019; Xie et al., 2021). + +# 7 CONCLUSION + +We have empirically demonstrated that the behavior of gradient decent on neural training objectives is both surprisingly consistent across architectures and tasks, and surprisingly different from that envisioned in the conventional wisdom. Our findings raise a number of questions. Why does progressive sharpening occur? At the Edge of Stability, by what mechanism does gradient descent avoid diverging entirely? Since the conventional wisdom for step size selection is wrong, how should the gradient descent step size be set during deep learning? Does the “Edge of Stability” effect generalize in some way to optimization algorithms beyond gradient descent, such as SGD? We hope to inspire future efforts aimed at addressing these questions. + +# 8 ACKNOWLEDGEMENTS + +This work was supported in part by DARPA FA875017C0141, the National Science Foundation grants IIS1705121 and IIS1838017, an Amazon Web Services Award, a JP Morgan A.I. Research Faculty Award, a Carnegie Bosch Institute Research Award, a Facebook Faculty Research Award, and a Block Center Grant. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA, the National Science Foundation, or any other funding agency. + +# REFERENCES + +Naman Agarwal, Zeyuan Allen-Zhu, Brian Bullins, Elad Hazan, and Tengyu Ma. Finding approximate local minima faster than gradient descent. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 1195–1199, 2017. + +Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via overparameterization. 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For shallow or wide networks, or on simple datasets, sharpness doesn’t rise that much When the network is shallow (Figure 16 - 17) or wide (Figure 12 - 14), or when the dataset is “easy” or small (Figure 18), the sharpness may rise only a small amount over the gradient flow trajectory. For these optimization problems, “stable step sizes” (those for which gradient descent never enters the Edge of Stability) may be quite reasonable, and the range of “Edge of Stability” step sizes may be quite small. + +3. Sharpness sometimes drops at the beginning of training We sometimes observe that the sharpness drops at the very beginning of training, as the network leaves its initialization. This was more common when training with MSE loss than with cross-entropy loss. For most networks, this drop was very slight. However, the combination of both batch normalization and MSE loss sometimes caused situations where the sharpness was considerably large at initialization, and dropped precipitously as soon as training began. Figure 8 illustrates one such network. + +![](images/4f65d59ff48c406a01dc0bef3cbe8d711ac190b425e7e02746021efd9dd37558.jpg) +Figure 8: With batch norm and MSE loss, the sharpness sometimes drops precipitously: We use gradient flow to train a CNN with ReLU activations and batch norm using the MSE loss. Observe that the sharpness drops substantially at the beginning of training, and never recovers to its initial value. This seems to occur due to the combination of both batch normalization and MSE loss. + +4. With batch normalization, need to look at sharpness between iterates As detailed in Appendix K, when we trained batch-normalized networks using very small step sizes $\eta$ , we sometimes observed that the sharpness at the gradient descent iterates themselves plateaued below $2 / \eta$ , even though the sharpness in between the iterates plateaued just above $2 / \eta$ , as expected. + +5. With non-differentiable components, instability sometimes begins when the sharpness is a bit less than $2 / \eta$ When training networks with ReLU or hardtanh activations, we sometimes observed that the sharpness started to plateau (and the training loss started to behave nonmonotonically) a bit before the instant when the sharpness crossed $2 / \eta$ . For example, see Figures 41, 43 or Figures 45, 47. One potential explanation is that for such networks, the training objective is not continuously differentiable, so the second-order Taylor approximation around a given iterate may be a poor local model for the training objective at even an tiny distance away in weight space, if that weight change causes some activations to switch “ReLU case” from $\leq 0$ to $> 0$ , or visa versa. + +# B STABILITY OF GRADIENT DESCENT ON QUADRATIC FUNCTIONS + +This appendix describes the stability properties of gradient descent (and its momentum variants) when optimizing the quadratic objective function + +$$ +f ( \mathbf { x } ) = { \frac { 1 } { 2 } } \mathbf { x } ^ { T } \mathbf { A } \mathbf { x } + \mathbf { b } ^ { T } \mathbf { x } + c +$$ + +starting from the initialization $\mathbf { x } _ { \mathrm { 0 } }$ . + +To review, vanilla gradient descent is defined by the iteration: + +$$ +\mathbf x _ { t + 1 } = \mathbf x _ { t } - \eta \nabla f ( \mathbf x _ { t } ) . +$$ + +Meanwhile, gradient descent with Polyak (also called “heavy ball”) momentum (Polyak, 1964; Sutskever et al., 2013; Goodfellow et al., 2016) is defined by the iteration: + +$$ +\begin{array} { r l } & { \mathbf { v } _ { t + 1 } = \beta \mathbf { v } _ { t } - \eta \nabla f ( \mathbf { x } _ { t } ) } \\ & { \mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } + \mathbf { v } _ { t + 1 } } \end{array} +$$ + +where $\mathbf { v } _ { t }$ is a “velocity” vector and $0 \leq \beta < 1$ is the momentum coefficient. For $\beta = 0$ the algorithm reduces to vanilla GD. + +Finally, Nesterov momentum Sutskever et al. (2013); Goodfellow et al. (2016) is an adaptation of Nesterov’s accelerated gradient (Nesterov, 1983) for deep learning defined by the iteration: + +$$ +\begin{array} { r l } & { \mathbf { v } _ { t + 1 } = \beta \mathbf { v } _ { t } - \eta \nabla f ( \mathbf { x } _ { t } + \beta \mathbf { v } _ { t } ) } \\ & { \mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } + \mathbf { v } _ { t + 1 } } \end{array} +$$ + +where $\mathbf { v } _ { t }$ is a “velocity” vector and $0 \leq \beta < 1$ is the momentum coefficient. For $\beta = 0$ the algorithm reduces to vanilla GD. + +All three of these algorithms share a special property: on quadratic functions, they act independently along each Hessian eigenvector. That is, if we express the iterates in the Hessian eigenvector basis, then in this basis the coordinates evolve independent from one another under gradient descent. + +Proposition 1. Consider running vanilla gradient descent on the quadratic objective (2) starting from $\mathbf { x } _ { \mathrm { 0 } }$ . Let $( \mathbf { q } , a )$ be an eigenvector/eigenvalue pair of A. If $a > 2 / \eta$ , then the sequence $\left\{ { \bf q } ^ { T } { \bf x } _ { t } \right\}$ will diverge. + +Proof. The update rule for gradient descent on this quadratic function is: + +$$ +\begin{array} { r } { \mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } - \eta ( \mathbf { A } \mathbf { x } _ { t } + \mathbf { b } ) } \\ { = ( \mathbf { I } - \eta \mathbf { A } ) \mathbf { x } _ { t } - \eta \mathbf { b } . } \end{array} +$$ + +Therefore, the quantity $\mathbf { q } ^ { T } \mathbf { x } _ { t }$ evolves under gradient descent as: + +$$ +\begin{array} { r l } & { \mathbf { q } ^ { T } \mathbf { x } _ { t + 1 } = \mathbf { q } ^ { T } ( \mathbf { I } - \eta \mathbf { A } ) \mathbf { x } _ { t } - \eta \mathbf { b } } \\ & { \qquad = ( 1 - \eta a ) \mathbf { q } ^ { T } \mathbf { x } _ { t } - \eta \mathbf { q } ^ { T } \mathbf { b } . } \end{array} +$$ + +$$ +( \mathbf { q } ^ { T } \mathbf { A } = a \mathbf { q } ) +$$ + +Define $\begin{array} { r } { \tilde { x } _ { t } = \mathbf q ^ { T } \mathbf x _ { t } + \frac { 1 } { a } \mathbf q ^ { T } \mathbf b } \end{array}$ , and note that $\left\{ { \bf q } ^ { T } { \bf x } _ { t } \right\}$ diverges if and only if $\{ \tilde { x } _ { t } \}$ diverges. + +The quantity $\tilde { x } _ { t }$ evolves under gradient descent according to the simple rule: + +$$ +\tilde { x } _ { t + 1 } = ( 1 - \eta a ) \tilde { x } _ { t } +$$ + +Since $\eta > 0$ , if $a > 2 / \eta$ then $( 1 - \eta a ) < - 1$ , so the sequence $\{ \tilde { x } _ { t } \}$ will diverge. + +We now prove analogous results for Nesterov and Polyak momentum. + +Theorem 1. Consider running Nesterov momentum on the quadratic objective (2) starting from any initialization. Let $( \mathbf { q } , a )$ be an eigenvector/eigenvalue pair of A. If $\begin{array} { r } { a > \frac { 1 } { \eta } \left( \frac { 2 + 2 \beta } { 1 + 2 \beta } \right) } \end{array}$ , then the sequence $\left\{ { \bf q } ^ { T } { \bf x } _ { t } \right\}$ will diverge. + +Proof. The update rules for Nesterov momentum on this quadratic function are: + +$$ +\begin{array} { r l } & { \mathbf { v } _ { t + 1 } = \beta ( \mathbf { I } - \eta \mathbf { A } ) \mathbf { v } _ { t } - \eta \mathbf { b } - \eta \mathbf { A } \mathbf { x } _ { t } } \\ & { \mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } + \mathbf { v } _ { t + 1 } . } \end{array} +$$ + +Using the fact that $\mathbf { v } _ { t } = \mathbf { x } _ { t } - \mathbf { x } _ { t - 1 }$ , we can rewrite this as a recursion in $\mathbf { x } _ { t }$ alone: + +$$ +\begin{array} { r } { \mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } + \beta ( \mathbf { I } - \eta \mathbf { A } ) ( \mathbf { x } _ { t } - \mathbf { x } _ { t - 1 } ) - \eta \mathbf { b } - \eta \mathbf { A } \mathbf { x } _ { t } } \\ { = ( 1 + \beta ) ( \mathbf { I } - \eta \mathbf { A } ) \mathbf { x } _ { t } - \beta ( \mathbf { I } - \eta \mathbf { A } ) \mathbf { x } _ { t - 1 } - \eta \mathbf { b } } \end{array} +$$ + +Define $\begin{array} { r } { \tilde { x } _ { t } = { \bf q } ^ { T } { \bf x } _ { t } + \frac { 1 } { a } { \bf q } ^ { T } { \bf b } } \end{array}$ , and note that $\mathbf { q } ^ { T } \mathbf { x } _ { t }$ diverges iff $\tilde { x } _ { t }$ diverges. It can be seen that $\tilde { x } _ { t }$ evolves as: + +$$ +\tilde { x } _ { t + 1 } = ( 1 + \beta ) ( 1 - \eta a ) \tilde { x } _ { t } - \beta ( 1 - \eta a ) \tilde { x } _ { t - 1 } +$$ + +This is a linear homogenous second-order difference equation. By Theorem 2.37 in Elaydi (2005), since $\eta > 0$ and $\beta < 1$ , if $\begin{array} { r } { a > \frac { 1 } { \eta } \left( \frac { 2 + 2 \beta } { 1 + 2 \beta } \right) } \end{array}$ then this recurrence diverges. + +The following result previously appeared in Goh (2017). + +Theorem 2. Consider running Polyak momentum on the quadratic objective (2) starting from any initialization. Let $( \mathbf { q } , a )$ be an eigenvector/eigenvalue pair of A. If $a ~ > ~ \frac { 1 } { \eta } \left( 2 + 2 \beta \right)$ , then the sequence $\left\{ \mathbf { q } ^ { T } \mathbf { x } _ { t } \right\}$ will diverge. + +Proof. Using the fact that $\mathbf v _ { t } = \mathbf x _ { t } - \mathbf x _ { t - 1 }$ , we can re-write the Polyak momentum recursion as a recursion in $\mathbf { x }$ alone: + +$$ +\begin{array} { r l } { \mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } + \beta \mathbf { v } _ { t } - \eta \nabla f ( \mathbf { x } _ { t } ) } & { } \\ { = \mathbf { x } _ { t } + \left( \beta ( \mathbf { x } _ { t } - \mathbf { x } _ { t - 1 } ) - \eta \nabla f ( \mathbf { x } _ { t } ) \right) } & { } \\ { = ( 1 + \beta ) \mathbf { x } _ { t } - \beta \mathbf { x } _ { t - 1 } - \eta \nabla f ( \mathbf { x } _ { t } ) . } \end{array} +$$ + +For the quadratic objective (2), this update rule amounts to: + +$$ +\begin{array} { r l } & { \mathbf { x } _ { t + 1 } = ( 1 + \beta ) \mathbf { x } _ { t } - \beta \mathbf { x } _ { t - 1 } - \eta ( \mathbf { A } \mathbf { x } _ { t } + \mathbf { b } ) } \\ & { \qquad = ( 1 + \beta - \eta \mathbf { A } ) \mathbf { x } _ { t } - \beta \mathbf { x } _ { t - 1 } - \eta \mathbf { b } . } \end{array} +$$ + +Multiplying by $\mathbf { q } ^ { T }$ , we obtain: + +$$ +\mathbf { q } ^ { T } \mathbf { x } _ { t + 1 } = ( 1 + \beta - \eta a ) \mathbf { q } ^ { T } \mathbf { x } _ { t } - \beta \mathbf { q } ^ { T } \mathbf { x } _ { t - 1 } - \eta \mathbf { q } ^ { T } \mathbf { b } . +$$ + +Now, define $\begin{array} { r } { \tilde { x } _ { t } = \mathbf q ^ { T } \mathbf x _ { t } + \frac { 1 } { a } \mathbf q ^ { T } \mathbf b } \end{array}$ . Note that $\mathbf { q } ^ { T } \mathbf { x } _ { t }$ diverges iff $\tilde { x } _ { t }$ diverges. It can be seen that $\tilde { x } _ { t }$ evolves as: + +$$ +\begin{array} { r } { \tilde { x } _ { t + 1 } = ( 1 + \beta - \eta a ) \tilde { x } _ { t } - \beta \tilde { x } _ { t - 1 } . } \end{array} +$$ + +This is a linear homogenous second-order difference equation. By Theorem 2.37 in Elaydi (2005), since $\eta > 0$ and $\beta < 1$ , if $\begin{array} { r } { a > \frac { 1 } { \eta } ( 2 + 2 \beta ) } \end{array}$ then this recurrence diverges. □ + +# C CROSS-ENTROPY LOSS + +In this appendix, we explain why the sharpness decreases at the end of training when the crossentropy loss is used. Before considering the full multiclass case, let us first consider the simpler case of binary classification with the logistic loss. + +Binary classification with logistic loss We consider a dataset $\{ \mathbf { x } _ { i } , y _ { i } \} \} _ { i = 1 } ^ { n } \subset \mathbb { R } ^ { d } \times \{ - 1 , 1 \}$ , where the examples are vectors in $\mathbb { R } ^ { d }$ and the labels are binary $\{ - 1 , 1 \}$ . We consider a neural network $h : \mathbb { R } ^ { d ^ { \star } } \times \mathbb { R } ^ { p } \mathbb { R }$ which maps an input $\mathbf { x } \in \mathbb { R } ^ { d }$ and a parameter vector $\theta \in \mathbb { R } ^ { p }$ to a prediction $h ( \mathbf { x } ; \theta ) \in \mathbb { R }$ . Let $\ell : \mathbb { R } \times \{ - 1 , 1 \} \to \bar { \mathbb { R } }$ be the logistic loss function: + +$$ +\begin{array} { l } { \displaystyle \ell ( z ; y ) = \log ( 1 + \exp ( - z y ) ) } \\ { \displaystyle = - \log p \quad \mathrm { w h e r e } \quad p = \frac { 1 } { 1 + \exp ( - z y ) } . } \end{array} +$$ + +The second derivative of this loss function w.r.t $z$ is: + +$$ +\ell ^ { \prime \prime } ( z ; y ) = p ( 1 - p ) . +$$ + +The full training objective is: + +$$ +f ( \theta ) = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } f _ { i } ( \theta ) \qquad { \mathrm { w h e r e } } \quad f _ { i } ( \theta ) = \ell ( f ( \mathbf { x } _ { i } ; \theta ) ; y _ { i } ) . +$$ + +The Hessian of this training objective is the average of per-example Hessians: + +$$ +\nabla ^ { 2 } f ( \theta ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \nabla ^ { 2 } f _ { i } ( \theta ) . +$$ + +For any arbitrary loss function $\ell$ , we have the so-called Gauss-Newton decomposition (Martens, 2016; Bottou et al., 2018) of the per-example Hessian $\nabla ^ { 2 } f _ { i } ( \theta )$ : + +$$ +\nabla ^ { 2 } f _ { i } ( \theta ) = \ell ^ { \prime \prime } ( z _ { i } ; y _ { i } ) \nabla _ { \theta } h ( \mathbf { x } _ { i } ; \theta ) \nabla _ { \theta } h ( \mathbf { x } _ { i } ; \theta ) ^ { T } + \ell ^ { \prime } ( z _ { i } ; y _ { i } ) \nabla _ { \theta } ^ { 2 } h ( \mathbf { x } _ { i } ; \theta ) \quad { \mathrm { w h e r e } } \quad z _ { i } = h ( \mathbf { x } _ { i } ; \theta ) +$$ + +where $\nabla _ { \theta } h ( \mathbf { x } _ { i } ; \theta ) \in \mathbb { R } ^ { p }$ is the gradient of the network output with respect to the weights, and $\ell ^ { \prime }$ refers to the derivative of $\ell$ with respect to its first argument, the score. + +Empirically, the first term in the decomposition (usually called the “Gauss-Newton matrix”) tends to dominate the second, which implies the following “Gauss-Newton approximation” to the Hessian: + +$$ +\nabla ^ { 2 } f ( \theta ) \approx \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime \prime } ( z _ { i } ; y _ { i } ) \nabla _ { \theta } h ( \mathbf { x } _ { i } ; \theta ) \nabla _ { \theta } h ( \mathbf { x } _ { i } ; \theta ) ^ { T } +$$ + +In our experience, progressive sharpening affects $\nabla _ { \boldsymbol { \theta } } h ( \mathbf { x } _ { i } ; \boldsymbol { \theta } ) \nabla _ { \boldsymbol { \theta } } h ( \mathbf { x } _ { i } ; \boldsymbol { \theta } ) ^ { T }$ . That is, $\nabla _ { \boldsymbol { \theta } } h ( \mathbf { x } _ { i } ; \boldsymbol { \theta } ) \dot { \nabla } _ { \boldsymbol { \theta } } h ( \mathbf { x } _ { i } ; \boldsymbol { \theta } ) ^ { \dot { T } }$ tends to grow in scale continually during training. For the square loss $\begin{array} { r } { \ell ( z ; y ) = \frac 1 2 ( z - \ y ) ^ { 2 } } \end{array}$ , the second derivative $\ell ^ { \prime \prime } ( z ; y ) = 1$ is the identity, so the $\nabla ^ { 2 } f _ { i } ( \theta )$ grows continually as well. In contrast, for the logistic loss, many of the $\ell ^ { \prime \prime } ( z _ { i } ; y _ { i } )$ decrease at the very end of training. Why is this? In Figure 9, we plot both the logistic loss $\ell$ , and its second derivative $\ell ^ { \prime \prime }$ , as a function of the quantity $y z$ , which is often called the “margin.” + +Crucially, observe that both $\ell$ and $\ell ^ { \prime \prime }$ are decreasing in $y z$ . Because the loss $\ell$ is decreasing in $y z$ , once an example $i$ is classified correctly (i.e. $y _ { i } z _ { i } > 0 ,$ ), the training objective can be optimized further by increasing the margin $y _ { i } z _ { i }$ . Because $\ell ^ { \prime \prime }$ is also decreasing in $y z$ , if the margin $y _ { i } z _ { i }$ increases, the term $\ell ^ { \prime \prime } ( z _ { i } ; y _ { i } )$ drops. Near the end of training, once most examples are classified correctly, gradient descent can easily increase the margins of all these examples by simply scaling up the final layer weight matrix. This causes the $\ell ^ { \prime \prime } ( z _ { i } ; y _ { i } )$ to drop. Therefore, even though progressive sharpening still applies to $\nabla _ { \boldsymbol { \theta } } h ( \mathbf { x } _ { i } ; \boldsymbol { \theta } ) \nabla _ { \boldsymbol { \theta } } h ( \mathbf { x } _ { i } ; \boldsymbol { \theta } ) ^ { T }$ , the decrease in the $\ell ^ { \prime \prime } ( z _ { i } ; y _ { i } )$ ’s pulls down the leading eigenvalue of the Gauss-Newton matrix in equation 3. + +This effect is illustrated in Figure 10. Here, we train a network on the binary classification task of CIFAR-10 airplane vs. automobile, using the logistic loss. In Figure 10(e), we plot the margin $y _ { i } z _ { i }$ for 10 examples in the dataset. Notice that at the end of training, these margins all continually rise; + +![](images/8a4e5c872e964d91be4d77db3e60be48539f3c043542b649e3eabe50088a495d.jpg) +Figure 9: Left: logistic loss function $\ell$ as a function of $y z$ . Right: its second derivative $\ell ^ { \prime \prime }$ as a function of $y z$ . + +this is because gradient descent “games” the objective by increasing the margins of successfully classified examples. When the margin $y _ { i } z _ { i }$ rises, the second derivative $\ell ^ { \prime \prime } ( z _ { i } ; y _ { i } )$ drops. This can be seen from Figure 10(f), where wleading eigenvalue of the matrix $\ell ^ { \prime \prime } ( z _ { i } ; y _ { i } )$ examples. Now, all the while, thekeeps rising, as can be seen from $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \nabla _ { \theta } \hat { h } ( \mathbf x _ { i } ; \theta ) \nabla _ { \theta } h ( \mathbf x _ { i } ; \theta ) ^ { T } } \end{array}$ Figure 10(d). However, because the ${ \ell ^ { \prime \prime } s }$ are dropping, the leading eigenvalue of the Gauss-Newton matrix $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ^ { \prime \prime } ( z _ { i } ; y _ { i } ) \nabla _ { \theta } h ( \mathbf { x } _ { i } ; \theta ) \nabla _ { \theta } h ( \mathbf { x } _ { i } ; \theta ) ^ { \hat { T } } } \end{array}$ starts to decrease at the end of training, as can be seen from the green line in Figure 10(c). Finally, since the leading eigenvalue of the Gauss-Newton matrix is an excellent approximation to the leading eigenvalue of the Hessian (i.e. the sharpness), the sharpness also drops at the end of training, as can be seen from the orange line in Figure 10(c). + +![](images/8a7d3d06f9688932a7466481fcbac84ae8ee614ef3e52a623ad44dbe0cf530b4.jpg) +Figure 10: We train a network using the logistic loss on the binary classification problem of CIFAR10 airplane vs. automobile. (a) The train loss. (b) The train accuracy. (c) The leading eigenvalue of the Hessian and of the Gauss-Nthe former. (d) The leading eigenv ton matrix; obse of the matrix $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \nabla _ { \theta } h ( \mathbf x _ { i } ; \theta ) \nabla _ { \theta } \bar { h } ( \mathbf x _ { i } ; \theta ) ^ { \dot { T } } } \end{array}$ roximation to, which is the $\ell ^ { \prime \prime }$ dip at the end of training. (e) The margin $y _ { i } z _ { i }$ of 10 examples; observe that all the margins rise at the end of training. (f) the value $\ell ^ { \prime \prime } ( z _ { i } ; \bar { y } _ { i } )$ for 10 examples; observe that all of these curves decline at the end of training. + +Multiclass classification with cross-entropy loss We consider a dataset $\{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } \subset \mathbb { R } ^ { d } \times$ $\{ 1 , \ldots , k \}$ , where the examples are vectors in $\mathbb { R } ^ { d }$ and the labels are in $\{ 1 , \ldots , k \}$ . We consider a neural network $h : \mathbb { R } ^ { d } \times \mathbb { R } ^ { j } \mathbb { R } ^ { k }$ which maps an input $\mathbf { x } \in \mathbb { R } ^ { d }$ and a parameter vector $\theta \in \mathbb { R } ^ { p }$ to a prediction $h ( x ; \theta ) \in \mathbb { R } ^ { k }$ . Let $\ell : \mathbb { R } ^ { k } \times \{ 1 , \ldots , k \} \overset { \left. } { \right. } \mathbb { R }$ be the cross-entropy loss function: + +$$ +\begin{array} { l } { \displaystyle \ell ( { \mathbf z } ; y ) = - \log \frac { \exp ( z _ { y } ) } { \sum _ { j = 1 } ^ { k } \exp ( z _ { j } ) } } \\ { = - \log p _ { y } \quad \mathrm { w h e r e } \quad { \mathbf p } = \frac { \exp ( { \mathbf z } ) } { \sum _ { j } \exp ( z _ { j } ) } . } \end{array} +$$ + +The Hessian $\nabla ^ { 2 } \ell ( \mathbf { z } ; y ) \in \mathbb { R } ^ { k \times k }$ of this loss function w.r.t the class scores $\mathbf { z }$ is: + +$$ +\begin{array} { r } { \nabla ^ { 2 } \ell ( \mathbf { z } ; y ) = \mathrm { d i a g } ( \mathbf { p } ) - \mathbf { p } \mathbf { p } ^ { T } . } \end{array} +$$ + +Now, for any loss function $\ell : \mathbb { R } ^ { k } \times \{ 1 , \dots , k \} \mathbb { R }$ , we have the Gauss-Newton decomposition: + +$$ +\nabla ^ { 2 } f _ { i } ( \theta ) = { \mathbf J } _ { i } ^ { T } [ \nabla _ { { \mathbf z } _ { i } } ^ { 2 } ( { \mathbf z } _ { i } ; y _ { i } ) ] { \mathbf J } _ { i } + \sum _ { j = 1 } ^ { k } [ \nabla _ { { \mathbf z } _ { i } } \ell ( { \mathbf z } _ { i } ; y _ { i } ) ] _ { j } \nabla _ { \theta } ^ { 2 } h _ { j } ( { \mathbf x } _ { i } ; \theta ) +$$ + +where $\mathbf { z } _ { i } ~ = ~ h _ { i } ( \mathbf { x } _ { i } ; \theta ) ~ \in ~ \mathbb { R } ^ { k }$ are the logits for example $i$ , $\mathbf { J } _ { i } ~ \in ~ \mathbb { R } ^ { k \times p }$ is the network output-toweights Jacobian for example $i$ , $\nabla _ { \mathbf { z } _ { i } } ^ { 2 } \ell ( \mathbf { \bar { z } } _ { i } ; y _ { i } ) \ \in \ \mathbb { R } ^ { k \times k }$ is the Hessian of $\ell$ w.r.t its input $\mathbf { z } _ { i }$ , and $\nabla _ { \theta } ^ { 2 } h _ { j } ( \mathbf { x } _ { i } ; \theta ) \in \mathbb { R } ^ { p \times p }$ is the Hessian matrix of the $j$ -th output of the network $h$ on the $i$ -th example. + +Dropping the second term yields the Gauss-Newton approximation: + +$$ +\nabla ^ { 2 } f _ { i } ( \boldsymbol { \theta } ) \approx \mathbf { J } _ { i } ^ { T } \left[ \nabla _ { \mathbf { z } _ { i } } ^ { 2 } \ell ( \mathbf { z } _ { i } ; y _ { i } ) \right] \mathbf { J } _ { i } . +$$ + +As in the binary classification case discussed above: at the end of training, for many examples $i$ , the $y _ { i }$ entry of $\mathbf { p } _ { i }$ will tend toward 1 and the other entries of $\mathbf { p } _ { i }$ will tend to 0. Once this occurs, the matrix $\mathrm { d i a g } ( \mathbf { p } _ { i } ) - \mathbf { p } _ { i } \mathbf { p } _ { i } ^ { T }$ will broadly decrease in scale: the diagonal entries of this matrix are of the form $p ( 1 - p )$ , which goes to zero as $p 0$ or $p 1$ ; and the off-diagonal entries are of the form $- p q$ , which also goes to zero if $p 0 , q 1$ or $p 0 , q 0$ or $p 1 , q 0$ . + +This effect is illustrated in Figure 11. Here, we train a network on CIFAR-10 using the cross-entropy loss. In Figure 11(e), for ten examples $i$ in the dataset (with output scores $\mathbf { z } _ { i } = \breve { h } ( \mathbf { x } _ { i } ) \in \mathbb { R } ^ { k } )$ , we plot the margin $\mathbf { z } _ { i } [ y _ { i } ] - \operatorname* { m a x } _ { j \neq y _ { i } } \mathbf { z } _ { i } [ j ]$ , which is the difference between the score of the correct class $y _ { i }$ and the score of the next-highest class. Observe that for all of these examples, this margin rises at the end of training. In Figure 11(f), for those same ten examples, we plot the quantity ${ \bf p } _ { i } [ y _ { i } ] ( 1 - { \bf p } _ { i } [ y _ { i } ] )$ .Observe that for all of these examples, this quantity con. Now, all the while, the leading eigenvalue of the matrix ases atkeeps $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n ^ { * } } \mathbf { J } _ { i } ^ { T } \mathbf { J } _ { i } } \end{array}$ i i eigenvalue of the Gauss-Newton matrix 1n Pni=1 JTi ∇2\`(zi; yi) Ji starts to decrease at the end of training, as can be seen from the green line in Figure 11(c). Finally, since the leading eigenvalue of the Gauss-Newton matrix is an excellent approximation to the leading eigenvalue of the Hessian (i.e. the sharpness), the sharpness also drops at the end of training, as can be seen from the orange line in Figure 11(c). + +![](images/3afa4026e93af581c17185584cf5281249dbef1d4b1b3a1e05e3bd52c68806bf.jpg) +Figure 11: We train a network using the cross-entropy loss on CIFAR-10. (a) The train loss. (b) The train accuracy. (c) The leading eigenvalue of the Hessian and of the Gauss-Newton matrix; observe that the latter is a great approximation to the former. (d) The leading eigenvalue of the matrix $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathbf { J } _ { i } ^ { T } \mathbf { J } _ { i } } \end{array}$ , which is the Gauss-Newton matrix except the $\nabla ^ { 2 } \ell$ terms; observe that this value constantly rises — it does not dip at the end of training. (e) The margin $\mathbf { z } _ { i } [ y _ { i } ] - \operatorname* { m a x } _ { j \neq y _ { i } } \mathbf { z } _ { i } [ j ]$ of 10 examples; observe that all these margins rise at the end of training. (f) the value $\mathbf { p } _ { i } [ y _ { i } ] ( 1 - \mathbf { p } _ { i } [ y _ { i } ] )$ for 10 examples; observe that all of these curves decline at the end of training. + +# D EMPIRICAL STUDY OF PROGRESSIVE SHARPENING + +In this appendix, we empirically study how problem parameters such as network width, network depth, and dataset size affect the degree to which progressive sharpening occurs. To study progressive sharpening on its own, without the confounding factor of instability, we train neural networks using gradient flow (Runge-Kutta) rather than gradient descent. Informally speaking, gradient flow does “what gradient descent would do if gradient descent didn’t have to worry about instability.” + +We observe that progressive sharpening occurs to a greater degree: (1) for narrower networks than for wider networks (which is consistent with infinite-width NTK theory), (2) for deeper networks than for shallower networks, and (3) for larger datasets than for smaller datasets. + +The effect of width When networks parameterized in a certain way (the “NTK parameterization”) are made infinitely wide and trained using gradient flow, the Hessian moves a vanishingly small amount during training, implying that no progressive sharpening occurs (Jacot et al., 2018; Lee et al., 2019; Jacot et al., 2020; Li & Liang, 2018). Therefore, a natural hypothesis is that progressive sharpening might attenuate as network width increases. We now run an experiment which supports this hypothesis. + +We consider fully-connected architectures with two hidden layers and tanh activations, with widths $\{ 3 2 , 6 4 , 1 2 8 , 2 5 6 , 5 1 2 , 1 0 2 4 \}$ . We train on a size-5,000 subset of CIFAR-10 using the cross-entropy loss. We train using gradient flow (details in $\ S 1 . 5 )$ . We consider both NTK parameterization and standard parameterization (Lee et al., 2019). + +In Figure 12, for each width, we train NTK-parameterized networks from five different random initializations, and plot the evolution of the sharpness during gradient flow. Observe that the maximum sharpness along the gradient flow trajectory is larger for narrow networks, and smaller for wide networks. (As elsewhere in this paper, note that the sharpness drops at the end of training due to the cross-entropy loss.) In Figure 13, we plot summary statistics from these training runs. Namely, define $\lambda _ { \mathrm { m a x } }$ as the maximum sharpness over the gradient flow trajectory, and define $\lambda _ { 0 }$ as the initial sharpness. In Figure 13(a), for each width we plot the mean and standard deviation of the maximum sharpness $\lambda _ { \mathrm { m a x } }$ over the five different random initializations. Observe that $\lambda _ { \mathrm { m a x } }$ becomes smaller, on average, as the width is made larger. In Figure 13(b), for each width we plot the mean and standard deviation of the maximum sharpness gain $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ over the five different random initializations. Observe that the maximum sharpness gain $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ also becomes smaller as the width is made larger. NTK theory suggests that $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ should deterministically tend to 1 as the width $ \infty$ , and Figure 13(b) is consistent with this prediction. + +In Figures 14 and 15, we conduct similar experiments, but with standard parameterization rather than NTK parameterization. Similar to NTK parameterization, we observe in Figure 14 that the sharpness rises more for narrow networks than for wide networks. + +![](images/2d1c17bc341619769646d030b8cee578dbcd4f24085e159a010c564f12d4c6fe.jpg) +Figure 12: NTK parameterization: evolution of the sharpness. We use gradient flow to train NTK-parameterized networks, and we track the evolution of the sharpness during training. For each width, we train from five different random initializations (different colors). Observe that the sharpness rises more when training narrow networks than when training wide networks. + +![](images/cd54a3c02b617d0cc0545f19c174842e2bab1dc73ef4ca3837fe2ab02c313ec8.jpg) +Figure 13: NTK parameterization: summary statistics. Left: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness $\lambda _ { \mathrm { m a x } }$ along the gradient flow trajectory. Observe that $\lambda _ { \mathrm { m a x } }$ decreases in expectation as the width increases. Right: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness gain $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ along the gradient flow trajectory. Observe that $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ decreases in expectation as the width increases. Indeed, NTK theory predicts that $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ should deterministically tend to 1 as width $ \infty$ and this plot is consistent with that prediction. + +![](images/bd1a970ecb7576a7154308af3a1348b43fe144b6a96f4f00b28b797b83855275.jpg) +Figure 14: Standard parameterization: evolution of the sharpness. We use gradient flow to train standard-parameterized networks, and we track the evolution of the sharpness during training. For each width, we train from five different random initializations (different colors). Observe that the sharpness rises more when training narrow networks than when training wide networks. + +![](images/f12a357b31bc1c8cc211c138d39bbf7fe4603d9ecdc4b6bf9c004c95787db477.jpg) +Figure 15: Standard parameterization: summary statistics. Left: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness $\lambda _ { \mathrm { m a x } }$ along the gradient flow trajectory. Observe that $\lambda _ { \mathrm { m a x } }$ tends to decrease in expectation as the width increases, though it is not clear whether this pattern still holds when moving from width 512 to width 1024 — more samples are needed. Right: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness gain $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ along the gradient flow trajectory. Observe that $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ decreases in expectation as the width increases. It is not clear whether or not $\lambda _ { \mathrm { m a x } } / \lambda _ { 0 }$ is deterministically tending to 1, but that does seem possible. + +Effect of depth We now explore the effect of network depth on progressive sharpening. We use gradient flow to train fully-connected tanh architectures of width 200 and varying depths — ranging from 1 hidden layer to 4 hidden layers. We train on a $5 \mathrm { k }$ subset of CIFAR-10 using both crossentropy loss (in Figure 16) and square loss (in Figure 17). For each depth, we train from five different random initializations (different colors). Observe that progressive sharpening occurs to a greater degree as network depth increases. + +![](images/ff731c8f4096c9935f202efe47d173e8c6b2cf7e9d6f15ec453dd1fb3420741c.jpg) +Figure 16: The effect of depth: cross-entropy. We use gradient flow to train networks of various depths, ranging from 1 hidden layer to 4 hidden layers, using cross-entropy loss. We train each network from five different random initializations (different colors). Observe that progressive sharpening occurs to a greater degree for deeper networks. + +![](images/16bb97f24b44d20f43fb40f9d5f18f72ef60f71b2ad5a00d776a6b98d5f2c7a4.jpg) +Figure 17: The effect of depth: mean squared error. We use gradient flow to train networks of various depths, ranging from 1 hidden layer to 4 hidden layers, using MSE loss. We train each network from five different random initializations (different colors). Observe that progressive sharpening occurs to a greater degree for deeper networks. + +Effect of dataset size We now explore the effect of dataset size on progressive sharpening. We use gradient flow to train a network on different-sized subsets of CIFAR-10. The network is a 2- hidden-layer, width-200 fully-connected tanh architecture, and we train using cross-entropy loss. The results are shown in Figure 18. Observe that progressive sharpening occurs to a greater degree as dataset size increases. + +![](images/d880f1b51cb504affeeafd27c3459939af183c6cb9ef5d32bdd6757998401d0e.jpg) +Figure 18: The effect of dataset size. We use gradient flow to train a network on varying-sized subsets of CIFAR-10. Observe that progressive sharpening occurs to a greater degree as the dataset size increases. + +# E SPEED OF DIVERGENCE ON QUADRATIC TAYLOR APPROXIMATION + +If the sharpness at some iterate is strictly greater than $2 / \eta$ , then gradient descent with step size $\eta$ is guaranteed to diverge if run on the quadratic Taylor approximation around that iterate. However, the speed of this divergence could conceivably be slow — in particular, the train loss might continue to decrease for many iterations before it starts to increase. In this appendix we empirically demonstrate, to the contrary, that at the Edge of Stability, gradient descent diverges quickly if, at some iterate, we start running gradient descent on the quadratic Taylor approximation around that iterate. + +We consider the fully-connected tanh network from section 3, trained on a 5,000-sized subsample of CIFAR-10 using both cross-entropy loss and MSE loss. At some timestep $t _ { 0 }$ during training, we suddenly switch from running gradient descent on the real neural training objective, to running gradient descent on the quadratic Taylor approximation around the iterate at step $t _ { 0 }$ . We do this for three timesteps before gradient descent has entered the Edge of Stability, and three afterwards. Figure 20 shows the results for cross-entropy loss, and Figure 21 shows the (similar) results for MSE loss. Before entering the Edge of Stability (top row), gradient descent on the quadratic Taylor approximation behaves similar to gradient descent on the real neural training objective — that is, the orange line almost overlaps the blue line. Yet after entering the Edge of Stability (bottom row), gradient descent on the quadratic Taylor approximation quickly diverges, whereas gradient descent on the real neural training objective makes consistent (if choppy) progress. + +In short, when gradient descent is not at the Edge of Stability, the quadratic Taylor approximation serves as a good model for the local progress of gradient descent. But when gradient descent is at the Edge of Stability, the quadratic Taylor approximation is an extremely poor model for the local progress of gradient descent. It is conceivable that there exists some simple modification to the quadratic Taylor model which would fix this issue (e.g. perhaps if one ignores a certain direction, the quadratic Taylor model is accurate). Nevertheless, unless/until such a fix is discovered, it is unclear why quadratic Taylor approximations should yield any insight into the local behavior of gradient descent. + +![](images/955346fd537d3f513f8126ca81891da22319daf5c8fd798c8c111382b7b6b357.jpg) +Figure 19: We train a neural network using cross-entropy loss (left) and MSE loss (right). In Figure 20 and 21, we show what happens when, at the iterations marked above by vertical dotted lines, we switch from gradient descent on the real neural training objective to gradient descent on the quadratic Taylor approximation. + +![](images/4614c000ab8bb8a0dc9aba4d60480907ab576f1310315b7f847bdf10fb234171.jpg) +Figure 20: Cross-entropy loss $( \eta = 2 / 6 0 )$ . At six different iterations during the training of the network from Figure 19 (marked by the vertical dotted black lines), we switch from running gradient descent on the real neural training objective (for which the train loss is plotted in blue) to running gradient descent on the quadratic Taylor approximation around the current iterate (for which the train loss is plotted in orange). Top row are timesteps $\{ 2 0 0 , 4 0 0 , 6 0 0 \}$ before gradient descent has entered the Edge of Stability; observe that the orange line (Taylor approximation) closely tracks the blue line (real objective). Bottom row are timesteps {800, 1000, 1200} during the Edge of Stability; observe that the orange line quickly diverges, whereas the blue line does not. + +![](images/d96e7be6e15e63ae8d6c0c90126924b85697ad5673bba27ca664692f011134fb.jpg) +Figure 21: MSE loss $( \eta = 2 / 2 0 0 )$ . At six different iterations during the training of the network from Figure 19 (marked by the vertical dotted black lines), we switch from running gradient descent on the real neural training objective (for which the train loss is plotted in blue) to running gradient descent on the quadratic Taylor approximation around the current iterate (for which the train loss is plotted in orange). Top row are timesteps (2000, 3000, 4000) before gradient descent has entered the Edge of Stability; observe that the orange line (Taylor approximation) closely tracks the blue line (real objective). Bottom row are timesteps (5000, 6000, 7000) during the Edge of Stability; observe that the orange line quickly diverges, whereas the blue line does not. + +# F “OPTIMAL” STEP SIZE SELECTION + +One heuristic for setting the step size of gradient descent is to set the step size at iteration $t$ to $\eta _ { t } =$ $1 / \lambda _ { t }$ , where $\lambda _ { t }$ is the sharpness at iteration $t$ . While this heuristic is computationally impractical due to the time required to compute the sharpness at each iteration, it is often regarded as an ideal, for instance in LeCun et al. (1998) (Eq. 39), LeCun et al. (1993), and Schaul et al. (2013) (Eq 8). The motivation for this heuristic is: if all that is known about the training objective is that the local sharpness is $\lambda$ , then a step size of $1 / \lambda$ maximizes the guaranteed decrease in the training objective that would result from taking a step. + +First, we demonstrate (on a single numerical example) that the dynamic step size $\eta _ { t } = 1 / \lambda _ { t }$ is outperformed by the baseline approach of gradient descent with a fixed step size $\eta _ { t } = 1 / \lambda _ { 0 }$ , where $\lambda _ { 0 }$ is the sharpness at initialization. In Figure 22, we train the network from $\ S 3$ using both the dynamic $\eta = 1 / \lambda _ { t }$ step size heuristic as well as the baseline fixed step size of $\eta = 1 / \lambda _ { 0 }$ . Observe that the $\eta \ : = \ : 1 / \lambda _ { 0 }$ baseline outperforms the $1 / \lambda _ { t }$ heuristic. Intuitively, because of progressive sharpening, the $\eta _ { t } = 1 / \lambda _ { t }$ heuristic anneals the step size, and therefore ends up taking steps that are suboptimally small. In contrast, while the $\eta _ { t } = 1 / \lambda _ { 0 }$ baseline quickly becomes unstable, this instability is apparently a worthwhile “price to pay” in return for the benefit of taking larger steps. + +![](images/514a14703006dd249fb03c2eb5a2e44d6f8dc50d85bed04cfb07f63ac05aa485.jpg) +Figure 22: A dynamic step size $\eta _ { t } = 1 / \lambda _ { t }$ underperforms a fixed step size of $\eta _ { t } = 1 / \lambda _ { 0 }$ + +Another natural idea is to dynamically set the step size at iteration $t$ to $\eta _ { t } = 1 . 9 / \lambda _ { t }$ . This step size rule takes larger steps than the $\eta = 1 / \lambda _ { t }$ rule while still remaining stable. In Figure 23, we compare this $\eta _ { t } = 1 . 9 / \lambda _ { t }$ rule to a baseline approach of gradient descent with a fixed step size $\eta _ { t } = 1 . 9 \bar { / } \lambda _ { 0 }$ , where $\lambda _ { 0 }$ is the sharpness at initialization. Observe that the baseline of a fixed $1 . 9 / \lambda _ { 0 }$ step size outperforms the dynamic $\eta _ { t } = 1 . 9 / \lambda _ { t }$ rule. + +![](images/c7d74ca5c721efee3fa1e367f439b3ac52ad82217cc62d2f294d77d1915428ca.jpg) +Figure 23: A dynamic step size $\eta _ { t } = 1 . 9 / \lambda _ { t }$ underperforms a fixed step size of $\eta _ { t } = 1 . 9 / \lambda _ { 0 }$ . + +# G EVOLUTION OF SHARPNESS DURING SGD + +In this appendix, we briefly illustrate how the sharpness evolves during stochastic gradient descent. In Figure 24, we train the tanh network from $\ S 3$ using SGD with both cross-entropy loss (top) and mean squared error (bottom). We train using a range of batch sizes (different colors). We observe the following: + +1. During large-batch SGD, the sharpness behaves similar to full-batch gradient descent: it rises to $2 / \eta$ (marked by the black horizontal dashed line) and then hovers just above that value. +2. Consistent with prior reports, we find that the smaller the batch size, the lower the sharpness (Keskar et al., 2016; Jastrz˛ebski et al., 2017; 2019; 2020). +3. Notice that when training with cross-entropy loss at batch size 8 (the blue line), the sharpness decreases throughout most of training. The train accuracy (not pictured) is only $6 6 \%$ when the sharpness starts to decrease, which suggests that the cause of this decrease is unrelated to the effect described in Appendix C, whereby the sharpness decreases at the end of training. Figure 5(a) of Jastrz˛ebski et al. (2020) also depicts a network where the sharpness decreases during SGD training. + +![](images/43c9eb37df93a822a4ae22a49f7be8053d7b68ffd685c8aaa8216120ea5ec9e4.jpg) +Figure 24: Evolution of sharpness during SGD. We train a network using SGD at various batch sizes, and plot the evolution of the sharpness. In the top row, we train with cross-entropy loss and step size $\eta = 0 . 0 2$ ; in the bottom row, we train with mean squared error and step size $\eta = 0 . 0 1$ . The leftmost column shows all batch sizes; the center column focuses on just the smallest; and the rightmost column focuses on just the largest. The black dashed horizontal line marks $2 / \eta$ . + +# H SGD ACCLIMATES TO THE HYPERPARAMETERS + +In this appendix, we conduct an experiment which suggests that some version of “Edge of Stability” may hold for SGD. + +One way to interpret our main findings is that gradient descent “acclimates” to the step size in such a way that each training update sometimes increases, and sometimes decreases, the training loss, yet an update with a smaller step size would always decrease the training loss. We now demonstrate that this interpretation may generalize to SGD. In particular, we will demonstrate that SGD seems to “acclimate” to the step size and batch size in such a way that an actual update sometimes increases and sometimes decreases the loss in expectation, yet a update with a larger step size or smaller batch size would almost always increase the loss in expectation, and a step with a smaller step size or a larger batch size would almost always decrease the loss in expectation. + +In Figure 25, we train the tanh network from $\ S 3$ with MSE loss, using SGD with step size 0.01 and batch size 32. We periodically compute the training loss (over the full dataset) and plot these on the left pane of Figure 25. Observe that the training loss does not decrease monotonically, but of course this is not surprising — SGD is a random algorithm. However, what may be more surprising is that SGD is not even decreasing the training loss in expectation. On the right pane of Figure 25, every 500 steps during training, we use the Monte Carlo method to approximately compute the expected change in training loss that would result from an SGD step (the expectation here is over the randomness involved in selecting the minibatch). Observe that at many points during training, an SGD step would decrease the loss (as desired) in expectation, but at other points, and SGD step would increase the loss in expectation. + +In Figure 26(a), while training that network, we compute the expected change in training loss that would result from taking an SGD step with the same step size used during training (i.e. 0.01), but half the batch size used during training (i.e. 16). We observe that an SGD step with half the batch size would consistently cause an increase in the training loss in expectation. In Figure 26(b) we repeat this experiment, but with twice the batch size used during training (i.e. 64). Notice that an SGD step with twice the batch size would consistently cause a decrease in the training loss in expectation. In Figure 26(c) and (d) repeat this experiment with the step size; we observe that an SGD step with a larger step size (0.02) would consistently increase the training loss in expectation, while an SGD step with a smaller step size (0.005) would consistently decrease the training loss in expectation. + +In Figure 27, as a “control” experiment, we both train and measure the expected loss change under the following four hyperparameter settings: (step size 0.01, batch size 16), (step size 0.01, batch size 64), (step size 0.02, batch size 32), and (step size 0.005, batch size 32). In each case, we observe that, after a brief period at the beginning of training, each SGD update sometime increases and sometimes decreases the training loss in expectation. + +Therefore, at least for this single network, we can conclude that no matter the hyperparameters, SGD quickly navigates to, and then lingers in, regions of the loss landscape in which an SGD update with those hyperparameters sometimes increases, and sometimes decreases the training loss in expectation, yet an SGD update with a smaller step size or larger batch size would consistently decrease the loss in expectation, and an SGD update with a larger step size or smaller batch size would consistently increase the loss in expectation. + +![](images/4e018c957930a5474c4334e97cd3b044261c25dc2f6b1ba664e7b8e5382ded7d.jpg) +Figure 25: SGD does not consistently decrease the training loss in expectation. We train the tanh network from section 3 with MSE loss, using SGD with step size 0.01 and batch size 32. Periodically during training, we compute (left) the full-batch training loss, and (right) the expected change in the full-batch training loss that would result from taking an SGD step (where the expectation is over the randomness in sampling the minibatch). Strikingly, note that after the very beginning of training, the expected loss change is sometimes negative (as desired) but oftentimes positive. See Figure 26. + +![](images/44ca4a3ebd286c2444ccd3e72ea0ff8569c9de959a3b0a9610b17cae9cd771cc.jpg) +Figure 26: An SGD step with a smaller learning rate or a larger batch size than the ones used during training would consistently decrease the loss in expectation. At regular intervals during the training run depicted in Figure 25 (with $\eta = 0 . 0 1$ and batch size 32), we measure the expected change in the full-batch training loss that would result from an SGD step with a different step size or batch size. Observe that taking an SGD step with a smaller step size or a larger batch size would consistently have decreased the loss in expectation, while taking an SGD step with a larger step size or a smaller batch size would have consistently increased the loss in expectation. + +![](images/f4e7c089edb03d37c47b5d8e944e1bde4562157461a6077650839c660ed4a356.jpg) +Figure 27: Control experiment. Above, in Figure 26(a), as we trained a network with step size 0.01 and batch size 32, we evaluated the expected change in training loss that would result from taking an SGD step with step size 0.01 and batch size 64. Here, in Figure 27(a), as a “control experiment,” we train the network with step size 0.01 and batch size 64, and evaluate the expected change in training loss that would result from taking an SGD step with the same step size and batch size. We observe that an SGD step using the same step size and batch size that are used during training would sometimes increase and sometimes decrease the training loss in expectation. The other three panes are analogous. + +# I EXPERIMENTAL DETAILS + +# I.1 VARYING ARCHITECTURES ON 5K SUBSET OF CIFAR-10 + +Dataset. The dataset consists of the first 5,000 examples from CIFAR-10. To preprocess the dataset, we subtracted the mean from each channel, and then divided each channel by the standard deviation (where both the mean and stddev were computed over the full CIFAR-10 dataset, not the 5k subset). + +Architectures. We experimented with two architecture families: fully-connected and convolutional. For each of these two families, we experimented with several different activation functions, and for convolutional networks we experimented with both max pooling and average pooling. + +The PyTorch code for e.g. the fully-connected ReLU network is as follows: + +nn.Sequential( nn.Flatten(), nn.Linear(3072, 200, bias $=$ True), nn.ReLU(), nn.Linear(200, 200, bias $=$ True) nn.ReLU(), nn.Linear(200, 10, bias $=$ True) +) + +Networks with other activation functions would have nn.ReLU() replaced by nn.ELU(), nn.Tanh(), nn.Softplus(), or nn.Hardtanh(). + +The PyTorch code for e.g. the convolutional ReLU network with max-pooling is as follows: + +nn.Sequential( nn.Conv2d(3, 32, bias $=$ True, kernel_siz ${ \tt 3 } = 3 { \tt 1 }$ , padding $^ { \cdot = 1 }$ ), nn.ReLU(), nn.MaxPool2d(2), nn.Conv2d(32, 32, bias $=$ True, kernel_size $= 3$ , padding $^ { = 1 }$ ), nn.ReLU(), nn.MaxPool2d(2), nn.Flatten(), nn.Linear(2048, 10, bias $=$ True) +) + +Networks with other activation functions would have nn.ReLU() replaced by nn.ELU() or nn.Tanh(), and networks with average pooling would have nn.MaxPool2d(2) replaced by nn.AvgPool2d(2). + +For all of these networks, we use the default PyTorch initialization. That is, both fully-connected layers and convolutional layers have the entries of their weight matrix and bias vector sampled i.i.d from Uniform(− √ 1fan in , √ 1fan in ). + +Loss functions. For a $k$ -class classification problem, if the network outputs are $\textbf { z } \in \mathbb { R } ^ { k }$ and the correct class is $\textit { i } \in \ [ k ]$ , then the mean squared error (MSE) loss is defined as $\begin{array} { r } { \frac { 1 } { 2 } \left[ ( z [ i ] - 1 ) ^ { 2 } + \sum _ { j \neq i } z [ j ] ^ { 2 } \right] } \end{array}$ . That is, we encode the correct class with a “1” and the other classes with $^ { 6 6 } 0$ .” The cross-entropy loss is defined as $- \log \left( \frac { \exp ( z [ i ] ) } { \sum _ { j \neq i } \exp ( z [ j ] ) } \right) .$ . + +# I.2 STANDARD ARCHITECTURES ON CIFAR-10 + +To preprocess the CIFAR-10 dataset, we subtracted the mean from each channel, and then divided each channel by the standard deviation. + +Since training with full-batch gradient descent is slow, we opted to experiment on relatively shallow networks. The VGG networks (both with and without BN) are VGG-11’s, from the implementation here: https://github.com/chengyangfu/pytorch-vgg-cifar10/ blob/master/vgg.py, with the dropout layers removed. The ResNet is the (non-fixup) ResNet32 implemented here: https://github.com/hongyi-zhang/Fixup. + +For the two networks with batch normalization, running gradient descent with full-dataset batch normalization would not have been feasible under our GPU memory constraints. Therefore, we instead used ghost batch normalization (Hoffer et al., 2017) with 50 ghost batches of size 1,000. This means that we divided the 50,000 examples in CIFAR-10 into 50 fixed groups of size 1,000 each, and defined the overall objective function to be the average of 50 fixed batch-wise objectives. To correctly compute the overall gradient of this training objective, we can just run backprop 50 times (once on each group) and average the resulting gradients. + +To compute the sharpness over the full CIFAR-10 dataset would have been computationally expensive. Therefore, in an approximation, we instead computed the sharpness over just the first 5,000 examples in the dataset (or, for the BN networks, over the first 5 batches out of 50). + +# I.3 BATCH NORMALIZATION EXPERIMENTS + +We used the CNN architecture from $\ S J$ (described above in $\ S \mathrm { I . 1 }$ ), but with a BatchNorm2d() layer inserted after each activation layer. + +Since our GPUs did not have enough memory to run batch normalization with the full dataset of size 5,000, we used ghost batch normalization with five ghost batches of size 1,000 (see I.2 for details). + +# I.4 TRANSFORMER ON WIKITEXT-2 + +We used both the Transformer architecture and the preprocessing setup from the official PyTorch (Paszke et al., 2019) word-level language modeling tutorial: https://github.com/ pytorch/examples/tree/master/word_language_model. We used the settings ninp $\scriptstyle \phantom { . } = 2 0 0$ , nhead $^ { = 2 }$ , nhid $= 2 0 0$ , nlayer ${ \tt S } = 2$ , dropout $= 0$ . We set bptt $\qquad = \ 3 5$ , which means that we divided the corpus into chunks of 35 tokens, and trained the network, using the negative log likelihood loss, to predict each token from the preceding tokens in the same chunk. Since computing the sharpness over the full dataset would not have been computationally practical, we computed the sharpness over a subset comprising 2500 training examples. + +# I.5 RUNGE-KUTTA + +We used the “RK4” fourth-order Runge-Kutta algorithm (Press et al., 1992) to numerically integrate the gradient flow ODE. The Runge-Kutta algorithm requires a step size. Rather than use a sophisticated algorithm for adaptive step size control, we decided to take advantage of the fact that we were already periodically computing the sharpness: at each step, we set the step size to $\alpha / \lambda$ , where $\alpha$ is a tunable parameter and $\lambda$ is the most recent value for the sharpness. We set $\alpha = 1$ or $\alpha = 0 . 5$ . + +# I.6 RANDOM PROJECTIONS + +In order to ascertain whether gradient descent at step size $\eta$ initially followed the gradient flow trajectory (and, if so, for how long), we monitored the $\ell _ { 2 }$ distance in weight space between the gradient flow solution at time $t$ and the gradient descent iterate at step $t / \eta$ . One way to do would be as follows: (a) when running gradient flow, save the weights after every $\Delta t$ units of time, for some parameter $\Delta t$ ; (b) when running gradient descent, save the weights at each $\big ( \frac { \Delta t } { \eta } \big )$ -th step; (c) plot the difference between these two sequences. (Note that this approach requires $\Delta t$ to be divisible by $\eta$ .) + +We essentially used this approach, but with one modification: regularly saving the entire network weight vector would have consumed a large amount of disk space, so we instead saved lowdimensional random projections of the network weights. To be clear, let $d$ be the number of network weights, and let $k$ be the number of random projections (a tunable parameter chosen such that $k \ll d ,$ ). Then we first generated a matrix $\mathbf { M } \in \breve { \mathbb { R } } ^ { k \times d }$ by sampling each entry i.i.d from the standard normal distribution. During training, rather than periodically save the whole weight vector (a $d$ -dimensional vector), we premultiplied this vector by the matrix $\mathbf { M }$ to obtain a $k$ -dimensional vector, and we periodically saved these vectors instead. Then we plotted the $\ell _ { 2 }$ distance between the low-dimensional vectors from gradient flow, and the low-dimensional vectors from gradient descent. + +# J EXPERIMENTS: VARY ARCHITECTURES + +In this appendix, we fix the task as that of fitting a 5,000-sized subset of CIFAR-10, and we verify our main findings across a broad range of architectures. + +Procedure We consider fully-connected networks and convolutional networks, the latter with both max-pooling and average pooling. For all of these, we consider tanh, ReLU, and ELU activations, and for fully-connected networks we moreover consider softplus and hardtanh activations. We train each network with both cross-entropy and MSE loss. See $\ S \mathrm { I . 1 }$ for full experimental details. + +In each case, we first use the Runge-Kutta method to numerically integrate the gradient flow ODE (see $\ S 1 . 5$ for details). For architectures that give rise to continuously differentiable training objectives, the gradient flow ODE is guaranteed to have a unique solution (which we call the gradient flow trajectory), and Runge-Kutta will return a numerical approximation to this solution. On the other hand, for architectures with ReLU, hardtanh, or max-pooling, the training objective is not continuously differentiable, so the gradient flow ODE does not necessarily have a unique solution, and there are no guarantees a priori on what Runge-Kutta will return (more on this below under the “findings” heading). Still, in both cases, since our implementation of Runge-Kutta automatically adjusts the step size based on the local sharpness in order to remain stable, the Runge-Kutta trajectory can be roughly viewed as “what gradient descent would do if instability was not an issue.” + +We then run gradient descent at a range of step sizes. These step sizes $\eta$ were chosen by hand so that the quantity $2 / \eta$ would be spaced uniformly between $\lambda _ { 0 }$ (the sharpness at initialization) and $\lambda _ { \mathrm { m a x } }$ (the maximum sharpness along the Runge-Kutta trajectory). + +Results During Runge-Kutta, we observe that the sharpness tends to continually increase during training (progressive sharpening), with the exception that when cross-entropy loss is used, the sharpness decreases at the very end of training, as explained in Appendix C. + +During gradient descent with step size $\eta$ , we observe that once the sharpness reaches $2 / \eta$ , it ceases to increase much further, and instead hovers right at, or just above, the value $2 / \eta$ . For reasons unknown, it tends to be true that for MSE loss, the sharpness hovers just a tiny bit above the value $2 / \eta$ , while for cross-entropy loss the gap between the sharpness and the value $2 / \eta$ is a bit larger. + +For each step size, we monitor the distance between the gradient descent trajectory and the RungeKutta trajectory — that is, we monitor the distance between the Runge-Kutta iterate at time $t$ and the gradient descent iterate at step $t / \eta$ (see $\ S 1 . 6$ for details). Empirically, for architectures that give rise to continuously differentiable training objectives, we observe that this distance is nearly zero before the sharpness hits $2 / \eta$ , and it starts to climb immediately afterwards. This means that gradient descent closely tracks the gradient flow trajectory so long as the sharpness remains less than $2 / \eta$ . Note that this finding was not a foregone conclusion: gradient descent is guaranteed to track the gradient flow trajectory in the limit of infinitesimal step sizes (since gradient descent is the forward Euler discretization of the gradient flow ODE), but for non-infinitesimal step sizes, there is discretization error, which is studied in Barrett & Dherin (2021). Our empirical finding is essentially that this discretization error is small compared to the difference between trajectories caused by instability. + +On the other hand, for architectures with non-differentiable components such as ReLU or maxpooling, we sometimes observe that gradient descent tracks the Runge-Kutta trajectory so long as the sharpness remains less than $2 / \eta$ , but we also sometimes observe that the gradient descent trajectories differ from one another (and from Runge-Kutta) from the beginning of training. In the former case, we can infer that the gradient flow trajectory apparently does exist, and is returned by Runge-Kutta; in the latter case, we can infer that either (a) the gradient flow trajectory does not exist, or (b) that it does exist (and is returned by Runge-Kutta), but the step sizes we used for gradient descent were too large to track it. + +# J.1 FULLY-CONNECTED TANH NETWORK + +# J.1.1 SQUARE LOSS + +![](images/6d0728c745c52a1364ffe39ce6e8f8dd29a4e209e89c23debe9bab6652a7db62.jpg) +Figure 28: Gradient flow. We train the network to $9 9 \%$ accuracy with gradient flow, by using the Runge-Kutta method to discretize the gradient flow ODE (details in $\ S 1 . 5 )$ ). We plot the train loss (left), sharpness (center), and train accuracy (right) over time. Observe that the sharpness tends to continually increase (except for a slight decrease at initialization). + +![](images/35dd58b8cdeebf43d0600dd08fe120012208937c690e830d55713a80255dffe1.jpg) +Figure 29: Gradient descent. We train the network to $9 9 \%$ accuracy using gradient descent at a range of step sizes $\eta$ . Top left: we plot the train loss curves, with a vertical line marking the iteration where the sharpness first crosses $2 / \eta$ . Observe that the train loss monotonically decreases before this point, but behaves non-monotonically afterwards. Top middle: we plot the sharpness (measured at regular intervals during training). For each step size, the horizontal dashed line of the appropriate color marks the maximum stable sharpness $2 / \eta$ . Observe that the sharpness tends to increase during training until reaching the value $2 / \eta$ , and then hovers just a bit above that value. Bottom left: we track the $\ell _ { 2 }$ distance between (random projections of) the gradient flow iterate at time $t$ and the gradient descent iterate at iteration $t / \eta$ (details in $\ S 1 . 6 )$ . For each step size $\eta$ , the vertical dotted line marks the time when the sharpness first crosses $2 / \eta$ . Observe that the distance is essentially zero until this time, and starts to grow afterwards. From this, we can conclude that gradient descent closely tracks the gradient flow trajectory (moving at a speed $\eta$ ) until the point on that trajectory where the sharpness reaches $2 / \eta$ . Bottom middle: to further visualize the previous point, we plot the evolution of sharpness during gradient descent, but with time $\underline { { \underline { { \mathbf { \Pi } } } } }$ iteration $\times$ step size) on the $\mathbf { X }$ -axis rather than iteration. We plot the gradient flow sharpness in black. Observe that the gradient descent sharpness matches the gradient flow sharpness until reaching the value $2 / \eta$ . + +![](images/9f1176dc6f69dfd4fa5b6d6db6c53ba62bde8c94190927b8a35edf279a89486b.jpg) +Figure 30: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/b81380c85e577848398396434eefa62bd8bb599a82a59eb78c6a957b33272662.jpg) +Figure 31: Gradient descent. Refer to the Figure 29 caption for more information. + +# J.2 FULLY-CONNECTED ELU NETWORK + +# J.2.1 SQUARE LOSS + +![](images/b3e94c174f923c6bdced758acf3a2ced2f1e7c66fc53dab900bcf78bf300965d.jpg) +Figure 32: Gradient flow. Refer to the Figure 28 caption for more information. + +![](images/4c0a86b8781bb31461dc1bce13886f593f43f99df4ce934a163bc05a705bb9a8.jpg) +Figure 33: Gradient descent. Refer to the Figure 29 caption for more information. + +![](images/8a86945e72f1fe25a4f3070e013421495ee08fa9ff33db6a60d041810c4e4687.jpg) +Figure 34: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/4a329418dd77653b7f29b39603d5dd44d54504b861cacf6f47e369f3b67d05ba.jpg) +Figure 35: Gradient descent. Refer to the Figure 29 caption for more information. + +# J.3.1 SQUARE LOSS + +![](images/8391669a360d732e7f83b1494325054c11e62f2f9235cb2126999368ceed0154.jpg) +Figure 36: Gradient flow. Refer to the Figure 28 caption for more information. + +![](images/d465735b68886c33d5c8841e0e2e1bb80c03a351a2dc855e3401d800ad91daba.jpg) +Figure 37: Gradient descent. Refer to the Figure 29 caption for more information. + +![](images/0e9bb698a4336e001ad81c8114f36400b529a81502da9475df49f39f3eb586c4.jpg) +Figure 38: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/f00372e0bd8673c2f3d9574c3cabd991903dff9a0bc1d339f01d1206dd7dfc9e.jpg) +Figure 39: Gradient descent. Refer to the Figure 29 caption for more information. + +# J.4 FULLY-CONNECTED RELU NETWORK + +Note that since the ReLU activation function is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. + +# J.4.1 SQUARE LOSS + +![](images/b65f18aee6331aef07845d67a2af73b7ba1cbdc19cfcdb0b6b9b48beab0e5596.jpg) +Figure 40: Runge-Kutta. We train the network to $9 9 \%$ accuracy using the Runge-Kutta algorithm. (Since a unique gradient flow trajectory is not guaranted to exist, we hesitate to call this “gradient flow”; Runge-Kutta should essentially be viewed as gradient descent with a very small step size.) Observe that the sharpness tends to continually increase. + +![](images/b1b8344d483a24a4f72f14784ebc40d2c8d2c9bde3f1f512b731dbdf4b9a4abb.jpg) +Figure 41: Gradient descent. All panes except bottom left: refer to the Figure 29 caption for more information. Bottom left: We track the $\ell _ { 2 }$ distance between (random projections of) the Runge-Kutta iterate at time $t$ and the gradient descent iterate at iteration $t / \eta$ . For each step size $\eta$ , the vertical dotted loss marks the time when the sharpness first crosses $2 / \eta$ . In contrast to Figure 29, here the distance between gradient descent and gradient flow starts to noticeably grow from the beginning of training. From this we conclude that for this architecture, gradient descent does not track the Runge-Kutta trajectory at first. Furthermore, for this architecture, observe that the training loss starts behaving non-monotonically (and the sharpness begins to plateau) before the sharpness hits $2 / \eta$ . As discussed in Appendix A (“Caveats”), we believe that this is also due to the fact that ReLU is non-differentiable. + +![](images/080e0e96ecf6c9e65d4b58a5e5a0f60aabcee34aaba1510ebe924b9b16f9d32b.jpg) +Figure 42: Runge-Kutta. Refer to the Figure 40 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/b95f7ea697fca38a44fbe0a655898d68852bae81cd422940ae5b3fd52bea86a3.jpg) +Figure 43: Gradient descent. Refer to the Figure 41 caption for more information. + +# J.5 FULLY-CONNECTED HARD TANH NETWORK + +Note that since the hardtanh function is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. + +# J.5.1 SQUARE LOSS + +![](images/0bc9768cf25d338c4e1b508e3813c7aa7030dc08065642675bc803c8b6a17e49.jpg) +Figure 44: Runge-Kutta. Refer to the Figure 40 caption for more information. + +![](images/ca1f76cfa2ac88da464072247c958261cbf89072c609dc6386a4cd57acbd6c37.jpg) +Figure 45: Gradient descent. Refer to the Figure 41 caption for more information. + +![](images/342e59c506d9595ac0d85b7e34e627ce8b7374c71b5574aa6a1cb0e9d26dd183.jpg) +Figure 46: Runge-Kutta. Refer to the Figure 40 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/af9781d3525550b67896d26c9d9bfa28b0b2dc47fa42586b6b87a5094e0afa19.jpg) +Figure 47: Gradient descent. Refer to the Figure 41 caption for more information. + +Note that since max-pooling is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. + +# J.6.1 SQUARE LOSS + +![](images/049f303ad138b110803f979b7cd1c1bc4addf624becba6394c052c1b040aebfc.jpg) +Figure 48: Runge-Kutta. We train the network to $9 9 \%$ accuracy using the Runge-Kutta algorithm. (Since a unique gradient flow trajectory is not guaranted to exist, we hesitate to call this “gradient flow.”) Observe that the sharpness tends to continually increase. + +![](images/ad0d0af556a7c12ef216ca82d353b0fd180bad9803fb4075e81f25cc0d936f0a.jpg) +Figure 49: All panes except bottom left: refer to the Figure 29 caption for more information. Bottom left: We track the $\ell _ { 2 }$ distance between (random projections of) the Runge-Kutta iterate at time $t$ and the gradient descent iterate at iteration $t / \eta$ . For each step size $\eta$ , the vertical dotted loss marks the time when the sharpness first crosses $2 / \eta$ . Observe that the distance is essentially zero until this time, and starts to grow afterwards. From this, we can conclude that even though the training objective is not differentiable, gradient descent does closely track the Runge-Kutta trajectory (moving at a speed $\eta$ ) until the point on that trajectory where the sharpness reaches $2 / \eta$ . + +![](images/390df9bc0caf092f1864bb885d53b2f559d540dec9c69f3c53b959cf46f51120.jpg) +Figure 50: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/73eeff454b86753bd8ec4fb9bf4078fa0e6a6b427d87c5bb2056f5089786859d.jpg) +Figure 51: Gradient descent. Refer to the Figure 49 caption for more information. + +# J.7 CONVOLUTIONAL ELU NETWORK WITH MAX POOLING + +Note that since max-pooling is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. + +# J.7.1 SQUARE LOSS + +![](images/2af547d1f32796d61084d68c2dbd95aa8e5becbcf3e3511a6a325b86163fd6a6.jpg) +Figure 52: Runge-Kutta. Refer to the Figure 48 caption for more information. + +![](images/2f08a3738c3c4f650dc342aaf0d3c51987cd69fa097750905fd6cafe13c7463f.jpg) +Figure 53: Gradient descent. Refer to the Figure 49 caption for more information. + +![](images/58012bb2f46014c9b3153fb53031b4ea27b0b52131061a6b4189c48550105c38.jpg) +Figure 54: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/ba9b5bfab7634947d642740eafd3d7dbcc6a20a7e108fd5690e2f8ceff1c12ff.jpg) +Figure 55: Gradient descent. Refer to the Figure 49 caption for more information. + +# J.8 CONVOLUTIONAL RELU NETWORK WITH MAX POOLING + +Note that since ReLU and max-pooling are not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. + +# J.8.1 SQUARE LOSS + +![](images/fd92020fa7ca8afa0810186f8a4121cf5fecb3ec7af7f5c7e7b1444e6a754bed.jpg) +Figure 56: Runge-Kutta. Refer to the Figure 48 caption for more information. + +![](images/91310eec236fbee78503c5743b32a57a566c19f6237a11f26c4b0219d3682bf7.jpg) +Figure 57: Gradient descent. Refer to the Figure 49 caption for more information. + +![](images/100d02a6124d6452aa4bb5dfe59c76dcde76accc738b225cd0bdece8197df904.jpg) +Figure 58: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/9409954b1c9ee42bb893685bac7124031f52738142063be2c4b738c4ac9c99a7.jpg) +Figure 59: Gradient descent. Refer to the Figure 49 caption for more information. + +# J.9.1 CROSS-ENTROPY LOSS + +![](images/5709a6237df282611c7baa12addbfcdbe90ba38b0000decf5a717257a402274b.jpg) +Figure 60: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/71d1906e291cd73f61a9d1bbaacf0091ecd95343bb8ad55ca5f4fa74b854c5b5.jpg) +Figure 61: Gradient descent. Refer to the Figure 29 caption for more information. + +# J.10.1 SQUARE LOSS + +![](images/53c280bcc23d16aba3e84c02fb6532fec9fc7de8d0160b84852f60c784ce8be2.jpg) +Figure 62: Gradient flow. Refer to the Figure 28 caption for more information. + +![](images/3d407f0114b83cd238a86ceaba42240b89c4b59c0bbfd7368cebfa35a4f101ea.jpg) +Figure 63: Gradient descent. Refer to the Figure 29 caption for more information. + +![](images/1894c7a666ffc7497d7d86e498b9a5e964fde2e347b6b74f1a3313dd868ba228.jpg) +Figure 64: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/7c724928bf06230da48366d4e1f0665b73f1c854dc5641acd31793f7356b8eb2.jpg) +Figure 65: Gradient descent. Refer to the Figure 29 caption for more information. + +# J.11 CONVOLUTIONAL RELU NETWORK WITH AVERAGE POOLING + +Note that since ReLU is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. + +# J.11.1 SQUARE LOSS + +![](images/458da1d795e651173e3672e15724d65cf0b83c8ef58dc3419ea3fd23c59e0b71.jpg) +Figure 66: Runge-Kutta. Refer to the Figure 48 caption for more information. + +![](images/27877445d07844f31aeff350ad3fdfd6a393b75126e4eab34b35f81e37b66164.jpg) +Figure 67: Gradient descent. Refer to the Figure 49 caption for more information. + +![](images/269f1c53f21513b6d64f12624096c06f96d2d6a84df447da4be523f77bbd2b45.jpg) +Figure 68: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. + +![](images/758289dfcef21a8f3104a03dcf8b3dede0108d64705bb8500712558ff1c63a18.jpg) +Figure 69: Gradient descent. Refer to the Figure 49 caption for more information. + +# K BATCH NORMALIZATION EXPERIMENTS + +In this appendix, we demonstrate that our findings hold for networks that are trained with batch normalization (BN) (Ioffe & Szegedy, 2015). We experiment on a size-5,000 subset of CIFAR10, and we consider convolutional networks with three different activation functions: ELU (Figure 70-71, tanh (Figure 72-73), and ReLU (Figure 74-75). See $\ S \mathrm { I } . 3$ for experimental details. + +Empirically, our findings hold for batch-normalized networks. The one catch is that when training batch-normalized networks at very small step sizes, it is apparently inadequate to measure the sharpness directly at the iterates themselves, as we do elsewhere in the paper. Namely, observe that in Figure 71(e), when we run gradient descent at the red step size, the sharpness (measured directly at the iterates) plateaus a bit beneath the value $2 / \eta$ . At first, this might sound puzzling: after all, if the sharpness is less than $2 / \eta$ then gradient descent should be stable. The explanation is that the sharpness in between successive iterates does in fact cross $2 / \eta$ . In Figure 71(b), we track the maximum sharpness on the path “in between” successive iterates. (To estimate the maximum sharpness between a pair of successive iterates, we compute the sharpness at a grid of eight points spaced evenly between them, and then take the maximum of these values.) Observe that this quantity does rise to $2 / \eta$ and hover there. We do not know why measuring the sharpness between iterates is necessary for batch-normalized networks, whereas for non-BN networks it suffices to measure the sharpness only at the iterates themselves. + +In $\ S \mathrm { K } . 1$ , we reconcile these findings with Santurkar et al. (2018). + +![](images/404db4afb0e9177c92c54850c7f0dc79184ef00be99f37df5201e730d36bdd39.jpg) +Figure 70: We train a ELU CNN $\left( + \mathbf { B } \mathbf { N } \right)$ using gradient flow. + +![](images/96afe39b6a9023f038910844c477d8fff9a2289d8886f7c8e921094afd86297d.jpg) +Figure 71: We train an ELU CNN $\mathbf { \left( + \delta B N \right) }$ using gradient decent at a range of step sizes. (a) we plot the train loss, with a vertical dotted line marking the iteration where the sharpness on the path first crosses $2 / \eta$ . The inset shows that the red curve is indeed behaving non-monotonically. (b) we track the sharpness “between iterates.” This means that instead of computing the sharpness right at the iterates themselves (as we do elsewhere in the paper, and in pane (e) here), we compute the maximum sharpness on the line between between successive iterates. Observe that this quantity rises to $2 / \eta$ (marked by the horizontal dashed line) and then hovers right at, or just above that value. (c) we plot the same quantity by “time” $\mathit { \Psi } = \mathit { \Psi }$ iteration $\times$ step size) rather than iteration. (e) we plot the sharpness at the iterates themselves. Note that for the red step size, this quantity plateaus at a value that is beneath $2 / \eta$ . (f) distance from the gradient flow trajectory. + +![](images/99c2e827a5b9bc9069cc98ecacbcadd8e0c8cad7d7ebefdc205ef47bc77c1202.jpg) +Figure 72: We train a tanh CNN $\left( + \mathbf { B } \mathbf { N } \right)$ using gradient flow. + +![](images/a441c23940bb29cc3aeda6162fb228cf2efce55bce062edaf7010ff2919766d8.jpg) +Figure 73: We train a tanh CNN $\left( + \mathbf { B } \mathbf { N } \right)$ using gradient decent. + +![](images/3738d254485ca8f1db197851a4745810e8540f78a1fe2f1d367dbdafc996fdc4.jpg) +Figure 74: We train a ReLU CNN $\left( + \mathbf { B } \mathbf { N } \right)$ using gradient flow. + +![](images/322fdc9b1f1799546e45643baf826573ce3bc5d4b767f0efbcb8c3be28c4e87e.jpg) +Figure 75: We train a ReLU CNN $\left( + \mathbf { B } \mathbf { N } \right)$ using gradient descent. + +# K.1 RELATION TO SANTURKAR ET AL. (2018) + +We have demonstrated that the sharpness hovers right at (or just above) the value $2 / \eta$ when both BN and non-BN networks are trained using gradient descent at reasonable step sizes. Therefore, at least in the case of full-batch gradient descent, it cannot be said that batch normalization decreases the sharpness (i.e. improves the local $L$ -smoothness) along the optimization trajectory. + +Santurkar et al. (2018) argued that batch normalization improves the effective smoothness along the optimization trajectory, where effective smoothness is defined as the Lipschitz constant of the gradient in the update direction (i.e. the negative gradient direction, for full-batch GD). That is, given an objective function $f$ , an iterate $\theta$ , and a distance $\alpha$ , the effective smoothness of $f$ at parameter $\theta$ and distance $\alpha$ is defined in Santurkar et al. (2018) as + +$$ +\operatorname* { s u p } _ { \gamma \in [ 0 , \alpha ] } \frac { \| \nabla f ( \theta ) - \nabla f ( \theta - \gamma \nabla f ( \theta ) ) \| _ { 2 } } { \| \gamma \nabla f ( \theta ) \| _ { 2 } } +$$ + +where the sup can be numerically approximated by evaluating the given ratio at several values $\gamma$ spaced uniformly between 0 and $\alpha$ . + +In Figure 76, we train two ReLU CNNs — one with BN, one without — at the same set of step sizes, and we monitor both the sharpness (i.e. the $L$ -smoothness) and the effective smoothness. When computing the effective smoothness, we use a distance $\alpha = \eta$ . Observe that for both the BN and the non-BN network, the effective smoothness initially hovers around zero, but once gradient descent enters the Edge of Stability, the effective smoothness jumps to the value $2 / \eta$ and then remains there. Thus, at least for full-batch gradient descent on this particular architecture, batch normalization does not improve the effective smoothness along the optimization trajectory. (Despite this, note that for each step size, the BN network trains faster than the non-BN network, confirming that BN does accelerate training.) + +![](images/eae8d2d243df8c1253b39e48a1ddcded8efeab7593d115d544011a9caeda745d.jpg) +Figure 76: On a 5,000-size subset of CIFAR-10, we train a ReLU CNN both with BN (top row) and without BN (bottom row) at the same grid of step sizes. We plot the sharpness/smoothness (center column) as well as the effective smoothness (Santurkar et al., 2018) (right column). Observe that for both networks, the effective smoothness hovers around zero initially, and jumps up to $2 / \eta$ once gradient descent enters the Edge of Stability. + +Note that this finding is actually consistent with Figure 4(c) in Santurkar et al. (2018), which is meant to show that BN improves effective smoothness when training a VGG network using SGD. Their Figure 4(c) shows that during SGD with step size $\eta = 0 . 1$ , the effective smoothness hovers around the value 20 for both the BN and the non-BN network. Since $2 0 = 2 / ( 0 . 1 )$ , this is fully consistent with our findings (though they use SGD rather than full-batch GD). Figure 4(c) does show that the effective smoothness behaves more regularly for the BN network than for the non-BN network. But we disagree with their interpretation of this figure as demonstrating that BN improves the effective smoothness during training. + +The other piece of evidence in Santurkar et al. (2018) in support of the argument that batch normalization improves the effective smoothness during training is their Figure 9(c). This figure shows that a deep linear network (DLN) trained without BN has a much larger (i.e. worse) effective smoothness during training than a DLN trained with BN. However, for this figure, the distance $\alpha$ used to compute effective smoothness was larger than the training step size $\eta$ by a factor of 30. The effective smoothness at distances larger than the step size does not affect training. We have verified that when effective smoothness is computed at a distance equal to the training step size (i.e. $\alpha = \eta$ ), the effective smoothness for the DLN with BN and for the DLN without BN both hover right at $2 / \eta$ . + +Specifically, in Figure 77 and Figure 78, we train a DLN both with and without BN (respectively), and we measure the effective smoothness at a distance $\alpha = 3 0 \eta$ , as done in Figure 9(c) of Santurkar et al. (2018). We use the same experimental setup and the same step size of $\eta = 1 \mathrm { e } { - 6 }$ as they do, and we repeat the experiment across four random seeds. Observe that when training the BN network, the effective smoothness hovers right at $2 / \eta$ (marked by the horizontal black line), whereas when training the non-BN network, the effective smoothness is much larger. This is consistent with Figure 9(c) in Santurkar et al. (2018). However, in Figure 79 and 80, we measure the effective smoothness at the actual step size $\alpha = \eta$ . When effective smoothness is computed in this way, we observe that for both the network with BN and the network without BN, the effective smoothness hovers right at $2 / \eta$ . Therefore, we conclude that there is no evidence that the use of batch normalization improves either the smoothness or the effective smoothness along the optimization trajectory. (That said, this experiment possibly explains why the batch-normalized network permits training with larger step sizes.) + +![](images/711d45e6c38cb8db57825974e6d118c638c192753afb1ad27d8b8f11db673d6c.jpg) +Figure 77: When training a deep linear network without BN, we measure effective smoothness at a distance $\alpha = 3 0 \eta$ that is $\mathbf { 3 0 x }$ larger than the step size. + +![](images/3bc9cf62b9fb4ee1881a277840f0bd332c0e1c1c324b2af253089f6e9e89439b.jpg) +Figure 78: When training a deep linear network with BN, we measure effective smoothness at a distance $\alpha = 3 0 \eta$ that is $\mathbf { 3 0 x }$ larger than the step size. + +![](images/d626344fdf7f9ec1855efc2601d1c5c489599e0a6d439755681a405dc577f6f8.jpg) +Figure 79: When training a deep linear network without BN, we measure effective smoothness at a distance $\alpha = \eta$ that is equal to the step size. + +![](images/7b6737fa587e9662dcb6397b65243ac836bc908a5ac2bd03931e816debfd9868.jpg) +Figure 80: When training a deep linear network with BN, we measure effective smoothness at a distance $\alpha = \eta$ that is equal to the step size. + +# L ADDITIONAL TASKS + +So far, we have verified our findings on image classification and language modeling. In this appendix, we verify our findings on three additional tasks: training a Transformer on the WikiText-2 language modeling dataset (L.1), training a one-hidden-layer network on a one-dimensional toy regression task (L.2), and training a deep linear network (L.3). + +# L.1 TRANSFORMER ON WIKITEXT-2 + +We consider the problem of training a Transformer on the WikText-2 word-level language modeling dataset (Merity et al., 2016). See $\ S 1 . 4$ for full experimental details. In Figure 81, we train using gradient flow (only partially, not to completion). Observe that the sharpness continually rises. In Figure 82, we train using gradient descent at a range of step sizes. Consistent with our general findings, for each step size $\eta$ , we observe that the sharpness rises to $2 / \eta$ and then hovers right at, or just above, that value. However, for this Transformer, we do not observe that gradient descent closely tracks the gradient flow trajectory at the beginning of training. + +![](images/12cad1f71acba6d23954d772c7367242e0c306c559e7b0d15a5a6edc67b72da6.jpg) +Figure 81: Training a Transformer using gradient flow. + +![](images/bffbb195c94c724c5b9c7619674a20d6367ab03d9eb252d20206a69b8297a98d.jpg) +Figure 82: Training a Transformer using gradient descent. We train a Transformer for 15,000 iterations on the WikiText-2 language modeling dataset. (For this problem, training to completion would not be computationally practical.) Top left: we plot the train loss curves, with a vertical dotted line marking the iteration where the sharpness first crosses $2 / \eta$ . The train loss decreases monotonically before this line, but behaves non-monotonically afterwards. Top right: we plot the evolution of the sharpness, with a horizontal dashed line marking the value $2 / \eta$ . Observe that for each step size, the sharpness rises to $2 / \eta$ and then hovers right at, or just above, that value. Bottom left: for the initial phase of training, we plot the distance between the gradient descent trajectory and the gradient flow trajectory, with a vertical dotted line marking the gradient descent iteration where the sharpness first crosses $2 / \eta$ . Observe that the distance begins to rise from the start of training, indicating that for this architecture, gradient descent does not track the gradient flow trajectory initially. Bottom right: for the initial phase of training, we plot the evolution of the sharpness by “time” $=$ iteration $\times \eta$ rather than iteration. The black dots are the sharpness during gradient flow. + +Task Our toy regression problem is to approximate a Chebyshev polynomial using a neural network. To generate a toy dataset, we take 20 points spaced uniformly on the interval $[ - 1 , 1 ]$ , and we label them noiselessly using the Chebyshev polynomial of some degree $k$ . Note that the Chebyshev polynomial of degree $k$ is a polynomial with $k$ zeros that maps the domain $[ - 1 , 1 ]$ to the range $[ - 1 , 1 ]$ . Figure 83 shows the Chebyshev datasets for degree 3, 4, and 5. + +![](images/924b6cd0121397ed2391aa23bcdb5462952dedd01348ef21d88eadd4015708fe.jpg) +Figure 83: The datasets for the toy one-dimensional regression task. + +Network For the network, we use a tanh architecture with one hidden layer of $h = 1 0 0$ units, initialized using Xavier initialization. We train using the MSE loss until the loss reaches 0.05. + +Results In Figure 84, we fit the Chebyshev degree 3, 4, and 5 datasets using gradient flow. Empirically, the higher the degree, the more the sharpness rises during the course of training: on the degree 3 polynomial, the sharpness rises by a factor of 1.2; on the degree 4 polynomial, by a factor of 3.2; and on the degree 5 polynomial, by a factor of 63.5. + +In Figure 85 and Figure 86, we fit the degree 4 and 5 datasets using gradient descent at a range of step sizes. We observe mostly the same Edge of Stability behavior as elsewhere in the paper. The only difference is that for the degree 5 dataset, after the sharpness hits $2 / \eta$ , the training loss first undergoes a temporary period of non-monotonicity in which no progress is made, and then it decreases monotonically until training is finished (in contrast to our other experiments where we observe that training loss behaves non-monotonically at the same time as it is consistently decreasing). + +![](images/402e98d48907bb48bb8cd932791b00422fdd9bb17a24b93290a6f5868d68c871.jpg) +Figure 84: Fitting Chebyshev polynomials using gradient flow. We use gradient flow to fit a one-hidden-layer tanh network to the Chebyshev polynomials of degrees 3, 4, and 5. We observe that the sharpness rises more when fitting a higher-degree polynomial, likely because the dataset is more complex. + +![](images/1fb3bdb91531e90387d4d5bbeec87d783a7f112d402f03037e318faf47f13d84.jpg) +Figure 85: Chebyshev degree 4 (gradient descent). We fit the Chebyshev polynomial of degree 4 using gradient descent at a range of step sizes (see legend). + +![](images/f863a93dc042fd5c3a02028c53f622ee65d3283ade9cfd4badfb4234ea590310.jpg) +Figure 86: Chebyshev degree 5 (gradient descent). We fit the Chebyshev polynomial of degree 5 using gradient descent at a range of step sizes (see legend). + +# L.3 DEEP LINEAR NETWORK + +is to map rror is me $n$ inputs ured u $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { n } \subseteq \mathbb { R } ^ { d }$ to ss, $n$ te. $\mathbf { y } _ { 1 } , \ldots , \mathbf { y } _ { n } \subseteq \mathbb { R } ^ { d }$ $f : \mathbb { R } ^ { d } \mathbb { R } ^ { d }$ $\begin{array} { r } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \| f ( \mathbf { { \bar { x } } } _ { i } ) - \mathbf { y } _ { i } \| _ { 2 } ^ { 2 } } \end{array}$ $\mathbf { X } \in \mathbb { R } ^ { n \times d }$ $\mathbf { Y } \in \mathbb { R } ^ { n \times d }$ targets. We first generate $\mathbf { X }$ as a random whitened matrix (i.e. $\textstyle { \frac { 1 } { n } } \mathbf { X } ^ { T } \mathbf { X } = \mathbf { I } )$ . To generate $\mathbf { X }$ as a random whitened matrix, we sample a $n \times d$ matrix of standard Gaussians, and then set $\mathbf { X }$ to be $\sqrt { n }$ times the Q factor in the QR factorization of that matrix. We then generate $\mathbf { Y }$ via $\mathbf { Y } = \mathbf { X } \mathbf { A } ^ { T }$ , where $\mathbf { A } \in \mathbb { R } ^ { d \times d }$ is a random matrix whose entries are sampled i.i.d from the standard normal distribution. + +We use $n = 5 0$ datapoints with a dimension of $d = 5 0$ . + +Network The function $f : \mathbb { R } ^ { d } \mathbb { R } ^ { d }$ is implemented as a $L$ -layer deep linear network: $f ( \mathbf { x } ) =$ $\mathbf { W } _ { L } \dots \mathbf { W } _ { 2 } \mathbf { W } _ { 1 } \mathbf { x }$ , with $\mathbf { W } _ { \ell } \in \mathbb { R } ^ { d \times d }$ . We initialize all layers of the deep linear network using Xavier initialization: all entries of each $\mathbf { W } _ { \ell }$ are drawn i.i.d from $\textstyle { \mathcal { N } } ( 0 , { \frac { 1 } { d } } )$ . + +We use a network with $L = 2 0$ layers. + +Results In Figure 87, we train the network using gradient flow. (Since it is unclear whether the network can be trained to zero loss, and how long this would take, we arbitrarily chose to stop training at time 100.) In Figure 88, we train the network using gradient descent at a range of step sizes. We observe mostly the same Edge of Stability behavior as elsewhere in the paper. The only difference is that in Figure 88, the train loss does not really behave non-monotonically — for each step size $\eta$ , there is a brief blip at some point, but otherwise, the train loss decreases monotonically. + +![](images/e326bc0690fd1050067b147d39b711c643bccb982cd01f9353f4e60775546f3a.jpg) +Figure 87: Training a deep linear network using gradient flow. We use gradient flow to train a deep linear network. Since it is unclear whether this network can be trained to zero loss (or how long that would take), we arbitrarily chose to stop training at time 100. + +![](images/2fe7e0caa74be9dfbde7de428e59112895a019374e9f9a1754cddfbe6cc6c1d9.jpg) +Figure 88: Training a deep linear network using gradient descent. We train a deep linear network using gradient descent at a range of step sizes (see legend). (Note that in the top left pane, the blue, orange and green dotted lines are directly on top of one another.) + +# M EXPERIMENTS: STANDARD ARCHITECTURES ON CIFAR-10 + +In this appendix, we demonstrate that our findings hold for three standard architectures on the standard dataset CIFAR-10. The three architectures are: a VGG with batch normalization (Figures 89-90), a VGG without batch normalization (Figures 91-92), and a ResNet with batch normalization (Figures 93 -94). See $\ S 1 . 2$ for full experimental details. + +For each of these three architectures, we confirm our main points: (1) so long as the sharpness is less than $2 / \eta$ , the sharpness tends to increase; and (2) if the sharpness reaches $2 / \eta$ , gradient descent enters a regime (the Edge of Stability) in which (a) the training loss behaves non-monotonically, yet consistently decreases over long timescales, and (b) the sharpness hovers right at, or just above, the value $2 / \eta$ . Moreover, we observe that even though these architectures use the ReLU activation function (which is not continuously differentiable), gradient descent closely tracks the Runge-Kutta trajectory until reaching the point on that trajectory where the sharpness hits $2 / \eta$ . + +Furthermore, we observe that for these three standard architectures, the following additional points hold: (1) progressive sharpening occurs to a dramatic degree, and (2) stable step sizes are so small as to be completely unreasonable. + +Progressive sharpening occurs to a dramatic degree To assess the degree of progressive sharpening, we train these networks using Runge-Kutta / gradient flow, which can be viewed as gradient descent with an infinitesimally small step size. (In practice, the Runge-Kutta algorithm does have a step size parameter, and throughout training, we periodically adjust this step size in order to ensure that the algorithm remains stable.) Intuitively, training with gradient flow tell us how far the sharpness would rise “if it didn’t have to worry about” instability caused by nonzero step sizes. In Figure 89, we train the VGG with BN to completion $9 9 \%$ training accuracy) using Runge-Kutta / gradient flow, and find that the sharpness rises from its initial value of 6.38 to a peak value of 2227.6. For the other two architectures, progressive sharpening occurs to such a degree that it is not computationally feasible for to train using Runge-Kutta / gradient flow all the way to completion. (The reason is that in regions where the sharpness is high, the Runge-Kutta step size must be made small, so Runge-Kutta requires very many iterations.) Therefore, we instead train these two networks only partially. In Figure 91, for the VGG without BN, we find that the sharpness rises from its initial value of 0.64 to the value 2461.78 at $3 7 . 1 \%$ accuracy, when we stop training. In Figure 93, for the ResNet, we find that the sharpness rises from its initial value of 1.07 to the value 760.6 at $4 3 . 2 \%$ accuracy, when we stop training. Thus, even though we observed in Appendix D that progressive sharpening attenuates as the width of fully-connected networks is made larger, it appears that either: (1) this does not happen for modern families of architectures such as ResNet and VGG, or (2) this does happen for modern families of architectures, but practical network widths lie on the narrow end of the scale. + +Stable step sizes are so small as to be unreasonable Recall from 3.3 that if $\lambda _ { \mathrm { m a x } }$ is the maximum sharpness along the gradient flow trajectory, then any stable step size must be less than $2 / \lambda _ { \mathrm { m a x } }$ . Therefore, for these three architectures, because progressive sharpening occurs to a dramatic degree (i.e. $\lambda _ { \mathrm { m a x } }$ is extraordinarily large), any stable step size must be extraordinarily small, which means that training will require many iterations. Yet, at the same time, we find that these three networks can be successfully trained in far fewer iterations by using larger step step size. This means that training at a stable step size is extremely suboptimal. We now elaborate on this point: + +VGG with BN. For this network, gradient flow terminates at time 15.66, and the maximum sharpness along the gradient flow trajectory is 2227.6. Therefore, the largest stable step size is $2 / 2 2 2 7 . 5 9 = 0 . 0 0 0 8 9 7$ , and training to completion at this step size would take $1 4 . 9 1 / 0 . 0 0 0 8 9 7 =$ 16622 iterations. Meanwhile, we empirically observe that the network can also be trained to completion at the much larger step size of $\eta = 0 . 1 6$ in just 329 iterations. Therefore, using a stable step size is suboptimal by a factor of at least $1 6 6 2 2 / 3 2 9 = 5 0 . 5 $ . + +For the other two architectures, since we are unable to train to completion using gradient flow, we are unable to obtain a tight lower bound for the number of iterations required to run gradient descent to completion at a stable step size. Therefore, by extension, we are unable to compute a tight lower bound for the suboptimality factor of stable step sizes. As a substitute, we will instead compute both: (1) a tight lower bound on the suboptimality of training partially at a stable step size, and (2) a very loose lower bound on the suboptimality of training to completion at a stable step size. + +VGG without BN. For this network, gradient flow reaches $3 7 . 1 \%$ accuracy at time 8, and the maximum sharpness up through this point on the gradient flow trajectory is 2461.8. Therefore, the largest stable step size is $2 / 2 4 6 1 . 8 = 0 . 0 0 0 8 1$ , and training to $3 7 . 1 \%$ accuracy at this step size would require $8 / 0 . 0 0 0 8 1 = 9 , \quad$ 876 iterations. Meanwhile, we empirically observe that the network can also be trained to $3 7 . 1 \%$ accuracy at the larger step size of $\eta = 0 . 1 6$ in just 355 iterations. Therefore, when training this network to $37 \%$ accuracy, stable step sizes are suboptimal by a factor of at least $9 8 7 6 / 3 5 5 = \mathrm { \bar { 2 7 } } . 8 .$ . This is a tight lower bound on the suboptimality of training to $3 7 . 1 \%$ accuracy at a stable step size. + +To obtain a loose lower bound on the suboptimality of training the VGG without BN to completion at a stable step size, we note that (a) since the maximum sharpness up through time 8 is 2461.8, the maximum sharpness along the entire gradient flow trajectory must be at least 2461.8; and (b) since by time 8 gradient flow has only attained $3 7 . 1 \%$ training accuracy, the time to reach $9 9 \%$ accuracy (i.e. completion) must be at least 8. (Note that both of these lower bounds are extremely loose.) Therefore, training this network to completion at a stable step size would require at least $8 / ( 2 / 2 4 6 1 . 8 ) \ = \ 9 , 8 7 6$ iterations (which is the same number of iterations as training to $3 7 . 1 \%$ accuracy at a stable step size). Meanwhile, we find that the network can be trained to completion at the larger step size of $\eta = 0 . 1 6$ in just 1782 iterations. Therefore, training to completion at a stable step size is suboptimal by a factor of at least $9 , 8 7 6 / 1 7 8 2 = 5 . 5 4$ . + +ResNet. For this network, gradient flow reaches $4 3 . 2 \%$ accuracy at time 70, and the maximum sharpness up through this point on the gradient flow trajectory is 760.6. Therefore, the largest stable step size is $2 / 7 6 0 . 6 = 0 . 0 0 2 6$ , and training to $4 3 . 2 \%$ accuracy at this step size would require $7 0 / 0 . 0 0 2 6 = 2 6 ,$ 923 iterations. Meanwhile, we empirically observe that the network can also be trained to $4 3 . 2 \%$ accuracy at the larger step size of $\eta = 2 . 0$ in just 99 iterations. Therefore, when training this network to $4 3 . 2 \%$ accuracy, stable step sizes are suboptimal by a factor of at least $2 6 , 9 2 { \bar { 3 } } / 9 9 = 2 7 1 . 9 .$ . This is a tight lower bound on the suboptimality of training to $4 3 . 2 \%$ accuracy at a stable step size. For the loose lower bound on the suboptimality of training to completion at a stable step size, note that (by similar reasoning as the VGG-without-BN above), training to completion at a stable step size must require at least 26, 923 iterations. Meanwhile, we find that the network can be trained to completion at the larger step size of $\eta = 2 . 0$ in just 807 iterations. Therefore, training to completion at a stable step size is suboptimal by a factor of at least $2 6 , 9 2 3 / 8 0 7 = 3 3 . 3$ . + +![](images/1aea1ce5c7e9dddceab313af8fc6f861b105d3164143d97c0beb229fb91197d9.jpg) +Figure 89: We train a VGG with BN to completion using gradient flow (Runge-Kutta). Observe that the sharpness rises dramatically from 6.38 at initialization to a peak of 2227.59. + +![](images/4b37034eb812e5a6f0e13a18abb8eed6b97188c126c689072faf59d018799676.jpg) +Figure 90: We train a VGG with BN to completion using gradient descent at different step sizes (see legend in the top right pane). Top left: we plot the train loss, with a vertical dotted line marking the iteration where the sharpness first crosses $2 / \eta$ . Top center: we plot the evolution of the sharpness, with a horizontal dashed line (of the appropriate color) marking the value $2 / \eta$ . Top right: we plot the train accuracy. Bottom left: for the initial phase of training, we monitor the distance between the gradient descent iterate at iteration $t / \eta$ , and the gradient flow solution at time $t$ , with a vertical dotted line (of the appropriate color) marking the time when the sharpness crosses $2 / \eta$ . Observe that the distance between gradient descent and gradient flow is almost zero before this instant, but starts to rise shortly afterwards. Bottom center: we plot the sharpness by time $\left( = \right.$ iteration $\times \eta$ ) rather than iteration. The black dots are the sharpness of the gradient flow trajectory, which shoots up immediately. Bottom right: we plot the test accuracy. + +![](images/2cba675712614b25e4b0e73c288854542afdda1705bd41eca425139b8bfcf0e0.jpg) +Figure 91: We train a VGG without BN to $3 7 . 1 \%$ accuracy using gradient flow (Runge-Kutta). Observe that the sharpness rises dramatically from 0.64 at initialization to 2461.78 at $3 7 . 1 \%$ accuracy. We train this network only partway because training this network to completion would be too computationally expensive: Runge-Kutta runs very slowly when the sharpness is high (because it is forced to take small steps) and for this network the sharpness is extremely high when the train accuracy is only $37 \%$ (which means that there is a long way to go). + +![](images/d4bfe9fcbdd1335d834762612e090edacd2eb089dd52c241375224a57890a8ba.jpg) +Figure 92: We train a VGG without BN to completion using gradient descent at a range of step sizes (see legend in the top right pane). Refer to the Figure 90 caption for more information. + +![](images/8fd5dfa00a6140106d5a5b7860d9738fe09f879d8763d36414f68080d6a1f13a.jpg) +Figure 93: We train a ResNet to $4 3 . 2 \%$ accuracy using gradient flow (Runge-Kutta). Observe that the sharpness rises dramatically from 1.07 at initialization to 760.63 at $4 3 . 2 \%$ accuracy. We train this network only partway because training this network to completion would be too computationally expensive: Runge-Kutta runs very slowly when the sharpness is high (because it is forced to take small steps) and for this network the sharpness is extremely high when the train accuracy is only $42 \%$ (which means that there is a long way to go). + +![](images/130d034a56276190c7b4f98f4f4c55a94d894e02d8caf6718923f78cc85394ad.jpg) +Figure 94: We train a ResNet to completion using gradient descent at a range of step sizes (see legend in the top right pane). Refer to the Figure 90 caption for more information. + +# N EXPERIMENTS: MOMENTUM + +This appendix contains systematic experiments for gradient descent with Polyak momentum and Nesterov momentum. Our aim is to demonstrate that the sharpness rises until reaching the maximum stable sharpness (MSS) given by equation 1, and then either plateaus just above that value, or oscillates around that value. + +We experiment on a $5 \mathrm { k }$ -sized subset of CIFAR-10, using four architectures: a tanh fully-connected network (section N.1), a ReLU fully-connected network (section N.2), a tanh convolutional network (section N.3), and a ReLU convolutional network (section N.4). For each of these four architectures, we experiment with both the square loss (for classification) and cross-entropy loss. For each architecture and loss function, we experiment with both Polyak momentum at $\beta = 0 . 9$ , and gradient descent with Nesterov momentum at $\beta = 0 . 9$ . We run gradient descent at a range of several step sizes which were chosen by hand so that the MSS’s are approximately spaced evenly. Note that for Polyak momentum with step size $\eta$ and momentum parameter $\beta = 0 . 9$ , the MSS is $\begin{array} { r } { \frac { \dot { 2 } + 2 \beta } { \eta } = \frac { 3 . 8 } { \eta } } \end{array}$ . For Netsterov momentum with step size $\eta$ and momentum parameter $\beta = 0 . 9$ , the MSS is η2+2 β $\begin{array} { r } { \frac { 2 + 2 \beta } { \eta ( 1 + 2 \beta ) } \approx \frac { 1 . 3 5 7 1 4 } { \eta } } \end{array}$ . We run gradient descent until reaching $9 9 \%$ accuracy. + +We find that the sharpness rises until reaching the maximum stable sharpness (MSS) given by equation 1, and then either plateaus just above that value, or oscillates around that value. Sometimes these oscillations are rapid (e.g. Figure 96), sometimes they are a bit slower (e.g. Figure 103), and sometimes they are slow (e.g. Figure 109). + +# N.1 FULLY-CONNECTED TANH NETWORK + +In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) In the middle plot, the horizontal dashed line marks the MSS. + +# N.1.1 SQUARE LOSS + +![](images/c635e15ac8da04def67411e9c66fea250f5acccf22c2f0c8445aac14ff7b7c11.jpg) +Figure 95: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/18b1d4e8ddd5dfcd4ac260780cfdb5e61e128a05bb4d176d635cf0dd95b00dc8.jpg) +Figure 96: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# N.1.2 CROSS-ENTROPY LOSS + +![](images/577b27608f5c5362d2236e7a2aeb96fe6ab7600bf3fd5b2a5e0cb51aca438af6.jpg) +Figure 97: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/9c8eaa747084c6fd0eee24a15f13fe7cab787990986b26ec928a5677672ddaf9.jpg) +Figure 98: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# N.2 FULLY-CONNECTED RELU NETWORK + +In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) + +In the middle plot, the horizontal dashed line marks the MSS. + +# N.2.1 SQUARE LOSS + +![](images/58f01faf3df568b2a611cf0d8ce60484e8dcc031f4f236b7649e0ac6bab8e324.jpg) +Figure 99: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/01212b19f8d82ee634814f18695512cc31c0c763ee6db34ef2cfee597613003f.jpg) +Figure 100: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# N.2.2 CROSS-ENTROPY LOSS + +![](images/5a0256dead0f65338f9bdc82b061f281d9cac406d9ae79ce3d539c957956a303.jpg) +Figure 101: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/63f0f077747f117a1df6884844d6a4a3b1a3276917f710cfb44b76989bb04095.jpg) +Figure 102: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# N.3 CONVOLUTIONAL TANH NETWORK + +In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) + +In the middle plot, the horizontal dashed line marks the MSS. + +# N.3.1 SQUARE LOSS + +![](images/16667ae2c68eb53ffd87e79e6fe4d3cde565358efaa0b64ad1256a29c7af8fbe.jpg) +Figure 103: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/c8bfb03ffd79b7881dc6f4c9ad5f6524d8e445c98d75c00bad920245ddf8a405.jpg) +Figure 104: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# N.3.2 CROSS-ENTROPY LOSS + +![](images/e921c61207bbe83724e6a142d823673404c7c0134a5e904bcd11dabbb95b70a7.jpg) +Figure 105: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/9fc7427c21335831099aaf8091c3c0268425939a5a01e4ff54dc4a93183cde98.jpg) +Figure 106: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# N.4 CONVOLUTIONAL RELU NETWORK + +In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) In the middle plot, the horizontal dashed line marks the MSS. + +# N.4.1 SQUARE LOSS + +![](images/6ab9b3351c47aa3ed6d83b7c6bb70ba04a789ccdb2cce6d05f735cb944472443.jpg) +Figure 107: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/86c7cdbe8905f56f022877678effcf54958de7fa8d763725e681bc7244609100.jpg) +Figure 108: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# N.4.2 CROSS-ENTROPY LOSS + +![](images/73f8de1c537a70383970b3f9589de4d80a14e524268e65eaafa99b19a2f620d2.jpg) +Figure 109: Gradient descent with Polyak momentum, $\beta = 0 . 9$ . + +![](images/bc70ca76e68f98d0817877e5c550174a8635e44bdf5fc5e1b97b862a0368b914.jpg) +Figure 110: Gradient descent with Nesterov momentum, $\beta = 0 . 9$ . + +# O EXPERIMENTS: LEARNING RATE DROP + +In this appendix, we run gradient descent until reaching the Edge of Stability, and then we cut the step size. We will see that the sharpness starts increasing as soon as the step size is cut, and only stops increasing once gradient descent is back at the Edge of Stability (or training is finished). As a consequence of this experiment, one can interpret the Edge of Stability as a regime in which gradient descent is constantly “trying” to increase the sharpness beyond $2 / \eta$ , but is constantly being blocked from doing so. Our experiments focus on image classification on a $5 \mathrm { k }$ -sized subset of CIFAR-10. We study two architectures (a fully-connected tanh network and a convolutional ReLU network) and two loss functions (squared loss and cross-entropy loss). + +![](images/63079ee2bf16f1303274a111a15b75f6e37ada1fe66fe1fef66f1cb5d153e95a.jpg) +Figure 111: Top row: we train a network using gradient descent at a range of step sizes $\eta$ (see the legend in the right pane). Bottom three rows: Once gradient descent has reached the Edge of Stability, we cut the step size. The black vertical dotted line marks the iteration where the step size is cut. This iteration was chosen by hand, but was not cherry-picked. We use different colors (consistent with the legend) to plot the train loss, sharpness, and train accuracy before and after the learning rate drop. In the middle sharpness plot, the two horizontal lines mark the maximum stable sharpness $2 / \eta$ for both the old and new step size. Takeaway: once we decrease the step size, the sharpness immediately starts to increase until it reaches the maximum stable sharpness $2 / \eta$ for the new step size $\eta$ (or training finishes). + +![](images/e08ca739281869a73b1c196159c8d03258ad2755b0bdd25baec5ac29e060ff67.jpg) +Figure 112: Refer to the caption for Figure 111. + +![](images/2aa1ff8f8343899eb878ab9e97c7a19948fb4248c1d7567ef426779380863791.jpg) +Figure 113: Refer to the caption for Figure 111. + +![](images/d2a98f1537dc53f080bbd9d04ff78a1b399991fb816ab1e705225c67befad5fd.jpg) +Figure 114: Refer to the caption for Figure 111. + +# P OTHER EIGENVALUES + +Throughout most of this paper, we have studied the evolution of the maximum Hessian eigenvalue during gradient descent. In this appendix, we examine the evolution of the top six eigenvalues. While training the network from $\ S 3$ , we monitor the evolution of the top six Hessian eigenvalues. For each of cross-entropy loss and MSE loss, we train at four different step sizes. The results are shown in Figure 115. Observe that each of the top six eigenvalues rises and then plateaus. The precise details differ between MSE loss and cross-entropy loss. For MSE loss, each eigenvalue rises past $2 / \eta$ , and then plateaus just above that value. In contrast, for cross-entropy loss, some of the lesser eigenvalues plateau below $2 / \eta$ . + +![](images/b4f593a216377f4dd8239fb942354819c685c31f0dfada89c725a888501f40d1.jpg) +Figure 115: Evolution of top six eigenvalues. For both cross-entropy loss (top) and MSE loss (bottom), we run gradient descent at four different step sizes (columns), and plot the evolution of the top six eigenvalues (different colors; see legend). \ No newline at end of file diff --git a/parse/train/jh-rTtvkGeM/jh-rTtvkGeM_content_list.json b/parse/train/jh-rTtvkGeM/jh-rTtvkGeM_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..f14e470cf8dd6a137e09693ece77e6349ea7e401 --- /dev/null +++ b/parse/train/jh-rTtvkGeM/jh-rTtvkGeM_content_list.json @@ -0,0 +1,5832 @@ +[ + { + "type": "text", + "text": "GRADIENT DESCENT ON NEURAL NETWORKS TYPICALLY OCCURS AT THE EDGE OF STABILITY ", + "text_level": 1, + "bbox": [ + 176, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jeremy Cohen Simran Kaur Yuanzhi Li J. Zico Kolter1 and Ameet Talwalkar2 \nCarnegie Mellon University and: 1Bosch AI 2 Determined AI \nCorrespondence to: jeremycohen@cmu.edu ", + "bbox": [ + 183, + 169, + 774, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 250, + 544, + 265 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We empirically demonstrate that full-batch gradient descent on neural network training objectives typically operates in a regime we call the Edge of Stability. In this regime, the maximum eigenvalue of the training loss Hessian hovers just above the value 2/(step size), and the training loss behaves non-monotonically over short timescales, yet consistently decreases over long timescales. Since this behavior is inconsistent with several widespread presumptions in the field of optimization, our findings raise questions as to whether these presumptions are relevant to neural network training. We hope that our findings will inspire future efforts aimed at rigorously understanding optimization at the Edge of Stability. ", + "bbox": [ + 233, + 281, + 764, + 406 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 433, + 336, + 448 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Neural networks are almost never trained using (full-batch) gradient descent, even though gradient descent is the conceptual basis for popular optimization algorithms such as SGD. In this paper, we train neural networks using gradient descent, and find two surprises. First, while little is known about the dynamics of neural network training in general, we find that in the special case of gradient descent, there is a simple characterization that holds across a broad range of network architectures and tasks. Second, this characterization is strongly at odds with prevailing beliefs in optimization. ", + "bbox": [ + 174, + 464, + 823, + 547 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In more detail, as we train neural networks using gradient descent with step size $\\eta$ , we measure the evolution of the sharpness — the maximum eigenvalue of the training loss Hessian. Empirically, the behavior of the sharpness is consistent across architectures and tasks: so long as the sharpness is less than the value $2 / \\eta$ , it tends to continually rise (§3.1). We call this phenomenon progressive sharpening. The significance of the value $2 / \\eta$ is that gradient descent on quadratic objectives is unstable if the sharpness exceeds this threshold (§2). Indeed, in neural network training, if the sharpness ever crosses $2 / \\eta$ , gradient descent quickly becomes destabilized — that is, the iterates start to oscillate with ever-increasing magnitude along the direction of greatest curvature. Yet once this happens, gradient descent does not diverge entirely or stall. Instead, it enters a new regime we call the Edge of Stability1 (§3.2), in which (1) the sharpness hovers right at, or just above, the value $2 / \\eta$ ; and (2) the train loss behaves non-monotonically, yet consistently decreases over long timescales. In this regime, gradient descent is constantly “trying” to increase the sharpness, but is constantly restrained from doing so. The net effect is that gradient descent continues to successfully optimize the training objective, but in such a way as to avoid further increasing the sharpness.2 ", + "bbox": [ + 173, + 554, + 825, + 666 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/3abb073f42e9b646f68463de4cad2806e2fd057b656681f0cbce75095a7b79b5.jpg", + "image_caption": [ + "Figure 1: Gradient descent typically occurs at the Edge of Stability. On three separate architectures, we run gradient descent at a range of step sizes $\\eta$ , and plot both the train loss (top row) and the sharpness (bottom row). For each step size $\\eta$ , observe that the sharpness rises to $2 / \\eta$ (marked by the horizontal dashed line of the appropriate color) and then hovers right at, or just above, this value. " + ], + "image_footnote": [], + "bbox": [ + 184, + 695, + 857, + 840 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In principle, it is possible to run gradient descent at step sizes $\\eta$ so small that the sharpness never rises to $2 / \\eta$ . However, these step sizes are suboptimal from the point of view of training speed, sometimes dramatically so. In particular, for standard architectures on the standard dataset CIFAR-10, such step sizes are so small as to be completely unreasonable — at all reasonable step sizes, gradient descent eventually enters the Edge of Stability (see $\\ S 4$ ). Thus, at least for standard networks on CIFAR-10, the Edge of Stability regime should be viewed as the “rule,” not the “exception.” ", + "bbox": [ + 174, + 194, + 825, + 277 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As we describe in $\\ S 5$ , the Edge of Stability regime is inconsistent with several pieces of conventional wisdom in optimization theory: convergence analyses based on $L$ -smoothness or monotone descent, quadratic Taylor approximations as a model for local progress, and certain heuristics for step size selection. We hope that our empirical findings will both nudge the optimization community away from widespread presumptions that appear to be untrue in the case of neural network training, and also point the way forward by identifying precise empirical phenomena suitable for further study. ", + "bbox": [ + 174, + 285, + 825, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Certain aspects of the Edge of Stability have been observed in previous empirical studies of fullbatch gradient descent (Xing et al., 2018; Wu et al., 2018); our paper provides a unified explanation for these observations. Furthermore, Jastrz˛ebski et al. (2020) proposed a simplified model for the evolution of the sharpness during stochastic gradient descent which matches our empirical observations in the special case of full-batch SGD (i.e. gradient descent). However, outside the full-batch special case, there is no evidence that their model matches experiments with any degree of quantitative precision, although their model does successfully predict the directional trend that large step sizes and/or small batch sizes steer SGD into regions of low sharpness. We discuss SGD at greater length in $\\ S 6$ . To summarize, while the sharpness does not obey simple dynamics during SGD (as it does during GD), there are indications that the “Edge of Stability” intuition might generalize somehow to SGD, just in a way that does not center around the sharpness. ", + "bbox": [ + 173, + 375, + 825, + 527 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND: STABILITY OF GRADIENT DESCENT ON QUADRATICS ", + "text_level": 1, + "bbox": [ + 173, + 554, + 781, + 571 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we review the stability properties of gradient descent on quadratic functions. Later, we will see that the stability of gradient descent on neural training objectives is partly well-modeled by the stability of gradient descent on the quadratic Taylor approximation. ", + "bbox": [ + 174, + 590, + 825, + 633 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "On a quadratic objective function $\\begin{array} { r } { f ( \\mathbf { x } ) = \\frac { 1 } { 2 } \\mathbf { x } ^ { T } \\mathbf { A } \\mathbf { x } + \\mathbf { b } ^ { T } \\mathbf { x } + c , } \\end{array}$ , gradient descent with step size $\\eta$ will diverge if3 any eigenvalue of A exceeds the threshold $2 / \\eta$ . To see why, consider first the onedimensional quadratic $\\textstyle f ( x ) = { \\frac { 1 } { 2 } } a x ^ { 2 } + b x + c$ , with $a > 0$ . This function has optimum $x ^ { * } = - b / a$ . Consider running gradient descent with step size $\\eta$ starting from $x _ { 0 }$ . The update rule is $x _ { t + 1 } =$ $x _ { t } - \\eta ( a x _ { t } + b )$ , which means that the error $x _ { t } - x ^ { * }$ evolves as $( x _ { t + 1 } - x ^ { * } ) = ( 1 - \\eta a ) ( x _ { t } - x ^ { * } )$ . Therefore, the error at step $t$ is $( x _ { t } - x ^ { * } ) = ( 1 - \\eta a ) ^ { t } ( x _ { 0 } - \\dot { x } ^ { * } )$ , and so the iterate at step $t$ is $x _ { t } = ( 1 - \\eta a ) ^ { t } ( x _ { 0 } - x ^ { * } ) \\bar { + } x ^ { * }$ . If $a > 2 / \\eta$ , then $( 1 - \\eta a ) < - 1$ , so the sequence $\\{ x _ { t } \\}$ will oscillate around $x ^ { * }$ with ever-increasing magnitude, and diverge. ", + "bbox": [ + 173, + 638, + 825, + 753 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Now consider the general $d$ -dimensional case. Let $\\left( a _ { i } , \\mathbf { q } _ { i } \\right)$ be the $i$ -th largest eigenvalue/eigenvector of $\\mathbf { A }$ . As shown in Appendix $\\mathbf { B }$ , when the gradient descent iterates $\\left\\{ \\mathbf { x } _ { t } \\right\\}$ are expressed in the special coordinate system whose axes are the eigenvectors of A, each coordinate evolves separately. In particular, the coordinate for each eigenvector $\\mathbf { q } _ { i }$ , namely $\\left. \\mathbf { q } _ { i } , \\mathbf { x } _ { t } \\right.$ , evolves according to the dynamics of gradient descent on a one-dimensional quadratic objective with second derivative $a _ { i }$ . ", + "bbox": [ + 173, + 761, + 823, + 830 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Therefore, if $a _ { i } > 2 / \\eta$ , then the sequence $\\left\\{ \\left. \\mathbf { q } _ { i } , \\mathbf { x } _ { t } \\right. \\right\\}$ will oscillate with ever-increasing magnitude; in this case, we say that the iterates $\\left\\{ { \\bf x } _ { t } \\right\\}$ diverge along the direction $\\mathbf { q } _ { i }$ . To illustrate, Figure 2 shows a quadratic function with eigenvalues $a _ { 1 } ~ = ~ 2 0$ and $a _ { 2 } ~ = ~ 1$ . In Figure 2(a), we run gradient descent with step size $\\eta = 0 . 0 9$ ; since $0 < a _ { 2 } < a _ { 1 } < 2 / \\eta$ , gradient descent converges along both ${ \\bf q } _ { 1 }$ and $\\mathbf { q } _ { 2 }$ . In Figure 2(b), we use step size $\\eta ~ = ~ 0 . 1 1$ ; since $0 < a _ { 2 } < 2 / \\eta < a _ { 1 }$ , gradient descent converges along $\\mathbf { q } _ { 2 }$ yet diverges along ${ \\bf q } _ { 1 }$ , so diverges overall. ", + "bbox": [ + 174, + 103, + 516, + 257 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/b1e8b3a66fc1d4a03837e613f4987a4ce7a363ef61bd9bcea1e3a8afc30dbe59.jpg", + "image_caption": [ + "Figure 2: Gradient descent on a quadratic with eigenvalues $a _ { 1 } = 2 0$ and $a _ { 2 } = 1$ . " + ], + "image_footnote": [], + "bbox": [ + 535, + 92, + 805, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Polyak momentum (Polyak, 1964) and Nesterov momentum (Nesterov, 1983; Sutskever et al., 2013) are notable variants of gradient descent which often improve the convergence speed. On quadratic functions, these two algorithms also diverge if the sharpness exceeds a certain threshold, which we call the “maximum stable sharpness,” or MSS. In particular, we prove in Appendix B that gradient descent with step size $\\eta$ and momentum parameter $\\beta$ diverges if the sharpness exceeds: ", + "bbox": [ + 174, + 263, + 825, + 333 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/191d8add630829e8afec6fec3d67a720312e04238cbdb8abeba3bf3daaa9b752.jpg", + "text": "$$\n\\mathbf { M S S } _ { \\mathrm { P o l y a k } } ( \\eta , \\beta ) = \\frac { 1 } { \\eta } \\left( 2 + 2 \\beta \\right) , \\quad \\mathbf { M S S } _ { \\mathrm { N e s t e r o v } } ( \\eta , \\beta ) = \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 266, + 340, + 730, + 375 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The Polyak result previously appeared in Goh (2017); the Nesterov one seems to be new. Note that this discussion only applies to full-batch gradient descent. As we discuss in $\\ S 6$ , several recent papers have proposed stability analyses for SGD (Wu et al., 2018; Jastrz˛ebski et al., 2020). ", + "bbox": [ + 174, + 381, + 825, + 422 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Neural network training objectives are not globally quadratic. However, the second-order Taylor approximation around any point $\\mathbf { x } _ { \\mathrm { 0 } }$ in parameter space is a quadratic function whose “A” matrix is the Hessian at $\\mathbf { x } _ { \\mathrm { 0 } }$ . If any eigenvalue of this Hessian exceeds $2 / \\eta$ , gradient descent with step size $\\eta$ would diverge if run on this quadratic function — the iterates would oscillate with ever-increasing magnitude along the corresponding eigenvector. Therefore, at any point $\\mathbf { x } _ { \\mathrm { 0 } }$ in parameter space where the sharpness exceeds $2 / \\eta$ , gradient descent with step size $\\eta$ would diverge if run on the quadratic Taylor approximation to the training objective around $\\mathbf { x } _ { \\mathrm { 0 } }$ . ", + "bbox": [ + 173, + 429, + 825, + 527 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 GRADIENT DESCENT ON NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 547, + 589, + 564 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we empirically characterize the behavior of gradient descent on neural network training objectives. Section 4 will show that this characterization holds broadly. ", + "bbox": [ + 173, + 579, + 821, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 PROGRESSIVE SHARPENING ", + "text_level": 1, + "bbox": [ + 176, + 626, + 406, + 638 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "When training neural networks, it seems to be a general rule that so long as the sharpness is small enough for gradient descent to be stable $( < 2 / \\eta$ , for vanilla gradient descent), gradient descent has an overwhelming tendency to continually increase the sharpness. We call this phenomenon progressive sharpening. By “overwhelming tendency,” we mean that gradient descent can occasionally decrease the sharpness (especially at the beginning of training), but these brief decreases always seem be followed by a return to continual increase. Jastrz˛ebski et al. (2020) previously hypothesized (in their Assumption 4) that a similar phenomenon may hold for SGD, but the evidence for, and the precise scope of, this effect are both currently far clearer for gradient descent than for SGD. ", + "bbox": [ + 173, + 651, + 825, + 763 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/56cc0c19b654465ab0c58a070a13f44506ef5b48cb3127451a26a57a6f217e3d.jpg", + "image_caption": [ + "Figure 3: So long as the sharpness is less than $2 / \\eta$ , it tends to continually increase during gradient descent. We train a network to completion $9 9 \\%$ accuracy) using gradient descent with a very small step size. We consider both MSE loss (left) and cross-entropy loss (right). " + ], + "image_footnote": [], + "bbox": [ + 179, + 785, + 828, + 863 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Progressive sharpening is illustrated in Figure 3. Here, we use (full-batch) gradient descent to train a network on a subset of 5,000 examples from CIFAR-10, and we monitor the evolution of the sharpness during training. The network is a fully-connected architecture with two hidden layers of width 200, and tanh activations. In Figure 3(a), we train using the mean squared error loss for classification (Hui & Belkin, 2020), encoding the correct class with 1 and the other classes with 0. We use the small step size of $\\eta \\ : = \\ : 2 / 6 0 0$ , and stop when the training accuracy reaches $9 9 \\%$ . We plot both the train loss and the sharpness, with a horizontal dashed line marking the stability threshold $2 / \\eta$ . Observe that the sharpness continually rises during training (except for a brief dip at the beginning). This is progressive sharpening. For this experiment, we intentionally chose a step size $\\eta$ small enough that the sharpness remained beneath $2 / \\bar { \\eta }$ for the entire duration of training. ", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Cross-entropy. When training with cross-entropy loss, there is an exception to the rule that the sharpness tends to continually increase: with cross-entropy loss, the sharpness typically drops at the end of training. This behavior can be seen in Figure 3(b), where we train the same network using the cross-entropy loss rather than MSE. This drop occurs because once most data points are classified correctly, gradient descent tries to drive the cross-entropy loss to zero by scaling up the margins, as detailed in Soudry et al. (2018). As we explain in Appendix C, this causes the sharpness to drop. ", + "bbox": [ + 174, + 250, + 825, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The effect of width. It is known that when networks parameterized in a certain way (the “NTK parameterization”) are made infinitely wide, the Hessian moves a vanishingly small amount during training (Jacot et al., 2018; Lee et al., 2019; Li & Liang, 2018), which implies that no progressive sharpening occurs. In Appendix D, we experiment with networks of varying width, under both NTK and standard parameterizations. We find that progressive sharpening occurs to a lesser degree as networks become increasingly wide. Nevertheless, our experiments in $\\ S 4$ demonstrate that progressive sharpening occurs to a dramatic degree for standard architectures on the standard dataset CIFAR-10. ", + "bbox": [ + 174, + 340, + 823, + 438 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We do not know why progressive sharpening occurs, or whether “sharp” solutions differ in any important way from “not sharp” solutions. These are important questions for future work. Note that Mulayoff & Michaeli (2020) studied the latter question in the context of deep linear networks. ", + "bbox": [ + 176, + 445, + 825, + 486 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 THE EDGE OF STABILITY ", + "text_level": 1, + "bbox": [ + 176, + 507, + 388, + 522 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the preceding section, we ran gradient descent using step sizes $\\eta$ so small that the sharpness never reached the stability threshold $2 / \\eta$ . In Figure 4(a), we start to train the same network at the larger step size of $\\eta = 0 . 0 1$ , and pause training once the sharpness rises to $2 / \\eta = 2 0 0$ . Recall from $\\ S 2$ that in any region where the sharpness exceeds $2 / \\eta$ , gradient descent with step size $\\eta$ would be unstable if run on the quadratic Taylor approximation to the training objective — the gradient descent iterates would oscillate with ever-increasing magnitude along the leading Hessian eigenvector. Empirically, we find that gradient descent on the real neural training objective behaves similarly — at first. Namely, let ${ \\bf q } _ { 1 }$ be the leading Hessian eigenvector at the iteration where the sharpness reaches $2 / \\eta$ . In Figure 4(b), we resume training the network, and we monitor both the train loss and the quantity $\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.$ for the next 215 iterations. Observe that $\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.$ oscillates with ever-increasing magnitude, similar to the divergent quadratic example in Figure 2(b). At first, these oscillations are too small to affect the objective appreciably, and so the train loss continues to monotonically decrease. But eventually, these oscillations grow big enough that the train loss spikes. ", + "bbox": [ + 173, + 535, + 825, + 715 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/d167549c8548e4247c63c0dafe3ea4b11804fc67c00c956313b55282c60c6c65.jpg", + "image_caption": [ + "Figure 4: Once the sharpness crosses $2 / \\eta$ , gradient descent becomes destabilized. We run gradient descent at $\\eta = 0 . 0 1$ . (a) The sharpness eventually reaches $2 / \\eta$ . (b) Once the sharpness crosses $2 / \\eta$ , the iterates start to oscillate along ${ \\bf q } _ { 1 }$ with ever-increasing magnitude. (c) Somehow, GD does not diverge entirely; instead, the train loss continues to decrease, albeit non-monotonically. " + ], + "image_footnote": [], + "bbox": [ + 179, + 746, + 825, + 848 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Once gradient descent becomes destabilized in this manner, classical optimization theory gives no clues as to what will happen next. One might imagine that perhaps gradient descent might diverge entirely, or that gradient descent might stall while failing to make progress, or that gradient descent might jump to a flatter region and remain there. In reality, none of these outcomes occurs. In Figure 4(c), we plot both the train loss and $\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.$ for 1000 iterations after the sharpness first crossed $2 / \\eta$ . Observe that gradient descent somehow avoids diverging entirely. Instead, after initially spiking around iteration 215, the train loss continues to decrease, albeit non-monotonically. ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This numerical example is representative. In general, after the sharpness initially crosses $2 / \\eta$ , gradient descent enters a regime we call the Edge of Stability, in which (1) the sharpness hovers right at, or just above, the value $2 / \\eta$ ; and (2) the train loss behaves non-monotonically over short timescales, yet decreases consistently over long timescales. Indeed, in Figure 5, we run gradient descent at a range of step sizes using both MSE and cross-entropy loss. The left plane plots the train loss curves, with a vertical dotted line (of the appropriate color) marking the iteration where the sharpness first crosses $2 / \\eta$ . Observe that the train loss decreases monotonically before this dotted line, but behaves non-monotonically afterwards. The middle plane plots the evolution of the sharpness, with a horizontal dashed line (of the appropriate color) at the value $2 / \\eta$ . Observe that once the sharpness reaches $2 / \\eta$ , it ceases to increase further, and instead hovers right at, or just above, the value $2 / \\eta$ for the remainder of training. (The precise meaning of “just above” varies: in Figure 5, for MSE loss, the sharpness hovers just a minuscule amount above $2 / \\eta$ , while for cross-entropy loss, the gap between the sharpness and $2 / \\eta$ is small yet non-miniscule.) ", + "bbox": [ + 174, + 208, + 825, + 388 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "At the Edge of Stability, gradient descent is “trying” to increase the sharpness further, but is being restrained from doing so. To demonstrate this, in Figure 7, we train at step size $2 / 2 0 0$ until reaching the Edge of Stability, and then at iteration 6,000 (marked by the vertical black line), we drop the step size to $\\eta = 2 / 3 0 0$ . Observe that after the learning rate drop, the sharpness immediately starts to increase, and only stops increasing once gradient descent is back at the Edge of Stability. Appendix O repeats this experiment on more architectures. Intuitively, gradient descent with fixed step sizes acts like a constrained optimization algorithm: the use of step size $\\eta$ imposes an implicit $2 / \\eta$ constraint on the sharpness (Nar & Sastry, 2018), and at the Edge of Stability this constraint is “active.” ", + "bbox": [ + 173, + 395, + 825, + 507 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Observe from Figure 5 that there do exist step sizes $\\eta$ (in purple) small enough that the sharpness never rises to $2 / \\eta$ . We call such a step size stable. However, observe that with cross-entropy loss, it takes 3700 iterations to train at the stable step size in purple, but only 1000 iterations to train at the larger step size in blue. In general, we always observe that stable step sizes are suboptimal in terms of convergence speed. In fact, in $\\ S 4$ we will see that for standard networks on CIFAR-10, stable step sizes are so suboptimally small that they are completely unreasonable. ", + "bbox": [ + 174, + 513, + 825, + 570 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/1d1a6cde8e4d00e092499a010db67cddcc397824f725bfbf3fd1dc2519962684.jpg", + "image_caption": [ + "Figure 5: After the sharpness reaches $2 / \\eta$ , gradient descent enters the Edge of Stability. A network is trained with gradient descent at a range of step sizes (see legend), using both MSE loss (top row) and cross-entropy (bottom row). Left: the train loss curves, with a vertical dotted line at the iteration where the sharpness first crosses $2 / \\eta$ . Center: the sharpness, with a horizontal dashed line at the value $2 / \\eta$ . Right: sharpness plotted by time $\\mathit { \\Psi } = \\mathit { \\Psi }$ iteration $\\times \\eta$ ) rather than iteration. " + ], + "image_footnote": [], + "bbox": [ + 171, + 618, + 857, + 833 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/1f252f03c147233621dfa8b75b997a2e37764b54085eddf968800144261a3aa1.jpg", + "image_caption": [ + "Figure 6: Momentum. We run GD with step size $\\eta = 0 . 0 1$ and Polyak or Nesterov momentum at various $\\beta$ . For each algorithm, the horizontal dashed line marks the MSS from Equation 1. " + ], + "image_footnote": [], + "bbox": [ + 184, + 107, + 421, + 195 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/46bb801641419ab649df218983ffaa1aa8aacf166e8d5d6ec575bf2ff10b7ccd.jpg", + "image_caption": [ + "Figure 7: After a learning rate drop, progressive sharpening resumes. We start training at $\\eta = 2 / 2 0 0$ (orange) and then after 6000 iterations (dotted vertical black line), we cut the step size to $\\eta = 2 / 3 0 0$ (green). Observe that as soon as the step size is cut, the sharpness starts to rise. " + ], + "image_footnote": [], + "bbox": [ + 449, + 107, + 843, + 196 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 313, + 821, + 340 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The “Edge of Stability” effect generalizes to gradient descent with momentum. In Figure 6, we train using gradient descent with step size $\\eta = 0 . 0 1$ , and varying amounts of either Polyak or Nesterov momentum. Observe that in each case, the sharpness rises until reaching the MSS given by Equation 1, and then plateaus there. Appendix $\\mathbf { N }$ has more momentum experiments. ", + "bbox": [ + 174, + 348, + 825, + 404 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Appendix P, we briefly examine the evolution of the next few Hessian eigenvalues during gradient descent. We find that each of these eigenvalues rises until plateauing near $\\bar { 2 } / \\eta$ . ", + "bbox": [ + 174, + 411, + 821, + 439 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Prior work. Aspects of the Edge of Stability have been observed previously in the literature. Wu et al. (2018) noted that the sharpness at the solution reached by full-batch gradient descent was not just less than $2 / \\eta$ , as was expected due to stability considerations, but was mysteriously approximately equal to $2 / \\eta$ . In retrospect, we can attribute this observation to progressive sharpening. Xing et al. (2018) observed that full-batch gradient descent eventually enters a regime (the Edge of Stability) in which the training loss behaves non-monotonically, and the iterates oscillate along the direction of largest curvature; however, they did not relate this regime to the sharpness. Lewkowycz et al. (2020) found that in neural network training, if the sharpness at initialization is larger than $2 / \\eta$ , then after becoming initially destabilized, gradient descent does not always diverge entirely (as the quadratic Taylor approximation would suggest), but rather sometimes “catapults” into a flatter region that is flat enough to stably accommodate the step size. It seems plausible that whichever properties of neural training objectives permit this so-called “catapult” behavior may also be the same properties that permit successful optimization at the Edge of Stability. Indeed, optimization at the Edge of Stability can conceivably be viewed as a never-ending series of micro-catapults. As we discuss at greater length in $\\ S 6$ , several papers (Jastrz˛ebski et al., 2017; 2019) have observed that large step sizes steer stochastic gradient descent into less sharp regions of the loss landscape, and Jastrz˛ebski et al. (2020) attributed this effect to the stability properties of SGD. Finally, our precise characterization of the behavior of the sharpness during full-batch gradient descent adds to a growing body of work that empirically investigates the Hessian spectrum of neural networks (Sagun et al., 2017; Ghorbani et al., 2019; Li et al., 2020a; Papyan, 2018; 2019; 2020). ", + "bbox": [ + 173, + 445, + 825, + 724 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 THE GRADIENT FLOW TRAJECTORY ", + "text_level": 1, + "bbox": [ + 176, + 743, + 457, + 757 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the right pane of Figure 5, we plot the evolution of the sharpness during gradient descent, with “time” $=$ iteration $\\times \\eta$ , rather than iteration, on the $\\mathbf { X }$ -axis. This allows us to directly compare the sharpness after, say, 100 iterations at $\\eta = 0 . 0 1$ to the sharpness after 50 iterations at $\\eta = 0 . 0 2$ ; both are time 1. Observe that when plotted by time, the sharpnesses for gradient descent at different step sizes coincide until the time where each reaches $2 / \\eta$ . This is because for this network, gradient descent at $\\eta = 0 . 0 1$ and gradient descent at $\\eta = 0 . 0 2$ initially travel the same path (moving at a speed proportional to $\\eta$ ) until each reaches the point on that path where the sharpness hits $2 / \\eta$ . This path is the gradient flow trajectory. The gradient flow solution at time $t$ is defined as the limit as $\\eta 0$ of the gradient descent iterate at iteration $t / \\eta$ (if this limit exists). The empirical finding of interest is that for this particular network, gradient descent does not only track the gradient flow trajectory in the limit of infinitesimally small step sizes, but for any step size that is less than 2/sharpness. ", + "bbox": [ + 173, + 770, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We can numerically approximate gradient flow trajectories by using the Runge-Kutta RK4 algorithm (Press et al., 1992) to numerically integrate the gradient flow ODE. Empirically, for many but not all networks studied in this paper, we find that gradient descent at any step size $\\eta$ closely tracks the Runge-Kutta trajectory until reaching the point on that trajectory where the sharpness hits $2 / \\eta$ . (This sometimes occurs even for networks with ReLU activations or max-pooling, which give rise to training objectives that are not continuously differentiable, which means that the gradient flow trajectory is not necessarily guaranteed to exist.) For such networks, the gradient flow trajectory provides a coherent framework for reasoning about which step sizes will eventually enter the Edge of Stability. Let $\\lambda _ { 0 }$ be the sharpness at initialization, and let $\\lambda _ { \\mathrm { m a x } }$ be the maximum sharpness along the gradient flow trajectory. If $\\eta < 2 / \\lambda _ { \\mathrm { m a x } }$ , then gradient descent will stably track the gradient flow trajectory for the entire duration of training, and will never enter the Edge of Stability. On the other hand, if $\\eta \\in [ 2 / \\lambda _ { \\operatorname* { m a x } } , 2 / \\lambda _ { 0 } ]$ , then gradient descent will stably track the gradient flow trajectory only until reaching the point on that trajectory where the sharpness hits $2 / \\eta$ ; shortly afterwards, gradient descent will become destabilized, depart the gradient flow trajectory, and enter the Edge of Stability. ", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 FURTHER EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 324, + 406, + 340 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Section 3 focused for exposition on a single architecture and task. In this section, we show that our characterization of gradient descent holds broadly across a wide range of architectures and tasks. We detail several known caveats and qualifications in Appendix A. ", + "bbox": [ + 174, + 359, + 823, + 401 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Architectures. In Appendix J, we fix the task of training a $5 \\mathrm { k }$ subset of CIFAR-10, and we systematically vary the network architecture. We consider fully-connected networks, as well as convolutional networks with both max-pooling and average pooling. For all of these architectures, we consider tanh, ReLU, and ELU activations, and for fully-connected networks we moreover consider softplus and hardtanh —- eleven networks in total. We train each network with both cross-entropy and MSE loss. In each case, we successfully reproduce Figure 5. ", + "bbox": [ + 174, + 409, + 823, + 492 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Since batch normalization (Ioffe & Szegedy, 2015) is known to have unusual optimization properties (Li & Arora, 2019), it is natural to wonder whether our findings still hold with batch normalization. In Appendix K, we confirm that they do, and we reconcile this point with Santurkar et al. (2018). ", + "bbox": [ + 174, + 500, + 823, + 541 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Tasks. In Appendix L, we verify our findings on: (1) a Transformer trained on the WikiText2 language modeling task; (2) fully-connected tanh networks with one hidden layer, trained on a synthetic one-dimensional toy regression task; and (3) deep linear networks trained on Gaussian data. In each case, the sharpness rises until hovering right at, or just above, the value $2 / \\eta$ . ", + "bbox": [ + 174, + 547, + 823, + 604 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Standard networks on CIFAR-10. In Appendix M, we verify our findings on three standard architectures trained on the full CIFAR-10 dataset: a ResNet with BN, a VGG with BN, and a VGG without BN. For all three architectures, we find that progressive sharpening occurs to a dramatic degree, and, relatedly, that stable step sizes are dramatically suboptimal. For example, when we train the VGG-BN to $9 9 \\%$ accuracy using gradient flow / Runge-Kutta, we find that the sharpness rises from 6.3 at initialization to a peak sharpness of 2227.6. Since this is an architecture for which gradient descent closely hews to the gradient flow trajectory, we can conclude that any stable step size for gradient descent would need to be less than $2 / 2 2 2 7 . 6 = 0 . 0 0 0 8 9 7 .$ . Training finishes at time 14.91, so gradient descent at any stable step size would require at least $1 4 . 9 1 / 0 . 0 \\bar { 0 } 0 8 9 7 = 1 6 , 6 2 2$ iterations. Yet empirically, this network can be trained to completion at the larger, “Edge of Stability” step size of $\\eta = 0 . 1 6$ in just 329 iterations. Therefore, training at a stable step size is suboptimal by a factor of at least $1 6 6 2 2 / 3 2 9 = 5 0 . 5 $ . The situation is similar for the other two architectures we consider. In short, for standard architectures on the standard dataset CIFAR-10, stable step sizes are not just suboptimally small, they are so suboptimal as to be completely unreasonable. For these networks, gradient descent at any reasonable step size eventually enters the Edge of Stability regime. ", + "bbox": [ + 174, + 611, + 825, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Tracking gradient flow. Recall that for some (but not all) architectures, gradient descent closely hews to the gradient flow trajectory so long as the sharpness is less than $2 / \\eta$ . Among the architectures considered in Appendix J, we found this to be true for the architectures with continuously differentiable components, as well as some, but not all, with ReLU, hardtanh, and max-pooling. Among the architectures in Appendix L, we found this to be true for the tanh network, but not for the deep linear network or the Transformer. Finally, we did find this to be true for the three standard architectures in Appendix M, even though those architectures use ReLU. ", + "bbox": [ + 174, + 825, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 310, + 118 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now explain why the behavior of gradient descent at the Edge of Stability contradicts several pieces of conventional wisdom in optimization. ", + "bbox": [ + 176, + 133, + 821, + 161 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "At reasonable step sizes, gradient descent cannot be analyzed using (even local) $L$ -smoothness Many convergence analyses of gradient descent assume a bound on the sharpness — either globally or, at the very least, along the optimization trajectory. This condition, called $L$ -smoothness (Nesterov, 1998), is intended to guarantee that each gradient step will decrease the training objective by a certain amount; the weakest guarantee is that if the local sharpness is less than $2 / \\eta$ , then a gradient step with size $\\eta$ is guaranteed to decrease (rather than increase) the training objective. At a bare minimum, any convergence analysis of gradient descent based on $L$ -smoothness will require the sharpness along the optimization trajectory to be less than $2 / \\eta$ . Yet to the contrary, at the Edge of Stability, the sharpness hovers just above $2 / \\eta$ . Therefore, at any step size for which gradient descent enters the Edge of Stability (which, on realistic architectures, includes any reasonable step size), gradient descent cannot be analyzed using $L$ -smoothness. Li et al. (2020b) previously argued that convergence analyses based on $L$ -smoothness do not apply to networks with both batch normalization and weight decay; our paper empirically extends this to neural networks without either. ", + "bbox": [ + 174, + 179, + 825, + 359 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$L$ -smoothness may be inappropriate when analyzing other optimization algorithms too It is common for optimization papers seemingly motivated by deep learning to analyze algorithms under the “non-convex but $L$ -smooth”’ setting (Reddi et al., 2016; Agarwal et al., 2017; Zaheer et al., 2018; Zhou et al., 2018; Chen et al., 2019; Li & Orabona, 2019; Ward et al., 2019; You et al., 2020; Vaswani et al., 2019; Sankararaman et al., 2019; Reddi et al., 2021; Défossez et al., 2020; Xie et al., 2020; Liu et al., 2020; Defazio, 2020). Since our experiments focus on gradient descent, it does not necessarily follow that $L$ -smoothness assumptions are unjustified when analyzing other optimization algorithms. However, gradient descent is arguably the simplest optimization algorithm, so we believe that the fact that (even local) $L$ -smoothness fails even there should raise serious questions about the suitability of the $L$ -smoothness assumption in neural network optimization more generally. In particular, the burden of proof should be on authors to empirically justify this assumption. ", + "bbox": [ + 174, + 375, + 825, + 527 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "At reasonable step sizes, gradient descent does not monotonically decrease the training loss In neural network training, SGD does not monotonically decrease the training objective, in part due to minibatch randomness. However, it is often assumed that full-batch gradient descent would monotonically decrease the training objective, were it used to train neural networks. For example, Zhang et al. (2020) proposed a “relaxed $L$ -smoothness” condition that is less restrictive than standard $L$ -smoothness, and proved a convergence guarantee for gradient descent under this condition which asserted that the training objective will decrease monotonically. Likewise, some neural network analyses such as Allen-Zhu et al. (2019) also assert that the training objective will monotonically decrease. Yet, at the Edge of Stability, the training loss behaves non-monotonically over short timescales even as it consistently decreases over long timescales. Therefore, convergence analyses which assert monotone descent cannot possibly apply to gradient descent at reasonable step sizes. ", + "bbox": [ + 174, + 544, + 825, + 696 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The Edge of Stability is inherently non-quadratic It is tempting to try to reason about the behavior of gradient descent on neural network training objectives by analyzing, as a proxy, the behavior of gradient descent on the local quadratic Taylor approximation (LeCun et al., 1998). However, at the Edge of Stability, the behavior of gradient descent on the real neural training objective is irreconcilably different from the behavior of gradient descent on the quadratic Taylor approximation: the former makes consistent (if choppy) progress, whereas the latter would diverge (and this divergence would happen quickly, as we demonstrate in Appendix E). Thus, the behavior of gradient descent at the Edge of Stability is inherently non-quadratic. ", + "bbox": [ + 174, + 713, + 825, + 824 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Dogma for step size selection may be unjustified An influential piece of conventional wisdom concerning step size selection has its roots in the quadratic Taylor approximation model of gradient descent. This conventional wisdom (LeCun et al., 1993; 1998; Schaul et al., 2013) holds that if the sharpness at step $t$ is $\\lambda _ { t }$ , then the current step size $\\eta _ { t }$ must be set no greater than $2 / \\lambda _ { t }$ (in order to prevent divergence); and furthermore, barring additional information about the objective function, that $\\eta _ { t }$ should optimally be set to $1 / \\lambda _ { t }$ . Our findings complicate this conventional wisdom. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To start, it is nearly impossible to satisfy these prescriptions with a fixed step size: for any fixed (and reasonable) step size $\\eta _ { t } = \\eta$ , progressive sharpening eventually drives gradient descent into regions where the sharpness is just a bit greater than $2 / \\eta$ — which means that the step size $\\eta$ is purportedly impermissible. Furthermore, in Appendix F, we try running gradient descent with the purportedly optimal $\\eta _ { t } = 1 / \\lambda _ { t }$ rule, and find that this algorithm is soundly outperformed by the purportedly impermissible baseline of gradient descent with a fixed $\\eta _ { t } = 1 / \\lambda _ { 0 }$ step size, where $\\lambda _ { 0 }$ is the sharpness at initialization. The $\\eta _ { t } = 1 / \\lambda _ { t }$ rule continually anneals the step size, and in so doing ensures that the training objective will decrease at each iteration, whereas the fixed $\\eta _ { t } = 1 / \\lambda _ { 0 }$ step size often increases the training objective. However, this non-monotonicity turns out to be a worthwhile price to pay in return for the ability to take larger steps. ", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 STOCHASTIC GRADIENT DESCENT ", + "text_level": 1, + "bbox": [ + 174, + 263, + 488, + 279 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our precise characterization of the behavior of the sharpness only applies to full-batch gradient descent. In contrast, during SGD, the sharpness does not always settle at any fixed value (Appendix G), let alone one that can be numerically predicted from the hyperparameters. Nevertheless, prior works (Jastrz˛ebski et al., 2017; 2019; 2020) have demonstrated that large step sizes do steer SGD into regions of the landscape with lower sharpness; the $2 / \\eta$ rule we have identified for full-batch gradient descent is a special case of this observation. Furthermore, small batch sizes also steer SGD into regions with lower sharpness (Keskar et al., 2016; Jastrz˛ebski et al., 2017), as we illustrate in Appendix G. Jastrz˛ebski et al. (2020) attributed this phenomenon to the stability properties of SGD. ", + "bbox": [ + 174, + 294, + 825, + 406 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Even though our findings only strictly hold for gradient descent, they may have relevance to SGD as well. First, since gradient descent is a special case of SGD, any general characterization of the dynamics of SGD must reduce to the Edge of Stability in the full-batch special case. Second, there are indications that the Edge of Stability may have some analogue for SGD. One way to interpret our main findings is that gradient descent “acclimates” to the step size in such a way that each update sometimes increases and sometimes decrease the train loss, yet an update with a smaller step size would consistently decrease the training loss. Along similar lines, in Appendix H we demonstrate that SGD “acclimates” to the step size and batch size in such a way that each SGD update sometimes increases and sometimes decreases the training loss in expectation, yet an SGD update with a smaller step size or larger batch size would consistently decrease the training loss in expectation. ", + "bbox": [ + 174, + 412, + 825, + 551 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In extending these findings to SGD, the question arises of how to model “stability” of SGD. This is a highly active area of research. Wu et al. (2018) proposed modeling stability in expectation, and gave a sufficient (but not necessary) criterion for the stability of SGD in expectation. Building on this framework, Jastrz˛ebski et al. (2020) argued that if the Hessian is aligned with the second moment matrix of per-example gradients, then SGD is stable so long as a certain expression (involving the sharpness) is below a certain threshold. In the special full-batch case, their criterion reduces to the sharpness being beneath $2 / \\eta$ — a constraint which we have shown is “tight” throughout training. However, in the general SGD case, there is no evidence that their stability constraint is tight throughout training. Giladi et al. (2020) showed that the generalization gap in asynchronous SGD can be mostly ameliorated by setting the step size so as to ensure that stability properties in expectation remain identical to those of a well-tuned implementation of synchronous SGD. Finally, a number of papers have attempted to mathematically model the propensity of SGD to “escape from sharp minima” (Hu et al., 2017; Zhu et al., 2019; Xie et al., 2021). ", + "bbox": [ + 174, + 559, + 825, + 739 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 758, + 318, + 775 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have empirically demonstrated that the behavior of gradient decent on neural training objectives is both surprisingly consistent across architectures and tasks, and surprisingly different from that envisioned in the conventional wisdom. Our findings raise a number of questions. Why does progressive sharpening occur? At the Edge of Stability, by what mechanism does gradient descent avoid diverging entirely? Since the conventional wisdom for step size selection is wrong, how should the gradient descent step size be set during deep learning? Does the “Edge of Stability” effect generalize in some way to optimization algorithms beyond gradient descent, such as SGD? We hope to inspire future efforts aimed at addressing these questions. ", + "bbox": [ + 174, + 790, + 825, + 902 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 397, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported in part by DARPA FA875017C0141, the National Science Foundation grants IIS1705121 and IIS1838017, an Amazon Web Services Award, a JP Morgan A.I. Research Faculty Award, a Carnegie Bosch Institute Research Award, a Facebook Faculty Research Award, and a Block Center Grant. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA, the National Science Foundation, or any other funding agency. 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On the convergence of adaptive gradient methods for nonconvex optimization. arXiv preprint arXiv:1808.05671, 2018. ", + "bbox": [ + 171, + 619, + 823, + 648 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Zhanxing Zhu, Jingfeng Wu, Bing Yu, Lei Wu, and Jinwen Ma. The anisotropic noise in stochastic gradient descent: Its behavior of escaping from sharp minima and regularization effects. In Proceedings of the 36th International Conference on Machine Learning. PMLR, 2019. ", + "bbox": [ + 174, + 656, + 825, + 700 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A CAVEATS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this appendix, we list several caveats to our generic characterization of the dynamics of full-batch gradient descent on neural network training objectives. ", + "bbox": [ + 174, + 133, + 823, + 161 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "1. With cross-entropy loss, the sharpness often drops at the end of training As mentioned in the main text, when neural networks are trained on classification tasks using the cross-entropy loss, the sharpness frequently drops near the end of training, once the classification accuracy begins to approach 1. We explain this effect in Appendix C. ", + "bbox": [ + 174, + 178, + 825, + 233 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2. For shallow or wide networks, or on simple datasets, sharpness doesn’t rise that much When the network is shallow (Figure 16 - 17) or wide (Figure 12 - 14), or when the dataset is “easy” or small (Figure 18), the sharpness may rise only a small amount over the gradient flow trajectory. For these optimization problems, “stable step sizes” (those for which gradient descent never enters the Edge of Stability) may be quite reasonable, and the range of “Edge of Stability” step sizes may be quite small. ", + "bbox": [ + 173, + 248, + 825, + 332 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "3. Sharpness sometimes drops at the beginning of training We sometimes observe that the sharpness drops at the very beginning of training, as the network leaves its initialization. This was more common when training with MSE loss than with cross-entropy loss. For most networks, this drop was very slight. However, the combination of both batch normalization and MSE loss sometimes caused situations where the sharpness was considerably large at initialization, and dropped precipitously as soon as training began. Figure 8 illustrates one such network. ", + "bbox": [ + 173, + 348, + 825, + 431 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/4f65d59ff48c406a01dc0bef3cbe8d711ac190b425e7e02746021efd9dd37558.jpg", + "image_caption": [ + "Figure 8: With batch norm and MSE loss, the sharpness sometimes drops precipitously: We use gradient flow to train a CNN with ReLU activations and batch norm using the MSE loss. Observe that the sharpness drops substantially at the beginning of training, and never recovers to its initial value. This seems to occur due to the combination of both batch normalization and MSE loss. " + ], + "image_footnote": [], + "bbox": [ + 217, + 448, + 781, + 534 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "4. With batch normalization, need to look at sharpness between iterates As detailed in Appendix K, when we trained batch-normalized networks using very small step sizes $\\eta$ , we sometimes observed that the sharpness at the gradient descent iterates themselves plateaued below $2 / \\eta$ , even though the sharpness in between the iterates plateaued just above $2 / \\eta$ , as expected. ", + "bbox": [ + 174, + 636, + 825, + 691 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "5. With non-differentiable components, instability sometimes begins when the sharpness is a bit less than $2 / \\eta$ When training networks with ReLU or hardtanh activations, we sometimes observed that the sharpness started to plateau (and the training loss started to behave nonmonotonically) a bit before the instant when the sharpness crossed $2 / \\eta$ . For example, see Figures 41, 43 or Figures 45, 47. One potential explanation is that for such networks, the training objective is not continuously differentiable, so the second-order Taylor approximation around a given iterate may be a poor local model for the training objective at even an tiny distance away in weight space, if that weight change causes some activations to switch “ReLU case” from $\\leq 0$ to $> 0$ , or visa versa. ", + "bbox": [ + 173, + 707, + 825, + 819 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B STABILITY OF GRADIENT DESCENT ON QUADRATIC FUNCTIONS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 736, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This appendix describes the stability properties of gradient descent (and its momentum variants) when optimizing the quadratic objective function ", + "bbox": [ + 174, + 132, + 823, + 161 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/b08f3fb6d94c4e99f30a8748c8f3362b60436c7cae05531e3748484082a4787b.jpg", + "text": "$$\nf ( \\mathbf { x } ) = { \\frac { 1 } { 2 } } \\mathbf { x } ^ { T } \\mathbf { A } \\mathbf { x } + \\mathbf { b } ^ { T } \\mathbf { x } + c\n$$", + "text_format": "latex", + "bbox": [ + 401, + 167, + 596, + 199 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "starting from the initialization $\\mathbf { x } _ { \\mathrm { 0 } }$ . ", + "bbox": [ + 174, + 204, + 397, + 219 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To review, vanilla gradient descent is defined by the iteration: ", + "bbox": [ + 173, + 224, + 576, + 241 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/be3e753aacdaf65d6bd046fde0850c2c27d5a97c3867beb45c543199b2f64d86.jpg", + "text": "$$\n\\mathbf x _ { t + 1 } = \\mathbf x _ { t } - \\eta \\nabla f ( \\mathbf x _ { t } ) .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 247, + 578, + 265 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Meanwhile, gradient descent with Polyak (also called “heavy ball”) momentum (Polyak, 1964; Sutskever et al., 2013; Goodfellow et al., 2016) is defined by the iteration: ", + "bbox": [ + 174, + 270, + 823, + 299 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/4b2dbb2dcfe8b7269da72de452d53cbcd37299bf3eac43901b6686fb742e4530.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { v } _ { t + 1 } = \\beta \\mathbf { v } _ { t } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 415, + 305, + 581, + 343 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\mathbf { v } _ { t }$ is a “velocity” vector and $0 \\leq \\beta < 1$ is the momentum coefficient. For $\\beta = 0$ the algorithm reduces to vanilla GD. ", + "bbox": [ + 174, + 347, + 823, + 376 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Finally, Nesterov momentum Sutskever et al. (2013); Goodfellow et al. (2016) is an adaptation of Nesterov’s accelerated gradient (Nesterov, 1983) for deep learning defined by the iteration: ", + "bbox": [ + 174, + 382, + 821, + 411 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2fc7900f4d1874e51fa52179567b0feb115f1698d8ef9f5e3ac6fa657621cf57.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { v } _ { t + 1 } = \\beta \\mathbf { v } _ { t } - \\eta \\nabla f ( \\mathbf { x } _ { t } + \\beta \\mathbf { v } _ { t } ) } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 392, + 417, + 604, + 454 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\mathbf { v } _ { t }$ is a “velocity” vector and $0 \\leq \\beta < 1$ is the momentum coefficient. For $\\beta = 0$ the algorithm reduces to vanilla GD. ", + "bbox": [ + 173, + 459, + 820, + 488 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "All three of these algorithms share a special property: on quadratic functions, they act independently along each Hessian eigenvector. That is, if we express the iterates in the Hessian eigenvector basis, then in this basis the coordinates evolve independent from one another under gradient descent. ", + "bbox": [ + 173, + 494, + 825, + 536 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proposition 1. Consider running vanilla gradient descent on the quadratic objective (2) starting from $\\mathbf { x } _ { \\mathrm { 0 } }$ . Let $( \\mathbf { q } , a )$ be an eigenvector/eigenvalue pair of A. If $a > 2 / \\eta$ , then the sequence $\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}$ will diverge. ", + "bbox": [ + 173, + 540, + 825, + 583 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. The update rule for gradient descent on this quadratic function is: ", + "bbox": [ + 173, + 598, + 651, + 614 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/cc33d683ab17cdba521de10972b9913d629706f5672fa14ed396a67497bcf2d9.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - \\eta ( \\mathbf { A } \\mathbf { x } _ { t } + \\mathbf { b } ) } \\\\ { = ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\eta \\mathbf { b } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 408, + 619, + 589, + 657 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore, the quantity $\\mathbf { q } ^ { T } \\mathbf { x } _ { t }$ evolves under gradient descent as: ", + "bbox": [ + 171, + 662, + 593, + 679 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9cab90bcb3a949bb22abc1fff25329327a36d8ce439c3b0a566b603eaa7c3cba.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { q } ^ { T } \\mathbf { x } _ { t + 1 } = \\mathbf { q } ^ { T } ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\eta \\mathbf { b } } \\\\ & { \\qquad = ( 1 - \\eta a ) \\mathbf { q } ^ { T } \\mathbf { x } _ { t } - \\eta \\mathbf { q } ^ { T } \\mathbf { b } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 379, + 684, + 619, + 727 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/4df55f00b4539a762d0888911b989e2ae1553327b3052871f8292528df217304.jpg", + "text": "$$\n( \\mathbf { q } ^ { T } \\mathbf { A } = a \\mathbf { q } )\n$$", + "text_format": "latex", + "bbox": [ + 733, + 705, + 825, + 724 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Define $\\begin{array} { r } { \\tilde { x } _ { t } = \\mathbf q ^ { T } \\mathbf x _ { t } + \\frac { 1 } { a } \\mathbf q ^ { T } \\mathbf b } \\end{array}$ , and note that $\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}$ diverges if and only if $\\{ \\tilde { x } _ { t } \\}$ diverges. ", + "bbox": [ + 171, + 739, + 751, + 757 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The quantity $\\tilde { x } _ { t }$ evolves under gradient descent according to the simple rule: ", + "bbox": [ + 176, + 762, + 676, + 777 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/b927b8e51febf2c128a19445c33dec5e8a7dd8448c5225dab175fc47f6e156d2.jpg", + "text": "$$\n\\tilde { x } _ { t + 1 } = ( 1 - \\eta a ) \\tilde { x } _ { t }\n$$", + "text_format": "latex", + "bbox": [ + 433, + 784, + 563, + 801 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since $\\eta > 0$ , if $a > 2 / \\eta$ then $( 1 - \\eta a ) < - 1$ , so the sequence $\\{ \\tilde { x } _ { t } \\}$ will diverge. ", + "bbox": [ + 173, + 814, + 700, + 832 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We now prove analogous results for Nesterov and Polyak momentum. ", + "bbox": [ + 174, + 844, + 632, + 861 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Theorem 1. Consider running Nesterov momentum on the quadratic objective (2) starting from any initialization. Let $( \\mathbf { q } , a )$ be an eigenvector/eigenvalue pair of A. If $\\begin{array} { r } { a > \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) } \\end{array}$ , then the sequence $\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}$ will diverge. ", + "bbox": [ + 173, + 102, + 825, + 157 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. The update rules for Nesterov momentum on this quadratic function are: ", + "bbox": [ + 171, + 171, + 696, + 188 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/f616dcc37ade3672fcc0d5f0ddd3fffc41aad18dd30b5e6fe7a6438f4b489daa.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { v } _ { t + 1 } = \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { v } _ { t } - \\eta \\mathbf { b } - \\eta \\mathbf { A } \\mathbf { x } _ { t } } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 375, + 193, + 620, + 229 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Using the fact that $\\mathbf { v } _ { t } = \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 }$ , we can rewrite this as a recursion in $\\mathbf { x } _ { t }$ alone: ", + "bbox": [ + 173, + 242, + 710, + 258 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/023dd9f49c717cdb578bdfa3754ebf0214d1b0c545df2e63a91a65620addb629.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) ( \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 } ) - \\eta \\mathbf { b } - \\eta \\mathbf { A } \\mathbf { x } _ { t } } \\\\ { = ( 1 + \\beta ) ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { b } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 263, + 674, + 301 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Define $\\begin{array} { r } { \\tilde { x } _ { t } = { \\bf q } ^ { T } { \\bf x } _ { t } + \\frac { 1 } { a } { \\bf q } ^ { T } { \\bf b } } \\end{array}$ , and note that $\\mathbf { q } ^ { T } \\mathbf { x } _ { t }$ diverges iff $\\tilde { x } _ { t }$ diverges. It can be seen that $\\tilde { x } _ { t }$ evolves as: ", + "bbox": [ + 168, + 314, + 823, + 344 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/86072fb8138371a0270d0e5423ca362d419de9fbe0334bd4af1a3e076fee7bcb.jpg", + "text": "$$\n\\tilde { x } _ { t + 1 } = ( 1 + \\beta ) ( 1 - \\eta a ) \\tilde { x } _ { t } - \\beta ( 1 - \\eta a ) \\tilde { x } _ { t - 1 }\n$$", + "text_format": "latex", + "bbox": [ + 348, + 351, + 650, + 368 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This is a linear homogenous second-order difference equation. By Theorem 2.37 in Elaydi (2005), since $\\eta > 0$ and $\\beta < 1$ , if $\\begin{array} { r } { a > \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) } \\end{array}$ then this recurrence diverges. ", + "bbox": [ + 173, + 381, + 823, + 420 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The following result previously appeared in Goh (2017). ", + "bbox": [ + 173, + 455, + 545, + 470 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Theorem 2. Consider running Polyak momentum on the quadratic objective (2) starting from any initialization. Let $( \\mathbf { q } , a )$ be an eigenvector/eigenvalue pair of A. If $a ~ > ~ \\frac { 1 } { \\eta } \\left( 2 + 2 \\beta \\right)$ , then the sequence $\\left\\{ \\mathbf { q } ^ { T } \\mathbf { x } _ { t } \\right\\}$ will diverge. ", + "bbox": [ + 173, + 474, + 826, + 522 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Using the fact that $\\mathbf v _ { t } = \\mathbf x _ { t } - \\mathbf x _ { t - 1 }$ , we can re-write the Polyak momentum recursion as a recursion in $\\mathbf { x }$ alone: ", + "bbox": [ + 174, + 536, + 823, + 565 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/b2e7fb1644cf9b35f467a98b39d645fdde2da527ee28ef5991b877863faaa65d.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\beta \\mathbf { v } _ { t } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) } & { } \\\\ { = \\mathbf { x } _ { t } + \\left( \\beta ( \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 } ) - \\eta \\nabla f ( \\mathbf { x } _ { t } ) \\right) } & { } \\\\ { = ( 1 + \\beta ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 359, + 570, + 637, + 627 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For the quadratic objective (2), this update rule amounts to: ", + "bbox": [ + 173, + 631, + 563, + 647 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/2baf7033190aa82ab06c7ec5f6b2926fcd86e37253a84a1305f697795aba186a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { x } _ { t + 1 } = ( 1 + \\beta ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta ( \\mathbf { A } \\mathbf { x } _ { t } + \\mathbf { b } ) } \\\\ & { \\qquad = ( 1 + \\beta - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { b } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 651, + 643, + 689 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Multiplying by $\\mathbf { q } ^ { T }$ , we obtain: ", + "bbox": [ + 174, + 695, + 377, + 712 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/a2d6477c3eb33e96773ff2908186d1aa508a8f03a3c5b6c188acaf52cf3a8f24.jpg", + "text": "$$\n\\mathbf { q } ^ { T } \\mathbf { x } _ { t + 1 } = ( 1 + \\beta - \\eta a ) \\mathbf { q } ^ { T } \\mathbf { x } _ { t } - \\beta \\mathbf { q } ^ { T } \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { q } ^ { T } \\mathbf { b } .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 717, + 673, + 736 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Now, define $\\begin{array} { r } { \\tilde { x } _ { t } = \\mathbf q ^ { T } \\mathbf x _ { t } + \\frac { 1 } { a } \\mathbf q ^ { T } \\mathbf b } \\end{array}$ . Note that $\\mathbf { q } ^ { T } \\mathbf { x } _ { t }$ diverges iff $\\tilde { x } _ { t }$ diverges. It can be seen that $\\tilde { x } _ { t }$ evolves as: ", + "bbox": [ + 173, + 743, + 823, + 773 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/6207d746a126d94a118c85d3743bd9436b973920f5defcc09e6f3c46fd43b8fd.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { x } _ { t + 1 } = ( 1 + \\beta - \\eta a ) \\tilde { x } _ { t } - \\beta \\tilde { x } _ { t - 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 385, + 780, + 612, + 797 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This is a linear homogenous second-order difference equation. By Theorem 2.37 in Elaydi (2005), since $\\eta > 0$ and $\\beta < 1$ , if $\\begin{array} { r } { a > \\frac { 1 } { \\eta } ( 2 + 2 \\beta ) } \\end{array}$ then this recurrence diverges. □ ", + "bbox": [ + 171, + 811, + 823, + 843 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C CROSS-ENTROPY LOSS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 398, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In this appendix, we explain why the sharpness decreases at the end of training when the crossentropy loss is used. Before considering the full multiclass case, let us first consider the simpler case of binary classification with the logistic loss. ", + "bbox": [ + 173, + 132, + 823, + 176 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Binary classification with logistic loss We consider a dataset $\\{ \\mathbf { x } _ { i } , y _ { i } \\} \\} _ { i = 1 } ^ { n } \\subset \\mathbb { R } ^ { d } \\times \\{ - 1 , 1 \\}$ , where the examples are vectors in $\\mathbb { R } ^ { d }$ and the labels are binary $\\{ - 1 , 1 \\}$ . We consider a neural network $h : \\mathbb { R } ^ { d ^ { \\star } } \\times \\mathbb { R } ^ { p } \\mathbb { R }$ which maps an input $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ and a parameter vector $\\theta \\in \\mathbb { R } ^ { p }$ to a prediction $h ( \\mathbf { x } ; \\theta ) \\in \\mathbb { R }$ . Let $\\ell : \\mathbb { R } \\times \\{ - 1 , 1 \\} \\to \\bar { \\mathbb { R } }$ be the logistic loss function: ", + "bbox": [ + 173, + 190, + 825, + 248 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/13e4eea3c0bb9f2e8216eb50cf49f492803a85a98e191158959dcabbd9f116fc.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\ell ( z ; y ) = \\log ( 1 + \\exp ( - z y ) ) } \\\\ { \\displaystyle = - \\log p \\quad \\mathrm { w h e r e } \\quad p = \\frac { 1 } { 1 + \\exp ( - z y ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 253, + 661, + 306 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The second derivative of this loss function w.r.t $z$ is: ", + "bbox": [ + 173, + 310, + 516, + 325 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2e572575ad892fffd793f234a5de49bd9989984531caac9824efba32903f4758.jpg", + "text": "$$\n\\ell ^ { \\prime \\prime } ( z ; y ) = p ( 1 - p ) .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 330, + 568, + 349 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The full training objective is: ", + "bbox": [ + 174, + 354, + 366, + 369 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/927dce7ced1996987428c5b420769c0020838b63531e6a7eb5277a8b0e6e855a.jpg", + "text": "$$\nf ( \\theta ) = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } f _ { i } ( \\theta ) \\qquad { \\mathrm { w h e r e } } \\quad f _ { i } ( \\theta ) = \\ell ( f ( \\mathbf { x } _ { i } ; \\theta ) ; y _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 375, + 686, + 416 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The Hessian of this training objective is the average of per-example Hessians: ", + "bbox": [ + 173, + 421, + 683, + 436 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/c01988a6c2a2becc291d850ccaccfd54eab8241f12d3969b2f317889cdcb8149.jpg", + "text": "$$\n\\nabla ^ { 2 } f ( \\theta ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla ^ { 2 } f _ { i } ( \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 441, + 588, + 483 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For any arbitrary loss function $\\ell$ , we have the so-called Gauss-Newton decomposition (Martens, 2016; Bottou et al., 2018) of the per-example Hessian $\\nabla ^ { 2 } f _ { i } ( \\theta )$ : ", + "bbox": [ + 173, + 488, + 821, + 517 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/06cf4cb23653fdf1956e73435121d01a337c981ed345194e99d2d153fc78b5b0.jpg", + "text": "$$\n\\nabla ^ { 2 } f _ { i } ( \\theta ) = \\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) ^ { T } + \\ell ^ { \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } ^ { 2 } h ( \\mathbf { x } _ { i } ; \\theta ) \\quad { \\mathrm { w h e r e } } \\quad z _ { i } = h ( \\mathbf { x } _ { i } ; \\theta )\n$$", + "text_format": "latex", + "bbox": [ + 189, + 522, + 807, + 541 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\in \\mathbb { R } ^ { p }$ is the gradient of the network output with respect to the weights, and $\\ell ^ { \\prime }$ refers to the derivative of $\\ell$ with respect to its first argument, the score. ", + "bbox": [ + 176, + 546, + 820, + 577 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Empirically, the first term in the decomposition (usually called the “Gauss-Newton matrix”) tends to dominate the second, which implies the following “Gauss-Newton approximation” to the Hessian: ", + "bbox": [ + 174, + 582, + 820, + 611 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/4072d81c2b0cfde599bfad7c8be7c96d9b8608c05fbb300466a53f97e8b781f6.jpg", + "text": "$$\n\\nabla ^ { 2 } f ( \\theta ) \\approx \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) ^ { T }\n$$", + "text_format": "latex", + "bbox": [ + 325, + 617, + 671, + 659 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In our experience, progressive sharpening affects $\\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) \\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) ^ { T }$ . That is, $\\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) \\dot { \\nabla } _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) ^ { \\dot { T } }$ tends to grow in scale continually during training. For the square loss $\\begin{array} { r } { \\ell ( z ; y ) = \\frac 1 2 ( z - \\ y ) ^ { 2 } } \\end{array}$ , the second derivative $\\ell ^ { \\prime \\prime } ( z ; y ) = 1$ is the identity, so the $\\nabla ^ { 2 } f _ { i } ( \\theta )$ grows continually as well. In contrast, for the logistic loss, many of the $\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )$ decrease at the very end of training. Why is this? In Figure 9, we plot both the logistic loss $\\ell$ , and its second derivative $\\ell ^ { \\prime \\prime }$ , as a function of the quantity $y z$ , which is often called the “margin.” ", + "bbox": [ + 173, + 671, + 825, + 757 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Crucially, observe that both $\\ell$ and $\\ell ^ { \\prime \\prime }$ are decreasing in $y z$ . Because the loss $\\ell$ is decreasing in $y z$ , once an example $i$ is classified correctly (i.e. $y _ { i } z _ { i } > 0 ,$ ), the training objective can be optimized further by increasing the margin $y _ { i } z _ { i }$ . Because $\\ell ^ { \\prime \\prime }$ is also decreasing in $y z$ , if the margin $y _ { i } z _ { i }$ increases, the term $\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )$ drops. Near the end of training, once most examples are classified correctly, gradient descent can easily increase the margins of all these examples by simply scaling up the final layer weight matrix. This causes the $\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )$ to drop. Therefore, even though progressive sharpening still applies to $\\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) \\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) ^ { T }$ , the decrease in the $\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )$ ’s pulls down the leading eigenvalue of the Gauss-Newton matrix in equation 3. ", + "bbox": [ + 173, + 762, + 825, + 876 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "This effect is illustrated in Figure 10. Here, we train a network on the binary classification task of CIFAR-10 airplane vs. automobile, using the logistic loss. In Figure 10(e), we plot the margin $y _ { i } z _ { i }$ for 10 examples in the dataset. Notice that at the end of training, these margins all continually rise; ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/8a4e5c872e964d91be4d77db3e60be48539f3c043542b649e3eabe50088a495d.jpg", + "image_caption": [ + "Figure 9: Left: logistic loss function $\\ell$ as a function of $y z$ . Right: its second derivative $\\ell ^ { \\prime \\prime }$ as a function of $y z$ . " + ], + "image_footnote": [], + "bbox": [ + 254, + 103, + 745, + 193 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "this is because gradient descent “games” the objective by increasing the margins of successfully classified examples. When the margin $y _ { i } z _ { i }$ rises, the second derivative $\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )$ drops. This can be seen from Figure 10(f), where wleading eigenvalue of the matrix $\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )$ examples. Now, all the while, thekeeps rising, as can be seen from $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla _ { \\theta } \\hat { h } ( \\mathbf x _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf x _ { i } ; \\theta ) ^ { T } } \\end{array}$ Figure 10(d). However, because the ${ \\ell ^ { \\prime \\prime } s }$ are dropping, the leading eigenvalue of the Gauss-Newton matrix $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) ^ { \\hat { T } } } \\end{array}$ starts to decrease at the end of training, as can be seen from the green line in Figure 10(c). Finally, since the leading eigenvalue of the Gauss-Newton matrix is an excellent approximation to the leading eigenvalue of the Hessian (i.e. the sharpness), the sharpness also drops at the end of training, as can be seen from the orange line in Figure 10(c). ", + "bbox": [ + 173, + 270, + 825, + 397 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/8a7d3d06f9688932a7466481fcbac84ae8ee614ef3e52a623ad44dbe0cf530b4.jpg", + "image_caption": [ + "Figure 10: We train a network using the logistic loss on the binary classification problem of CIFAR10 airplane vs. automobile. (a) The train loss. (b) The train accuracy. (c) The leading eigenvalue of the Hessian and of the Gauss-Nthe former. (d) The leading eigenv ton matrix; obse of the matrix $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla _ { \\theta } h ( \\mathbf x _ { i } ; \\theta ) \\nabla _ { \\theta } \\bar { h } ( \\mathbf x _ { i } ; \\theta ) ^ { \\dot { T } } } \\end{array}$ roximation to, which is the $\\ell ^ { \\prime \\prime }$ dip at the end of training. (e) The margin $y _ { i } z _ { i }$ of 10 examples; observe that all the margins rise at the end of training. (f) the value $\\ell ^ { \\prime \\prime } ( z _ { i } ; \\bar { y } _ { i } )$ for 10 examples; observe that all of these curves decline at the end of training. " + ], + "image_footnote": [], + "bbox": [ + 183, + 420, + 823, + 619 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Multiclass classification with cross-entropy loss We consider a dataset $\\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n } \\subset \\mathbb { R } ^ { d } \\times$ $\\{ 1 , \\ldots , k \\}$ , where the examples are vectors in $\\mathbb { R } ^ { d }$ and the labels are in $\\{ 1 , \\ldots , k \\}$ . We consider a neural network $h : \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { j } \\mathbb { R } ^ { k }$ which maps an input $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ and a parameter vector $\\theta \\in \\mathbb { R } ^ { p }$ to a prediction $h ( x ; \\theta ) \\in \\mathbb { R } ^ { k }$ . Let $\\ell : \\mathbb { R } ^ { k } \\times \\{ 1 , \\ldots , k \\} \\overset { \\left. } { \\right. } \\mathbb { R }$ be the cross-entropy loss function: ", + "bbox": [ + 174, + 780, + 823, + 838 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ab02ec7d37805eb30c86bf1e5fe5eeb24bb8410c39ebea92b87a440fdc2e8515.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\ell ( { \\mathbf z } ; y ) = - \\log \\frac { \\exp ( z _ { y } ) } { \\sum _ { j = 1 } ^ { k } \\exp ( z _ { j } ) } } \\\\ { = - \\log p _ { y } \\quad \\mathrm { w h e r e } \\quad { \\mathbf p } = \\frac { \\exp ( { \\mathbf z } ) } { \\sum _ { j } \\exp ( z _ { j } ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 845, + 656, + 922 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The Hessian $\\nabla ^ { 2 } \\ell ( \\mathbf { z } ; y ) \\in \\mathbb { R } ^ { k \\times k }$ of this loss function w.r.t the class scores $\\mathbf { z }$ is: ", + "bbox": [ + 173, + 102, + 684, + 118 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/df819ea29e39a1249c4552e66fc056ef59f328ef82731dfc1e7d3bdc4d36894e.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla ^ { 2 } \\ell ( \\mathbf { z } ; y ) = \\mathrm { d i a g } ( \\mathbf { p } ) - \\mathbf { p } \\mathbf { p } ^ { T } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 400, + 125, + 598, + 145 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Now, for any loss function $\\ell : \\mathbb { R } ^ { k } \\times \\{ 1 , \\dots , k \\} \\mathbb { R }$ , we have the Gauss-Newton decomposition: ", + "bbox": [ + 173, + 151, + 810, + 169 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/6432a4acaa83ba3b33ec5c8b0e1fdce6c2c85d4a4fe494cf17e12091f784e622.jpg", + "text": "$$\n\\nabla ^ { 2 } f _ { i } ( \\theta ) = { \\mathbf J } _ { i } ^ { T } [ \\nabla _ { { \\mathbf z } _ { i } } ^ { 2 } ( { \\mathbf z } _ { i } ; y _ { i } ) ] { \\mathbf J } _ { i } + \\sum _ { j = 1 } ^ { k } [ \\nabla _ { { \\mathbf z } _ { i } } \\ell ( { \\mathbf z } _ { i } ; y _ { i } ) ] _ { j } \\nabla _ { \\theta } ^ { 2 } h _ { j } ( { \\mathbf x } _ { i } ; \\theta )\n$$", + "text_format": "latex", + "bbox": [ + 272, + 174, + 725, + 219 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\mathbf { z } _ { i } ~ = ~ h _ { i } ( \\mathbf { x } _ { i } ; \\theta ) ~ \\in ~ \\mathbb { R } ^ { k }$ are the logits for example $i$ , $\\mathbf { J } _ { i } ~ \\in ~ \\mathbb { R } ^ { k \\times p }$ is the network output-toweights Jacobian for example $i$ , $\\nabla _ { \\mathbf { z } _ { i } } ^ { 2 } \\ell ( \\mathbf { \\bar { z } } _ { i } ; y _ { i } ) \\ \\in \\ \\mathbb { R } ^ { k \\times k }$ is the Hessian of $\\ell$ w.r.t its input $\\mathbf { z } _ { i }$ , and $\\nabla _ { \\theta } ^ { 2 } h _ { j } ( \\mathbf { x } _ { i } ; \\theta ) \\in \\mathbb { R } ^ { p \\times p }$ is the Hessian matrix of the $j$ -th output of the network $h$ on the $i$ -th example. ", + "bbox": [ + 174, + 226, + 826, + 273 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Dropping the second term yields the Gauss-Newton approximation: ", + "bbox": [ + 173, + 279, + 617, + 294 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/94be3c6cb19bcf469ef15d9fa64e611719e80532f3abb21a82d1481c08a97e8f.jpg", + "text": "$$\n\\nabla ^ { 2 } f _ { i } ( \\boldsymbol { \\theta } ) \\approx \\mathbf { J } _ { i } ^ { T } \\left[ \\nabla _ { \\mathbf { z } _ { i } } ^ { 2 } \\ell ( \\mathbf { z } _ { i } ; y _ { i } ) \\right] \\mathbf { J } _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 299, + 612, + 320 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "As in the binary classification case discussed above: at the end of training, for many examples $i$ , the $y _ { i }$ entry of $\\mathbf { p } _ { i }$ will tend toward 1 and the other entries of $\\mathbf { p } _ { i }$ will tend to 0. Once this occurs, the matrix $\\mathrm { d i a g } ( \\mathbf { p } _ { i } ) - \\mathbf { p } _ { i } \\mathbf { p } _ { i } ^ { T }$ will broadly decrease in scale: the diagonal entries of this matrix are of the form $p ( 1 - p )$ , which goes to zero as $p 0$ or $p 1$ ; and the off-diagonal entries are of the form $- p q$ , which also goes to zero if $p 0 , q 1$ or $p 0 , q 0$ or $p 1 , q 0$ . ", + "bbox": [ + 173, + 332, + 825, + 404 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "This effect is illustrated in Figure 11. Here, we train a network on CIFAR-10 using the cross-entropy loss. In Figure 11(e), for ten examples $i$ in the dataset (with output scores $\\mathbf { z } _ { i } = \\breve { h } ( \\mathbf { x } _ { i } ) \\in \\mathbb { R } ^ { k } )$ , we plot the margin $\\mathbf { z } _ { i } [ y _ { i } ] - \\operatorname* { m a x } _ { j \\neq y _ { i } } \\mathbf { z } _ { i } [ j ]$ , which is the difference between the score of the correct class $y _ { i }$ and the score of the next-highest class. Observe that for all of these examples, this margin rises at the end of training. In Figure 11(f), for those same ten examples, we plot the quantity ${ \\bf p } _ { i } [ y _ { i } ] ( 1 - { \\bf p } _ { i } [ y _ { i } ] )$ .Observe that for all of these examples, this quantity con. Now, all the while, the leading eigenvalue of the matrix ases atkeeps $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n ^ { * } } \\mathbf { J } _ { i } ^ { T } \\mathbf { J } _ { i } } \\end{array}$ i i eigenvalue of the Gauss-Newton matrix 1n Pni=1 JTi ∇2\\`(zi; yi) Ji starts to decrease at the end of training, as can be seen from the green line in Figure 11(c). Finally, since the leading eigenvalue of the Gauss-Newton matrix is an excellent approximation to the leading eigenvalue of the Hessian (i.e. the sharpness), the sharpness also drops at the end of training, as can be seen from the orange line in Figure 11(c). ", + "bbox": [ + 173, + 409, + 825, + 593 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/3afa4026e93af581c17185584cf5281249dbef1d4b1b3a1e05e3bd52c68806bf.jpg", + "image_caption": [ + "Figure 11: We train a network using the cross-entropy loss on CIFAR-10. (a) The train loss. (b) The train accuracy. (c) The leading eigenvalue of the Hessian and of the Gauss-Newton matrix; observe that the latter is a great approximation to the former. (d) The leading eigenvalue of the matrix $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbf { J } _ { i } ^ { T } \\mathbf { J } _ { i } } \\end{array}$ , which is the Gauss-Newton matrix except the $\\nabla ^ { 2 } \\ell$ terms; observe that this value constantly rises — it does not dip at the end of training. (e) The margin $\\mathbf { z } _ { i } [ y _ { i } ] - \\operatorname* { m a x } _ { j \\neq y _ { i } } \\mathbf { z } _ { i } [ j ]$ of 10 examples; observe that all these margins rise at the end of training. (f) the value $\\mathbf { p } _ { i } [ y _ { i } ] ( 1 - \\mathbf { p } _ { i } [ y _ { i } ] )$ for 10 examples; observe that all of these curves decline at the end of training. " + ], + "image_footnote": [], + "bbox": [ + 179, + 354, + 825, + 554 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D EMPIRICAL STUDY OF PROGRESSIVE SHARPENING ", + "text_level": 1, + "bbox": [ + 174, + 103, + 627, + 117 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In this appendix, we empirically study how problem parameters such as network width, network depth, and dataset size affect the degree to which progressive sharpening occurs. To study progressive sharpening on its own, without the confounding factor of instability, we train neural networks using gradient flow (Runge-Kutta) rather than gradient descent. Informally speaking, gradient flow does “what gradient descent would do if gradient descent didn’t have to worry about instability.” ", + "bbox": [ + 174, + 133, + 825, + 204 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We observe that progressive sharpening occurs to a greater degree: (1) for narrower networks than for wider networks (which is consistent with infinite-width NTK theory), (2) for deeper networks than for shallower networks, and (3) for larger datasets than for smaller datasets. ", + "bbox": [ + 174, + 210, + 823, + 252 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The effect of width When networks parameterized in a certain way (the “NTK parameterization”) are made infinitely wide and trained using gradient flow, the Hessian moves a vanishingly small amount during training, implying that no progressive sharpening occurs (Jacot et al., 2018; Lee et al., 2019; Jacot et al., 2020; Li & Liang, 2018). Therefore, a natural hypothesis is that progressive sharpening might attenuate as network width increases. We now run an experiment which supports this hypothesis. ", + "bbox": [ + 174, + 267, + 825, + 351 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We consider fully-connected architectures with two hidden layers and tanh activations, with widths $\\{ 3 2 , 6 4 , 1 2 8 , 2 5 6 , 5 1 2 , 1 0 2 4 \\}$ . We train on a size-5,000 subset of CIFAR-10 using the cross-entropy loss. We train using gradient flow (details in $\\ S 1 . 5 )$ . We consider both NTK parameterization and standard parameterization (Lee et al., 2019). ", + "bbox": [ + 174, + 358, + 825, + 414 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In Figure 12, for each width, we train NTK-parameterized networks from five different random initializations, and plot the evolution of the sharpness during gradient flow. Observe that the maximum sharpness along the gradient flow trajectory is larger for narrow networks, and smaller for wide networks. (As elsewhere in this paper, note that the sharpness drops at the end of training due to the cross-entropy loss.) In Figure 13, we plot summary statistics from these training runs. Namely, define $\\lambda _ { \\mathrm { m a x } }$ as the maximum sharpness over the gradient flow trajectory, and define $\\lambda _ { 0 }$ as the initial sharpness. In Figure 13(a), for each width we plot the mean and standard deviation of the maximum sharpness $\\lambda _ { \\mathrm { m a x } }$ over the five different random initializations. Observe that $\\lambda _ { \\mathrm { m a x } }$ becomes smaller, on average, as the width is made larger. In Figure 13(b), for each width we plot the mean and standard deviation of the maximum sharpness gain $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ over the five different random initializations. Observe that the maximum sharpness gain $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ also becomes smaller as the width is made larger. NTK theory suggests that $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ should deterministically tend to 1 as the width $ \\infty$ , and Figure 13(b) is consistent with this prediction. ", + "bbox": [ + 174, + 421, + 825, + 602 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In Figures 14 and 15, we conduct similar experiments, but with standard parameterization rather than NTK parameterization. Similar to NTK parameterization, we observe in Figure 14 that the sharpness rises more for narrow networks than for wide networks. ", + "bbox": [ + 176, + 608, + 823, + 650 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/2d1c17bc341619769646d030b8cee578dbcd4f24085e159a010c564f12d4c6fe.jpg", + "image_caption": [ + "Figure 12: NTK parameterization: evolution of the sharpness. We use gradient flow to train NTK-parameterized networks, and we track the evolution of the sharpness during training. For each width, we train from five different random initializations (different colors). Observe that the sharpness rises more when training narrow networks than when training wide networks. " + ], + "image_footnote": [], + "bbox": [ + 179, + 109, + 859, + 292 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/cd54a3c02b617d0cc0545f19c174842e2bab1dc73ef4ca3837fe2ab02c313ec8.jpg", + "image_caption": [ + "Figure 13: NTK parameterization: summary statistics. Left: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness $\\lambda _ { \\mathrm { m a x } }$ along the gradient flow trajectory. Observe that $\\lambda _ { \\mathrm { m a x } }$ decreases in expectation as the width increases. Right: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness gain $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ along the gradient flow trajectory. Observe that $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ decreases in expectation as the width increases. Indeed, NTK theory predicts that $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ should deterministically tend to 1 as width $ \\infty$ and this plot is consistent with that prediction. " + ], + "image_footnote": [], + "bbox": [ + 184, + 392, + 825, + 503 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/bd1a970ecb7576a7154308af3a1348b43fe144b6a96f4f00b28b797b83855275.jpg", + "image_caption": [ + "Figure 14: Standard parameterization: evolution of the sharpness. We use gradient flow to train standard-parameterized networks, and we track the evolution of the sharpness during training. For each width, we train from five different random initializations (different colors). Observe that the sharpness rises more when training narrow networks than when training wide networks. " + ], + "image_footnote": [], + "bbox": [ + 178, + 659, + 859, + 843 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/f12a357b31bc1c8cc211c138d39bbf7fe4603d9ecdc4b6bf9c004c95787db477.jpg", + "image_caption": [ + "Figure 15: Standard parameterization: summary statistics. Left: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness $\\lambda _ { \\mathrm { m a x } }$ along the gradient flow trajectory. Observe that $\\lambda _ { \\mathrm { m a x } }$ tends to decrease in expectation as the width increases, though it is not clear whether this pattern still holds when moving from width 512 to width 1024 — more samples are needed. Right: For each network width, we plot the mean and standard deviation (over the five different random initializations) of the maximum sharpness gain $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ along the gradient flow trajectory. Observe that $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ decreases in expectation as the width increases. It is not clear whether or not $\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }$ is deterministically tending to 1, but that does seem possible. " + ], + "image_footnote": [], + "bbox": [ + 184, + 386, + 820, + 497 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Effect of depth We now explore the effect of network depth on progressive sharpening. We use gradient flow to train fully-connected tanh architectures of width 200 and varying depths — ranging from 1 hidden layer to 4 hidden layers. We train on a $5 \\mathrm { k }$ subset of CIFAR-10 using both crossentropy loss (in Figure 16) and square loss (in Figure 17). For each depth, we train from five different random initializations (different colors). Observe that progressive sharpening occurs to a greater degree as network depth increases. ", + "bbox": [ + 173, + 103, + 825, + 188 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/ff731c8f4096c9935f202efe47d173e8c6b2cf7e9d6f15ec453dd1fb3420741c.jpg", + "image_caption": [ + "Figure 16: The effect of depth: cross-entropy. We use gradient flow to train networks of various depths, ranging from 1 hidden layer to 4 hidden layers, using cross-entropy loss. We train each network from five different random initializations (different colors). Observe that progressive sharpening occurs to a greater degree for deeper networks. " + ], + "image_footnote": [], + "bbox": [ + 181, + 205, + 764, + 315 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/16bb97f24b44d20f43fb40f9d5f18f72ef60f71b2ad5a00d776a6b98d5f2c7a4.jpg", + "image_caption": [ + "Figure 17: The effect of depth: mean squared error. We use gradient flow to train networks of various depths, ranging from 1 hidden layer to 4 hidden layers, using MSE loss. We train each network from five different random initializations (different colors). Observe that progressive sharpening occurs to a greater degree for deeper networks. " + ], + "image_footnote": [], + "bbox": [ + 183, + 412, + 764, + 523 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Effect of dataset size We now explore the effect of dataset size on progressive sharpening. We use gradient flow to train a network on different-sized subsets of CIFAR-10. The network is a 2- hidden-layer, width-200 fully-connected tanh architecture, and we train using cross-entropy loss. The results are shown in Figure 18. Observe that progressive sharpening occurs to a greater degree as dataset size increases. ", + "bbox": [ + 173, + 619, + 825, + 690 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/d880f1b51cb504affeeafd27c3459939af183c6cb9ef5d32bdd6757998401d0e.jpg", + "image_caption": [ + "Figure 18: The effect of dataset size. We use gradient flow to train a network on varying-sized subsets of CIFAR-10. Observe that progressive sharpening occurs to a greater degree as the dataset size increases. " + ], + "image_footnote": [], + "bbox": [ + 181, + 705, + 787, + 815 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E SPEED OF DIVERGENCE ON QUADRATIC TAYLOR APPROXIMATION ", + "text_level": 1, + "bbox": [ + 171, + 102, + 763, + 118 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "If the sharpness at some iterate is strictly greater than $2 / \\eta$ , then gradient descent with step size $\\eta$ is guaranteed to diverge if run on the quadratic Taylor approximation around that iterate. However, the speed of this divergence could conceivably be slow — in particular, the train loss might continue to decrease for many iterations before it starts to increase. In this appendix we empirically demonstrate, to the contrary, that at the Edge of Stability, gradient descent diverges quickly if, at some iterate, we start running gradient descent on the quadratic Taylor approximation around that iterate. ", + "bbox": [ + 174, + 133, + 825, + 217 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "We consider the fully-connected tanh network from section 3, trained on a 5,000-sized subsample of CIFAR-10 using both cross-entropy loss and MSE loss. At some timestep $t _ { 0 }$ during training, we suddenly switch from running gradient descent on the real neural training objective, to running gradient descent on the quadratic Taylor approximation around the iterate at step $t _ { 0 }$ . We do this for three timesteps before gradient descent has entered the Edge of Stability, and three afterwards. Figure 20 shows the results for cross-entropy loss, and Figure 21 shows the (similar) results for MSE loss. Before entering the Edge of Stability (top row), gradient descent on the quadratic Taylor approximation behaves similar to gradient descent on the real neural training objective — that is, the orange line almost overlaps the blue line. Yet after entering the Edge of Stability (bottom row), gradient descent on the quadratic Taylor approximation quickly diverges, whereas gradient descent on the real neural training objective makes consistent (if choppy) progress. ", + "bbox": [ + 173, + 223, + 825, + 377 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In short, when gradient descent is not at the Edge of Stability, the quadratic Taylor approximation serves as a good model for the local progress of gradient descent. But when gradient descent is at the Edge of Stability, the quadratic Taylor approximation is an extremely poor model for the local progress of gradient descent. It is conceivable that there exists some simple modification to the quadratic Taylor model which would fix this issue (e.g. perhaps if one ignores a certain direction, the quadratic Taylor model is accurate). Nevertheless, unless/until such a fix is discovered, it is unclear why quadratic Taylor approximations should yield any insight into the local behavior of gradient descent. ", + "bbox": [ + 173, + 383, + 826, + 496 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/955346fd537d3f513f8126ca81891da22319daf5c8fd798c8c111382b7b6b357.jpg", + "image_caption": [ + "Figure 19: We train a neural network using cross-entropy loss (left) and MSE loss (right). In Figure 20 and 21, we show what happens when, at the iterations marked above by vertical dotted lines, we switch from gradient descent on the real neural training objective to gradient descent on the quadratic Taylor approximation. " + ], + "image_footnote": [], + "bbox": [ + 302, + 512, + 694, + 616 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/4614c000ab8bb8a0dc9aba4d60480907ab576f1310315b7f847bdf10fb234171.jpg", + "image_caption": [ + "Figure 20: Cross-entropy loss $( \\eta = 2 / 6 0 )$ . At six different iterations during the training of the network from Figure 19 (marked by the vertical dotted black lines), we switch from running gradient descent on the real neural training objective (for which the train loss is plotted in blue) to running gradient descent on the quadratic Taylor approximation around the current iterate (for which the train loss is plotted in orange). Top row are timesteps $\\{ 2 0 0 , 4 0 0 , 6 0 0 \\}$ before gradient descent has entered the Edge of Stability; observe that the orange line (Taylor approximation) closely tracks the blue line (real objective). Bottom row are timesteps {800, 1000, 1200} during the Edge of Stability; observe that the orange line quickly diverges, whereas the blue line does not. " + ], + "image_footnote": [], + "bbox": [ + 181, + 132, + 856, + 344 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/d96e7be6e15e63ae8d6c0c90126924b85697ad5673bba27ca664692f011134fb.jpg", + "image_caption": [ + "Figure 21: MSE loss $( \\eta = 2 / 2 0 0 )$ . At six different iterations during the training of the network from Figure 19 (marked by the vertical dotted black lines), we switch from running gradient descent on the real neural training objective (for which the train loss is plotted in blue) to running gradient descent on the quadratic Taylor approximation around the current iterate (for which the train loss is plotted in orange). Top row are timesteps (2000, 3000, 4000) before gradient descent has entered the Edge of Stability; observe that the orange line (Taylor approximation) closely tracks the blue line (real objective). Bottom row are timesteps (5000, 6000, 7000) during the Edge of Stability; observe that the orange line quickly diverges, whereas the blue line does not. " + ], + "image_footnote": [], + "bbox": [ + 179, + 546, + 856, + 760 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "F “OPTIMAL” STEP SIZE SELECTION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 488, + 117 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "One heuristic for setting the step size of gradient descent is to set the step size at iteration $t$ to $\\eta _ { t } =$ $1 / \\lambda _ { t }$ , where $\\lambda _ { t }$ is the sharpness at iteration $t$ . While this heuristic is computationally impractical due to the time required to compute the sharpness at each iteration, it is often regarded as an ideal, for instance in LeCun et al. (1998) (Eq. 39), LeCun et al. (1993), and Schaul et al. (2013) (Eq 8). The motivation for this heuristic is: if all that is known about the training objective is that the local sharpness is $\\lambda$ , then a step size of $1 / \\lambda$ maximizes the guaranteed decrease in the training objective that would result from taking a step. ", + "bbox": [ + 173, + 133, + 825, + 231 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "First, we demonstrate (on a single numerical example) that the dynamic step size $\\eta _ { t } = 1 / \\lambda _ { t }$ is outperformed by the baseline approach of gradient descent with a fixed step size $\\eta _ { t } = 1 / \\lambda _ { 0 }$ , where $\\lambda _ { 0 }$ is the sharpness at initialization. In Figure 22, we train the network from $\\ S 3$ using both the dynamic $\\eta = 1 / \\lambda _ { t }$ step size heuristic as well as the baseline fixed step size of $\\eta = 1 / \\lambda _ { 0 }$ . Observe that the $\\eta \\ : = \\ : 1 / \\lambda _ { 0 }$ baseline outperforms the $1 / \\lambda _ { t }$ heuristic. Intuitively, because of progressive sharpening, the $\\eta _ { t } = 1 / \\lambda _ { t }$ heuristic anneals the step size, and therefore ends up taking steps that are suboptimally small. In contrast, while the $\\eta _ { t } = 1 / \\lambda _ { 0 }$ baseline quickly becomes unstable, this instability is apparently a worthwhile “price to pay” in return for the benefit of taking larger steps. ", + "bbox": [ + 173, + 237, + 825, + 351 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/514a14703006dd249fb03c2eb5a2e44d6f8dc50d85bed04cfb07f63ac05aa485.jpg", + "image_caption": [ + "Figure 22: A dynamic step size $\\eta _ { t } = 1 / \\lambda _ { t }$ underperforms a fixed step size of $\\eta _ { t } = 1 / \\lambda _ { 0 }$ " + ], + "image_footnote": [], + "bbox": [ + 215, + 363, + 782, + 546 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Another natural idea is to dynamically set the step size at iteration $t$ to $\\eta _ { t } = 1 . 9 / \\lambda _ { t }$ . This step size rule takes larger steps than the $\\eta = 1 / \\lambda _ { t }$ rule while still remaining stable. In Figure 23, we compare this $\\eta _ { t } = 1 . 9 / \\lambda _ { t }$ rule to a baseline approach of gradient descent with a fixed step size $\\eta _ { t } = 1 . 9 \\bar { / } \\lambda _ { 0 }$ , where $\\lambda _ { 0 }$ is the sharpness at initialization. Observe that the baseline of a fixed $1 . 9 / \\lambda _ { 0 }$ step size outperforms the dynamic $\\eta _ { t } = 1 . 9 / \\lambda _ { t }$ rule. ", + "bbox": [ + 173, + 597, + 826, + 667 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/c7d74ca5c721efee3fa1e367f439b3ac52ad82217cc62d2f294d77d1915428ca.jpg", + "image_caption": [ + "Figure 23: A dynamic step size $\\eta _ { t } = 1 . 9 / \\lambda _ { t }$ underperforms a fixed step size of $\\eta _ { t } = 1 . 9 / \\lambda _ { 0 }$ . " + ], + "image_footnote": [], + "bbox": [ + 215, + 681, + 782, + 863 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "G EVOLUTION OF SHARPNESS DURING SGD ", + "text_level": 1, + "bbox": [ + 173, + 102, + 560, + 117 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "In this appendix, we briefly illustrate how the sharpness evolves during stochastic gradient descent. In Figure 24, we train the tanh network from $\\ S 3$ using SGD with both cross-entropy loss (top) and mean squared error (bottom). We train using a range of batch sizes (different colors). We observe the following: ", + "bbox": [ + 174, + 133, + 825, + 189 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "1. During large-batch SGD, the sharpness behaves similar to full-batch gradient descent: it rises to $2 / \\eta$ (marked by the black horizontal dashed line) and then hovers just above that value. \n2. Consistent with prior reports, we find that the smaller the batch size, the lower the sharpness (Keskar et al., 2016; Jastrz˛ebski et al., 2017; 2019; 2020). \n3. Notice that when training with cross-entropy loss at batch size 8 (the blue line), the sharpness decreases throughout most of training. The train accuracy (not pictured) is only $6 6 \\%$ when the sharpness starts to decrease, which suggests that the cause of this decrease is unrelated to the effect described in Appendix C, whereby the sharpness decreases at the end of training. Figure 5(a) of Jastrz˛ebski et al. (2020) also depicts a network where the sharpness decreases during SGD training. ", + "bbox": [ + 210, + 200, + 825, + 364 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/43c9eb37df93a822a4ae22a49f7be8053d7b68ffd685c8aaa8216120ea5ec9e4.jpg", + "image_caption": [ + "Figure 24: Evolution of sharpness during SGD. We train a network using SGD at various batch sizes, and plot the evolution of the sharpness. In the top row, we train with cross-entropy loss and step size $\\eta = 0 . 0 2$ ; in the bottom row, we train with mean squared error and step size $\\eta = 0 . 0 1$ . The leftmost column shows all batch sizes; the center column focuses on just the smallest; and the rightmost column focuses on just the largest. The black dashed horizontal line marks $2 / \\eta$ . " + ], + "image_footnote": [], + "bbox": [ + 181, + 385, + 825, + 632 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "H SGD ACCLIMATES TO THE HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 173, + 103, + 602, + 117 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "In this appendix, we conduct an experiment which suggests that some version of “Edge of Stability” may hold for SGD. ", + "bbox": [ + 174, + 133, + 821, + 161 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "One way to interpret our main findings is that gradient descent “acclimates” to the step size in such a way that each training update sometimes increases, and sometimes decreases, the training loss, yet an update with a smaller step size would always decrease the training loss. We now demonstrate that this interpretation may generalize to SGD. In particular, we will demonstrate that SGD seems to “acclimate” to the step size and batch size in such a way that an actual update sometimes increases and sometimes decreases the loss in expectation, yet a update with a larger step size or smaller batch size would almost always increase the loss in expectation, and a step with a smaller step size or a larger batch size would almost always decrease the loss in expectation. ", + "bbox": [ + 173, + 167, + 825, + 280 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "In Figure 25, we train the tanh network from $\\ S 3$ with MSE loss, using SGD with step size 0.01 and batch size 32. We periodically compute the training loss (over the full dataset) and plot these on the left pane of Figure 25. Observe that the training loss does not decrease monotonically, but of course this is not surprising — SGD is a random algorithm. However, what may be more surprising is that SGD is not even decreasing the training loss in expectation. On the right pane of Figure 25, every 500 steps during training, we use the Monte Carlo method to approximately compute the expected change in training loss that would result from an SGD step (the expectation here is over the randomness involved in selecting the minibatch). Observe that at many points during training, an SGD step would decrease the loss (as desired) in expectation, but at other points, and SGD step would increase the loss in expectation. ", + "bbox": [ + 173, + 286, + 825, + 425 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "In Figure 26(a), while training that network, we compute the expected change in training loss that would result from taking an SGD step with the same step size used during training (i.e. 0.01), but half the batch size used during training (i.e. 16). We observe that an SGD step with half the batch size would consistently cause an increase in the training loss in expectation. In Figure 26(b) we repeat this experiment, but with twice the batch size used during training (i.e. 64). Notice that an SGD step with twice the batch size would consistently cause a decrease in the training loss in expectation. In Figure 26(c) and (d) repeat this experiment with the step size; we observe that an SGD step with a larger step size (0.02) would consistently increase the training loss in expectation, while an SGD step with a smaller step size (0.005) would consistently decrease the training loss in expectation. ", + "bbox": [ + 173, + 433, + 825, + 571 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "In Figure 27, as a “control” experiment, we both train and measure the expected loss change under the following four hyperparameter settings: (step size 0.01, batch size 16), (step size 0.01, batch size 64), (step size 0.02, batch size 32), and (step size 0.005, batch size 32). In each case, we observe that, after a brief period at the beginning of training, each SGD update sometime increases and sometimes decreases the training loss in expectation. ", + "bbox": [ + 173, + 579, + 825, + 648 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Therefore, at least for this single network, we can conclude that no matter the hyperparameters, SGD quickly navigates to, and then lingers in, regions of the loss landscape in which an SGD update with those hyperparameters sometimes increases, and sometimes decreases the training loss in expectation, yet an SGD update with a smaller step size or larger batch size would consistently decrease the loss in expectation, and an SGD update with a larger step size or smaller batch size would consistently increase the loss in expectation. ", + "bbox": [ + 174, + 655, + 825, + 739 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/4e018c957930a5474c4334e97cd3b044261c25dc2f6b1ba664e7b8e5382ded7d.jpg", + "image_caption": [ + "Figure 25: SGD does not consistently decrease the training loss in expectation. We train the tanh network from section 3 with MSE loss, using SGD with step size 0.01 and batch size 32. Periodically during training, we compute (left) the full-batch training loss, and (right) the expected change in the full-batch training loss that would result from taking an SGD step (where the expectation is over the randomness in sampling the minibatch). Strikingly, note that after the very beginning of training, the expected loss change is sometimes negative (as desired) but oftentimes positive. See Figure 26. " + ], + "image_footnote": [], + "bbox": [ + 178, + 131, + 852, + 248 + ], + "page_idx": 29 + }, + { + "type": "image", + "img_path": "images/44ca4a3ebd286c2444ccd3e72ea0ff8569c9de959a3b0a9610b17cae9cd771cc.jpg", + "image_caption": [ + "Figure 26: An SGD step with a smaller learning rate or a larger batch size than the ones used during training would consistently decrease the loss in expectation. At regular intervals during the training run depicted in Figure 25 (with $\\eta = 0 . 0 1$ and batch size 32), we measure the expected change in the full-batch training loss that would result from an SGD step with a different step size or batch size. Observe that taking an SGD step with a smaller step size or a larger batch size would consistently have decreased the loss in expectation, while taking an SGD step with a larger step size or a smaller batch size would have consistently increased the loss in expectation. " + ], + "image_footnote": [], + "bbox": [ + 174, + 412, + 854, + 503 + ], + "page_idx": 29 + }, + { + "type": "image", + "img_path": "images/f4e7c089edb03d37c47b5d8e944e1bde4562157461a6077650839c660ed4a356.jpg", + "image_caption": [ + "Figure 27: Control experiment. Above, in Figure 26(a), as we trained a network with step size 0.01 and batch size 32, we evaluated the expected change in training loss that would result from taking an SGD step with step size 0.01 and batch size 64. Here, in Figure 27(a), as a “control experiment,” we train the network with step size 0.01 and batch size 64, and evaluate the expected change in training loss that would result from taking an SGD step with the same step size and batch size. We observe that an SGD step using the same step size and batch size that are used during training would sometimes increase and sometimes decrease the training loss in expectation. The other three panes are analogous. " + ], + "image_footnote": [], + "bbox": [ + 178, + 675, + 856, + 766 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "I EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 411, + 118 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "I.1 VARYING ARCHITECTURES ON 5K SUBSET OF CIFAR-10 ", + "text_level": 1, + "bbox": [ + 173, + 133, + 606, + 148 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Dataset. The dataset consists of the first 5,000 examples from CIFAR-10. To preprocess the dataset, we subtracted the mean from each channel, and then divided each channel by the standard deviation (where both the mean and stddev were computed over the full CIFAR-10 dataset, not the 5k subset). ", + "bbox": [ + 174, + 160, + 826, + 202 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Architectures. We experimented with two architecture families: fully-connected and convolutional. For each of these two families, we experimented with several different activation functions, and for convolutional networks we experimented with both max pooling and average pooling. ", + "bbox": [ + 174, + 208, + 825, + 252 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "The PyTorch code for e.g. the fully-connected ReLU network is as follows: ", + "bbox": [ + 174, + 257, + 668, + 272 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "nn.Sequential( nn.Flatten(), nn.Linear(3072, 200, bias $=$ True), nn.ReLU(), nn.Linear(200, 200, bias $=$ True) nn.ReLU(), nn.Linear(200, 10, bias $=$ True) \n) ", + "bbox": [ + 174, + 285, + 526, + 396 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Networks with other activation functions would have nn.ReLU() replaced by nn.ELU(), nn.Tanh(), nn.Softplus(), or nn.Hardtanh(). ", + "bbox": [ + 173, + 409, + 825, + 436 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "The PyTorch code for e.g. the convolutional ReLU network with max-pooling is as follows: ", + "bbox": [ + 169, + 443, + 776, + 458 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "nn.Sequential( nn.Conv2d(3, 32, bias $=$ True, kernel_siz ${ \\tt 3 } = 3 { \\tt 1 }$ , padding $^ { \\cdot = 1 }$ ), nn.ReLU(), nn.MaxPool2d(2), nn.Conv2d(32, 32, bias $=$ True, kernel_size $= 3$ , padding $^ { = 1 }$ ), nn.ReLU(), nn.MaxPool2d(2), nn.Flatten(), nn.Linear(2048, 10, bias $=$ True) \n) ", + "bbox": [ + 174, + 472, + 750, + 609 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Networks with other activation functions would have nn.ReLU() replaced by nn.ELU() or nn.Tanh(), and networks with average pooling would have nn.MaxPool2d(2) replaced by nn.AvgPool2d(2). ", + "bbox": [ + 173, + 625, + 823, + 666 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "For all of these networks, we use the default PyTorch initialization. That is, both fully-connected layers and convolutional layers have the entries of their weight matrix and bias vector sampled i.i.d from Uniform(− √ 1fan in , √ 1fan in ). ", + "bbox": [ + 173, + 672, + 825, + 719 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Loss functions. For a $k$ -class classification problem, if the network outputs are $\\textbf { z } \\in \\mathbb { R } ^ { k }$ and the correct class is $\\textit { i } \\in \\ [ k ]$ , then the mean squared error (MSE) loss is defined as $\\begin{array} { r } { \\frac { 1 } { 2 } \\left[ ( z [ i ] - 1 ) ^ { 2 } + \\sum _ { j \\neq i } z [ j ] ^ { 2 } \\right] } \\end{array}$ . That is, we encode the correct class with a “1” and the other classes with $^ { 6 6 } 0$ .” The cross-entropy loss is defined as $- \\log \\left( \\frac { \\exp ( z [ i ] ) } { \\sum _ { j \\neq i } \\exp ( z [ j ] ) } \\right) .$ . ", + "bbox": [ + 173, + 727, + 825, + 805 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "I.2 STANDARD ARCHITECTURES ON CIFAR-10 ", + "text_level": 1, + "bbox": [ + 173, + 819, + 516, + 835 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "To preprocess the CIFAR-10 dataset, we subtracted the mean from each channel, and then divided each channel by the standard deviation. ", + "bbox": [ + 174, + 845, + 825, + 875 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Since training with full-batch gradient descent is slow, we opted to experiment on relatively shallow networks. The VGG networks (both with and without BN) are VGG-11’s, from the implementation here: https://github.com/chengyangfu/pytorch-vgg-cifar10/ blob/master/vgg.py, with the dropout layers removed. The ResNet is the (non-fixup) ResNet32 implemented here: https://github.com/hongyi-zhang/Fixup. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 821, + 132 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "For the two networks with batch normalization, running gradient descent with full-dataset batch normalization would not have been feasible under our GPU memory constraints. Therefore, we instead used ghost batch normalization (Hoffer et al., 2017) with 50 ghost batches of size 1,000. This means that we divided the 50,000 examples in CIFAR-10 into 50 fixed groups of size 1,000 each, and defined the overall objective function to be the average of 50 fixed batch-wise objectives. To correctly compute the overall gradient of this training objective, we can just run backprop 50 times (once on each group) and average the resulting gradients. ", + "bbox": [ + 174, + 138, + 825, + 236 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "To compute the sharpness over the full CIFAR-10 dataset would have been computationally expensive. Therefore, in an approximation, we instead computed the sharpness over just the first 5,000 examples in the dataset (or, for the BN networks, over the first 5 batches out of 50). ", + "bbox": [ + 176, + 243, + 825, + 285 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "I.3 BATCH NORMALIZATION EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 303, + 485, + 316 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We used the CNN architecture from $\\ S J$ (described above in $\\ S \\mathrm { I . 1 }$ ), but with a BatchNorm2d() layer inserted after each activation layer. ", + "bbox": [ + 173, + 328, + 818, + 357 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Since our GPUs did not have enough memory to run batch normalization with the full dataset of size 5,000, we used ghost batch normalization with five ghost batches of size 1,000 (see I.2 for details). ", + "bbox": [ + 173, + 363, + 823, + 392 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "I.4 TRANSFORMER ON WIKITEXT-2 ", + "text_level": 1, + "bbox": [ + 176, + 409, + 436, + 422 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We used both the Transformer architecture and the preprocessing setup from the official PyTorch (Paszke et al., 2019) word-level language modeling tutorial: https://github.com/ pytorch/examples/tree/master/word_language_model. We used the settings ninp $\\scriptstyle \\phantom { . } = 2 0 0$ , nhead $^ { = 2 }$ , nhid $= 2 0 0$ , nlayer ${ \\tt S } = 2$ , dropout $= 0$ . We set bptt $\\qquad = \\ 3 5$ , which means that we divided the corpus into chunks of 35 tokens, and trained the network, using the negative log likelihood loss, to predict each token from the preceding tokens in the same chunk. Since computing the sharpness over the full dataset would not have been computationally practical, we computed the sharpness over a subset comprising 2500 training examples. ", + "bbox": [ + 174, + 435, + 825, + 546 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "I.5 RUNGE-KUTTA ", + "text_level": 1, + "bbox": [ + 174, + 564, + 316, + 577 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We used the “RK4” fourth-order Runge-Kutta algorithm (Press et al., 1992) to numerically integrate the gradient flow ODE. The Runge-Kutta algorithm requires a step size. Rather than use a sophisticated algorithm for adaptive step size control, we decided to take advantage of the fact that we were already periodically computing the sharpness: at each step, we set the step size to $\\alpha / \\lambda$ , where $\\alpha$ is a tunable parameter and $\\lambda$ is the most recent value for the sharpness. We set $\\alpha = 1$ or $\\alpha = 0 . 5$ . ", + "bbox": [ + 174, + 589, + 825, + 659 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "I.6 RANDOM PROJECTIONS ", + "text_level": 1, + "bbox": [ + 174, + 676, + 375, + 690 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "In order to ascertain whether gradient descent at step size $\\eta$ initially followed the gradient flow trajectory (and, if so, for how long), we monitored the $\\ell _ { 2 }$ distance in weight space between the gradient flow solution at time $t$ and the gradient descent iterate at step $t / \\eta$ . One way to do would be as follows: (a) when running gradient flow, save the weights after every $\\Delta t$ units of time, for some parameter $\\Delta t$ ; (b) when running gradient descent, save the weights at each $\\big ( \\frac { \\Delta t } { \\eta } \\big )$ -th step; (c) plot the difference between these two sequences. (Note that this approach requires $\\Delta t$ to be divisible by $\\eta$ .) ", + "bbox": [ + 174, + 702, + 823, + 787 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We essentially used this approach, but with one modification: regularly saving the entire network weight vector would have consumed a large amount of disk space, so we instead saved lowdimensional random projections of the network weights. To be clear, let $d$ be the number of network weights, and let $k$ be the number of random projections (a tunable parameter chosen such that $k \\ll d ,$ ). Then we first generated a matrix $\\mathbf { M } \\in \\breve { \\mathbb { R } } ^ { k \\times d }$ by sampling each entry i.i.d from the standard normal distribution. During training, rather than periodically save the whole weight vector (a $d$ -dimensional vector), we premultiplied this vector by the matrix $\\mathbf { M }$ to obtain a $k$ -dimensional vector, and we periodically saved these vectors instead. Then we plotted the $\\ell _ { 2 }$ distance between the low-dimensional vectors from gradient flow, and the low-dimensional vectors from gradient descent. ", + "bbox": [ + 174, + 795, + 825, + 920 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "J EXPERIMENTS: VARY ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 173, + 103, + 527, + 117 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "In this appendix, we fix the task as that of fitting a 5,000-sized subset of CIFAR-10, and we verify our main findings across a broad range of architectures. ", + "bbox": [ + 174, + 133, + 821, + 161 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Procedure We consider fully-connected networks and convolutional networks, the latter with both max-pooling and average pooling. For all of these, we consider tanh, ReLU, and ELU activations, and for fully-connected networks we moreover consider softplus and hardtanh activations. We train each network with both cross-entropy and MSE loss. See $\\ S \\mathrm { I . 1 }$ for full experimental details. ", + "bbox": [ + 174, + 178, + 825, + 233 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "In each case, we first use the Runge-Kutta method to numerically integrate the gradient flow ODE (see $\\ S 1 . 5$ for details). For architectures that give rise to continuously differentiable training objectives, the gradient flow ODE is guaranteed to have a unique solution (which we call the gradient flow trajectory), and Runge-Kutta will return a numerical approximation to this solution. On the other hand, for architectures with ReLU, hardtanh, or max-pooling, the training objective is not continuously differentiable, so the gradient flow ODE does not necessarily have a unique solution, and there are no guarantees a priori on what Runge-Kutta will return (more on this below under the “findings” heading). Still, in both cases, since our implementation of Runge-Kutta automatically adjusts the step size based on the local sharpness in order to remain stable, the Runge-Kutta trajectory can be roughly viewed as “what gradient descent would do if instability was not an issue.” ", + "bbox": [ + 174, + 239, + 825, + 378 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "We then run gradient descent at a range of step sizes. These step sizes $\\eta$ were chosen by hand so that the quantity $2 / \\eta$ would be spaced uniformly between $\\lambda _ { 0 }$ (the sharpness at initialization) and $\\lambda _ { \\mathrm { m a x } }$ (the maximum sharpness along the Runge-Kutta trajectory). ", + "bbox": [ + 176, + 386, + 823, + 428 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Results During Runge-Kutta, we observe that the sharpness tends to continually increase during training (progressive sharpening), with the exception that when cross-entropy loss is used, the sharpness decreases at the very end of training, as explained in Appendix C. ", + "bbox": [ + 174, + 444, + 825, + 486 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "During gradient descent with step size $\\eta$ , we observe that once the sharpness reaches $2 / \\eta$ , it ceases to increase much further, and instead hovers right at, or just above, the value $2 / \\eta$ . For reasons unknown, it tends to be true that for MSE loss, the sharpness hovers just a tiny bit above the value $2 / \\eta$ , while for cross-entropy loss the gap between the sharpness and the value $2 / \\eta$ is a bit larger. ", + "bbox": [ + 174, + 492, + 825, + 549 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "For each step size, we monitor the distance between the gradient descent trajectory and the RungeKutta trajectory — that is, we monitor the distance between the Runge-Kutta iterate at time $t$ and the gradient descent iterate at step $t / \\eta$ (see $\\ S 1 . 6$ for details). Empirically, for architectures that give rise to continuously differentiable training objectives, we observe that this distance is nearly zero before the sharpness hits $2 / \\eta$ , and it starts to climb immediately afterwards. This means that gradient descent closely tracks the gradient flow trajectory so long as the sharpness remains less than $2 / \\eta$ . Note that this finding was not a foregone conclusion: gradient descent is guaranteed to track the gradient flow trajectory in the limit of infinitesimal step sizes (since gradient descent is the forward Euler discretization of the gradient flow ODE), but for non-infinitesimal step sizes, there is discretization error, which is studied in Barrett & Dherin (2021). Our empirical finding is essentially that this discretization error is small compared to the difference between trajectories caused by instability. ", + "bbox": [ + 174, + 555, + 825, + 722 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "On the other hand, for architectures with non-differentiable components such as ReLU or maxpooling, we sometimes observe that gradient descent tracks the Runge-Kutta trajectory so long as the sharpness remains less than $2 / \\eta$ , but we also sometimes observe that the gradient descent trajectories differ from one another (and from Runge-Kutta) from the beginning of training. In the former case, we can infer that the gradient flow trajectory apparently does exist, and is returned by Runge-Kutta; in the latter case, we can infer that either (a) the gradient flow trajectory does not exist, or (b) that it does exist (and is returned by Runge-Kutta), but the step sizes we used for gradient descent were too large to track it. ", + "bbox": [ + 173, + 729, + 825, + 840 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "J.1 FULLY-CONNECTED TANH NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 104, + 467, + 117 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "J.1.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 130, + 325, + 143 + ], + "page_idx": 33 + }, + { + "type": "image", + "img_path": "images/6d0728c745c52a1364ffe39ce6e8f8dd29a4e209e89c23debe9bab6652a7db62.jpg", + "image_caption": [ + "Figure 28: Gradient flow. We train the network to $9 9 \\%$ accuracy with gradient flow, by using the Runge-Kutta method to discretize the gradient flow ODE (details in $\\ S 1 . 5 )$ ). We plot the train loss (left), sharpness (center), and train accuracy (right) over time. Observe that the sharpness tends to continually increase (except for a slight decrease at initialization). " + ], + "image_footnote": [], + "bbox": [ + 179, + 164, + 856, + 266 + ], + "page_idx": 33 + }, + { + "type": "image", + "img_path": "images/35dd58b8cdeebf43d0600dd08fe120012208937c690e830d55713a80255dffe1.jpg", + "image_caption": [ + "Figure 29: Gradient descent. We train the network to $9 9 \\%$ accuracy using gradient descent at a range of step sizes $\\eta$ . Top left: we plot the train loss curves, with a vertical line marking the iteration where the sharpness first crosses $2 / \\eta$ . Observe that the train loss monotonically decreases before this point, but behaves non-monotonically afterwards. Top middle: we plot the sharpness (measured at regular intervals during training). For each step size, the horizontal dashed line of the appropriate color marks the maximum stable sharpness $2 / \\eta$ . Observe that the sharpness tends to increase during training until reaching the value $2 / \\eta$ , and then hovers just a bit above that value. Bottom left: we track the $\\ell _ { 2 }$ distance between (random projections of) the gradient flow iterate at time $t$ and the gradient descent iterate at iteration $t / \\eta$ (details in $\\ S 1 . 6 )$ . For each step size $\\eta$ , the vertical dotted line marks the time when the sharpness first crosses $2 / \\eta$ . Observe that the distance is essentially zero until this time, and starts to grow afterwards. From this, we can conclude that gradient descent closely tracks the gradient flow trajectory (moving at a speed $\\eta$ ) until the point on that trajectory where the sharpness reaches $2 / \\eta$ . Bottom middle: to further visualize the previous point, we plot the evolution of sharpness during gradient descent, but with time $\\underline { { \\underline { { \\mathbf { \\Pi } } } } }$ iteration $\\times$ step size) on the $\\mathbf { X }$ -axis rather than iteration. We plot the gradient flow sharpness in black. Observe that the gradient descent sharpness matches the gradient flow sharpness until reaching the value $2 / \\eta$ . " + ], + "image_footnote": [], + "bbox": [ + 181, + 401, + 857, + 614 + ], + "page_idx": 33 + }, + { + "type": "image", + "img_path": "images/9f1176dc6f69dfd4fa5b6d6db6c53ba62bde8c94190927b8a35edf279a89486b.jpg", + "image_caption": [ + "Figure 30: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 137, + 857, + 238 + ], + "page_idx": 34 + }, + { + "type": "image", + "img_path": "images/b81380c85e577848398396434eefa62bd8bb599a82a59eb78c6a957b33272662.jpg", + "image_caption": [ + "Figure 31: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 857, + 522 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "J.2 FULLY-CONNECTED ELU NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 464, + 117 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "J.2.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 130, + 325, + 143 + ], + "page_idx": 35 + }, + { + "type": "image", + "img_path": "images/b3e94c174f923c6bdced758acf3a2ced2f1e7c66fc53dab900bcf78bf300965d.jpg", + "image_caption": [ + "Figure 32: Gradient flow. Refer to the Figure 28 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 164, + 857, + 266 + ], + "page_idx": 35 + }, + { + "type": "image", + "img_path": "images/4c0a86b8781bb31461dc1bce13886f593f43f99df4ce934a163bc05a705bb9a8.jpg", + "image_caption": [ + "Figure 33: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 321, + 859, + 536 + ], + "page_idx": 35 + }, + { + "type": "image", + "img_path": "images/8a86945e72f1fe25a4f3070e013421495ee08fa9ff33db6a60d041810c4e4687.jpg", + "image_caption": [ + "Figure 34: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 136, + 856, + 238 + ], + "page_idx": 36 + }, + { + "type": "image", + "img_path": "images/4a329418dd77653b7f29b39603d5dd44d54504b861cacf6f47e369f3b67d05ba.jpg", + "image_caption": [ + "Figure 35: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 857, + 522 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "J.3.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 130, + 325, + 143 + ], + "page_idx": 37 + }, + { + "type": "image", + "img_path": "images/8391669a360d732e7f83b1494325054c11e62f2f9235cb2126999368ceed0154.jpg", + "image_caption": [ + "Figure 36: Gradient flow. Refer to the Figure 28 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 164, + 856, + 266 + ], + "page_idx": 37 + }, + { + "type": "image", + "img_path": "images/d465735b68886c33d5c8841e0e2e1bb80c03a351a2dc855e3401d800ad91daba.jpg", + "image_caption": [ + "Figure 37: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 176, + 321, + 857, + 536 + ], + "page_idx": 37 + }, + { + "type": "image", + "img_path": "images/0e9bb698a4336e001ad81c8114f36400b529a81502da9475df49f39f3eb586c4.jpg", + "image_caption": [ + "Figure 38: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 137, + 857, + 238 + ], + "page_idx": 38 + }, + { + "type": "image", + "img_path": "images/f00372e0bd8673c2f3d9574c3cabd991903dff9a0bc1d339f01d1206dd7dfc9e.jpg", + "image_caption": [ + "Figure 39: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 857, + 522 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "J.4 FULLY-CONNECTED RELU NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 472, + 118 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Note that since the ReLU activation function is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. ", + "bbox": [ + 171, + 128, + 825, + 159 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "J.4.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 172, + 325, + 188 + ], + "page_idx": 39 + }, + { + "type": "image", + "img_path": "images/b65f18aee6331aef07845d67a2af73b7ba1cbdc19cfcdb0b6b9b48beab0e5596.jpg", + "image_caption": [ + "Figure 40: Runge-Kutta. We train the network to $9 9 \\%$ accuracy using the Runge-Kutta algorithm. (Since a unique gradient flow trajectory is not guaranted to exist, we hesitate to call this “gradient flow”; Runge-Kutta should essentially be viewed as gradient descent with a very small step size.) Observe that the sharpness tends to continually increase. " + ], + "image_footnote": [], + "bbox": [ + 181, + 207, + 857, + 309 + ], + "page_idx": 39 + }, + { + "type": "image", + "img_path": "images/b1b8344d483a24a4f72f14784ebc40d2c8d2c9bde3f1f512b731dbdf4b9a4abb.jpg", + "image_caption": [ + "Figure 41: Gradient descent. All panes except bottom left: refer to the Figure 29 caption for more information. Bottom left: We track the $\\ell _ { 2 }$ distance between (random projections of) the Runge-Kutta iterate at time $t$ and the gradient descent iterate at iteration $t / \\eta$ . For each step size $\\eta$ , the vertical dotted loss marks the time when the sharpness first crosses $2 / \\eta$ . In contrast to Figure 29, here the distance between gradient descent and gradient flow starts to noticeably grow from the beginning of training. From this we conclude that for this architecture, gradient descent does not track the Runge-Kutta trajectory at first. Furthermore, for this architecture, observe that the training loss starts behaving non-monotonically (and the sharpness begins to plateau) before the sharpness hits $2 / \\eta$ . As discussed in Appendix A (“Caveats”), we believe that this is also due to the fact that ReLU is non-differentiable. " + ], + "image_footnote": [], + "bbox": [ + 179, + 406, + 857, + 621 + ], + "page_idx": 39 + }, + { + "type": "image", + "img_path": "images/080e0e96ecf6c9e65d4b58a5e5a0f60aabcee34aaba1510ebe924b9b16f9d32b.jpg", + "image_caption": [ + "Figure 42: Runge-Kutta. Refer to the Figure 40 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 136, + 856, + 238 + ], + "page_idx": 40 + }, + { + "type": "image", + "img_path": "images/b95f7ea697fca38a44fbe0a655898d68852bae81cd422940ae5b3fd52bea86a3.jpg", + "image_caption": [ + "Figure 43: Gradient descent. Refer to the Figure 41 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 857, + 522 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "J.5 FULLY-CONNECTED HARD TANH NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 509, + 117 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Note that since the hardtanh function is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. ", + "bbox": [ + 173, + 128, + 825, + 159 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "J.5.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 173, + 172, + 325, + 188 + ], + "page_idx": 41 + }, + { + "type": "image", + "img_path": "images/0bc9768cf25d338c4e1b508e3813c7aa7030dc08065642675bc803c8b6a17e49.jpg", + "image_caption": [ + "Figure 44: Runge-Kutta. Refer to the Figure 40 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 207, + 857, + 310 + ], + "page_idx": 41 + }, + { + "type": "image", + "img_path": "images/ca1f76cfa2ac88da464072247c958261cbf89072c609dc6386a4cd57acbd6c37.jpg", + "image_caption": [ + "Figure 45: Gradient descent. Refer to the Figure 41 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 179, + 364, + 859, + 579 + ], + "page_idx": 41 + }, + { + "type": "image", + "img_path": "images/342e59c506d9595ac0d85b7e34e627ce8b7374c71b5574aa6a1cb0e9d26dd183.jpg", + "image_caption": [ + "Figure 46: Runge-Kutta. Refer to the Figure 40 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 136, + 857, + 238 + ], + "page_idx": 42 + }, + { + "type": "image", + "img_path": "images/af9781d3525550b67896d26c9d9bfa28b0b2dc47fa42586b6b87a5094e0afa19.jpg", + "image_caption": [ + "Figure 47: Gradient descent. Refer to the Figure 41 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 857, + 522 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Note that since max-pooling is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. ", + "bbox": [ + 173, + 128, + 823, + 159 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "J.6.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 172, + 325, + 188 + ], + "page_idx": 43 + }, + { + "type": "image", + "img_path": "images/049f303ad138b110803f979b7cd1c1bc4addf624becba6394c052c1b040aebfc.jpg", + "image_caption": [ + "Figure 48: Runge-Kutta. We train the network to $9 9 \\%$ accuracy using the Runge-Kutta algorithm. (Since a unique gradient flow trajectory is not guaranted to exist, we hesitate to call this “gradient flow.”) Observe that the sharpness tends to continually increase. " + ], + "image_footnote": [], + "bbox": [ + 179, + 207, + 857, + 310 + ], + "page_idx": 43 + }, + { + "type": "image", + "img_path": "images/ad0d0af556a7c12ef216ca82d353b0fd180bad9803fb4075e81f25cc0d936f0a.jpg", + "image_caption": [ + "Figure 49: All panes except bottom left: refer to the Figure 29 caption for more information. Bottom left: We track the $\\ell _ { 2 }$ distance between (random projections of) the Runge-Kutta iterate at time $t$ and the gradient descent iterate at iteration $t / \\eta$ . For each step size $\\eta$ , the vertical dotted loss marks the time when the sharpness first crosses $2 / \\eta$ . Observe that the distance is essentially zero until this time, and starts to grow afterwards. From this, we can conclude that even though the training objective is not differentiable, gradient descent does closely track the Runge-Kutta trajectory (moving at a speed $\\eta$ ) until the point on that trajectory where the sharpness reaches $2 / \\eta$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 393, + 857, + 607 + ], + "page_idx": 43 + }, + { + "type": "image", + "img_path": "images/390df9bc0caf092f1864bb885d53b2f559d540dec9c69f3c53b959cf46f51120.jpg", + "image_caption": [ + "Figure 50: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 136, + 856, + 238 + ], + "page_idx": 44 + }, + { + "type": "image", + "img_path": "images/73eeff454b86753bd8ec4fb9bf4078fa0e6a6b427d87c5bb2056f5089786859d.jpg", + "image_caption": [ + "Figure 51: Gradient descent. Refer to the Figure 49 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 179, + 308, + 857, + 522 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "J.7 CONVOLUTIONAL ELU NETWORK WITH MAX POOLING ", + "text_level": 1, + "bbox": [ + 173, + 103, + 596, + 117 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Note that since max-pooling is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. ", + "bbox": [ + 174, + 128, + 823, + 159 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "J.7.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 172, + 325, + 188 + ], + "page_idx": 45 + }, + { + "type": "image", + "img_path": "images/2af547d1f32796d61084d68c2dbd95aa8e5becbcf3e3511a6a325b86163fd6a6.jpg", + "image_caption": [ + "Figure 52: Runge-Kutta. Refer to the Figure 48 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 207, + 857, + 310 + ], + "page_idx": 45 + }, + { + "type": "image", + "img_path": "images/2f08a3738c3c4f650dc342aaf0d3c51987cd69fa097750905fd6cafe13c7463f.jpg", + "image_caption": [ + "Figure 53: Gradient descent. Refer to the Figure 49 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 364, + 859, + 579 + ], + "page_idx": 45 + }, + { + "type": "image", + "img_path": "images/58012bb2f46014c9b3153fb53031b4ea27b0b52131061a6b4189c48550105c38.jpg", + "image_caption": [ + "Figure 54: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 137, + 857, + 238 + ], + "page_idx": 46 + }, + { + "type": "image", + "img_path": "images/ba9b5bfab7634947d642740eafd3d7dbcc6a20a7e108fd5690e2f8ceff1c12ff.jpg", + "image_caption": [ + "Figure 55: Gradient descent. Refer to the Figure 49 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 857, + 522 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "J.8 CONVOLUTIONAL RELU NETWORK WITH MAX POOLING ", + "text_level": 1, + "bbox": [ + 173, + 103, + 606, + 117 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Note that since ReLU and max-pooling are not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. ", + "bbox": [ + 173, + 128, + 825, + 159 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "J.8.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 172, + 325, + 188 + ], + "page_idx": 47 + }, + { + "type": "image", + "img_path": "images/fd92020fa7ca8afa0810186f8a4121cf5fecb3ec7af7f5c7e7b1444e6a754bed.jpg", + "image_caption": [ + "Figure 56: Runge-Kutta. Refer to the Figure 48 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 207, + 857, + 310 + ], + "page_idx": 47 + }, + { + "type": "image", + "img_path": "images/91310eec236fbee78503c5743b32a57a566c19f6237a11f26c4b0219d3682bf7.jpg", + "image_caption": [ + "Figure 57: Gradient descent. Refer to the Figure 49 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 364, + 857, + 579 + ], + "page_idx": 47 + }, + { + "type": "image", + "img_path": "images/100d02a6124d6452aa4bb5dfe59c76dcde76accc738b225cd0bdece8197df904.jpg", + "image_caption": [ + "Figure 58: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 137, + 856, + 238 + ], + "page_idx": 48 + }, + { + "type": "image", + "img_path": "images/9409954b1c9ee42bb893685bac7124031f52738142063be2c4b738c4ac9c99a7.jpg", + "image_caption": [ + "Figure 59: Gradient descent. Refer to the Figure 49 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 859, + 522 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "J.9.1 CROSS-ENTROPY LOSS ", + "text_level": 1, + "bbox": [ + 174, + 130, + 385, + 143 + ], + "page_idx": 49 + }, + { + "type": "image", + "img_path": "images/5709a6237df282611c7baa12addbfcdbe90ba38b0000decf5a717257a402274b.jpg", + "image_caption": [ + "Figure 60: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 161, + 857, + 265 + ], + "page_idx": 49 + }, + { + "type": "image", + "img_path": "images/71d1906e291cd73f61a9d1bbaacf0091ecd95343bb8ad55ca5f4fa74b854c5b5.jpg", + "image_caption": [ + "Figure 61: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 179, + 334, + 857, + 547 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "J.10.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 130, + 334, + 143 + ], + "page_idx": 50 + }, + { + "type": "image", + "img_path": "images/53c280bcc23d16aba3e84c02fb6532fec9fc7de8d0160b84852f60c784ce8be2.jpg", + "image_caption": [ + "Figure 62: Gradient flow. Refer to the Figure 28 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 164, + 856, + 266 + ], + "page_idx": 50 + }, + { + "type": "image", + "img_path": "images/3d407f0114b83cd238a86ceaba42240b89c4b59c0bbfd7368cebfa35a4f101ea.jpg", + "image_caption": [ + "Figure 63: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 179, + 321, + 859, + 536 + ], + "page_idx": 50 + }, + { + "type": "image", + "img_path": "images/1894c7a666ffc7497d7d86e498b9a5e964fde2e347b6b74f1a3313dd868ba228.jpg", + "image_caption": [ + "Figure 64: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 137, + 856, + 238 + ], + "page_idx": 51 + }, + { + "type": "image", + "img_path": "images/7c724928bf06230da48366d4e1f0665b73f1c854dc5641acd31793f7356b8eb2.jpg", + "image_caption": [ + "Figure 65: Gradient descent. Refer to the Figure 29 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 308, + 857, + 522 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "J.11 CONVOLUTIONAL RELU NETWORK WITH AVERAGE POOLING ", + "text_level": 1, + "bbox": [ + 173, + 103, + 647, + 118 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Note that since ReLU is not continuously differentiable, the training objective is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist. ", + "bbox": [ + 173, + 128, + 823, + 159 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "J.11.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 172, + 333, + 188 + ], + "page_idx": 52 + }, + { + "type": "image", + "img_path": "images/458da1d795e651173e3672e15724d65cf0b83c8ef58dc3419ea3fd23c59e0b71.jpg", + "image_caption": [ + "Figure 66: Runge-Kutta. Refer to the Figure 48 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 207, + 856, + 310 + ], + "page_idx": 52 + }, + { + "type": "image", + "img_path": "images/27877445d07844f31aeff350ad3fdfd6a393b75126e4eab34b35f81e37b66164.jpg", + "image_caption": [ + "Figure 67: Gradient descent. Refer to the Figure 49 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 364, + 857, + 579 + ], + "page_idx": 52 + }, + { + "type": "image", + "img_path": "images/269f1c53f21513b6d64f12624096c06f96d2d6a84df447da4be523f77bbd2b45.jpg", + "image_caption": [ + "Figure 68: Runge-Kutta. Refer to the Figure 48 caption for more information. Additionally, in this figure the sharpness drops at the end of training due to the cross-entropy loss. " + ], + "image_footnote": [], + "bbox": [ + 179, + 137, + 856, + 238 + ], + "page_idx": 53 + }, + { + "type": "image", + "img_path": "images/758289dfcef21a8f3104a03dcf8b3dede0108d64705bb8500712558ff1c63a18.jpg", + "image_caption": [ + "Figure 69: Gradient descent. Refer to the Figure 49 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 181, + 309, + 857, + 522 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "K BATCH NORMALIZATION EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 540, + 117 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "In this appendix, we demonstrate that our findings hold for networks that are trained with batch normalization (BN) (Ioffe & Szegedy, 2015). We experiment on a size-5,000 subset of CIFAR10, and we consider convolutional networks with three different activation functions: ELU (Figure 70-71, tanh (Figure 72-73), and ReLU (Figure 74-75). See $\\ S \\mathrm { I } . 3$ for experimental details. ", + "bbox": [ + 174, + 138, + 825, + 194 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Empirically, our findings hold for batch-normalized networks. The one catch is that when training batch-normalized networks at very small step sizes, it is apparently inadequate to measure the sharpness directly at the iterates themselves, as we do elsewhere in the paper. Namely, observe that in Figure 71(e), when we run gradient descent at the red step size, the sharpness (measured directly at the iterates) plateaus a bit beneath the value $2 / \\eta$ . At first, this might sound puzzling: after all, if the sharpness is less than $2 / \\eta$ then gradient descent should be stable. The explanation is that the sharpness in between successive iterates does in fact cross $2 / \\eta$ . In Figure 71(b), we track the maximum sharpness on the path “in between” successive iterates. (To estimate the maximum sharpness between a pair of successive iterates, we compute the sharpness at a grid of eight points spaced evenly between them, and then take the maximum of these values.) Observe that this quantity does rise to $2 / \\eta$ and hover there. We do not know why measuring the sharpness between iterates is necessary for batch-normalized networks, whereas for non-BN networks it suffices to measure the sharpness only at the iterates themselves. ", + "bbox": [ + 173, + 200, + 825, + 381 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "In $\\ S \\mathrm { K } . 1$ , we reconcile these findings with Santurkar et al. (2018). ", + "bbox": [ + 173, + 388, + 598, + 402 + ], + "page_idx": 54 + }, + { + "type": "image", + "img_path": "images/404db4afb0e9177c92c54850c7f0dc79184ef00be99f37df5201e730d36bdd39.jpg", + "image_caption": [ + "Figure 70: We train a ELU CNN $\\left( + \\mathbf { B } \\mathbf { N } \\right)$ using gradient flow. " + ], + "image_footnote": [], + "bbox": [ + 243, + 425, + 761, + 506 + ], + "page_idx": 54 + }, + { + "type": "image", + "img_path": "images/96afe39b6a9023f038910844c477d8fff9a2289d8886f7c8e921094afd86297d.jpg", + "image_caption": [ + "Figure 71: We train an ELU CNN $\\mathbf { \\left( + \\delta B N \\right) }$ using gradient decent at a range of step sizes. (a) we plot the train loss, with a vertical dotted line marking the iteration where the sharpness on the path first crosses $2 / \\eta$ . The inset shows that the red curve is indeed behaving non-monotonically. (b) we track the sharpness “between iterates.” This means that instead of computing the sharpness right at the iterates themselves (as we do elsewhere in the paper, and in pane (e) here), we compute the maximum sharpness on the line between between successive iterates. Observe that this quantity rises to $2 / \\eta$ (marked by the horizontal dashed line) and then hovers right at, or just above that value. (c) we plot the same quantity by “time” $\\mathit { \\Psi } = \\mathit { \\Psi }$ iteration $\\times$ step size) rather than iteration. (e) we plot the sharpness at the iterates themselves. Note that for the red step size, this quantity plateaus at a value that is beneath $2 / \\eta$ . (f) distance from the gradient flow trajectory. " + ], + "image_footnote": [], + "bbox": [ + 240, + 578, + 754, + 744 + ], + "page_idx": 54 + }, + { + "type": "image", + "img_path": "images/99c2e827a5b9bc9069cc98ecacbcadd8e0c8cad7d7ebefdc205ef47bc77c1202.jpg", + "image_caption": [ + "Figure 72: We train a tanh CNN $\\left( + \\mathbf { B } \\mathbf { N } \\right)$ using gradient flow. " + ], + "image_footnote": [], + "bbox": [ + 243, + 103, + 761, + 185 + ], + "page_idx": 55 + }, + { + "type": "image", + "img_path": "images/a441c23940bb29cc3aeda6162fb228cf2efce55bce062edaf7010ff2919766d8.jpg", + "image_caption": [ + "Figure 73: We train a tanh CNN $\\left( + \\mathbf { B } \\mathbf { N } \\right)$ using gradient decent. " + ], + "image_footnote": [], + "bbox": [ + 238, + 241, + 754, + 409 + ], + "page_idx": 55 + }, + { + "type": "image", + "img_path": "images/3738d254485ca8f1db197851a4745810e8540f78a1fe2f1d367dbdafc996fdc4.jpg", + "image_caption": [ + "Figure 74: We train a ReLU CNN $\\left( + \\mathbf { B } \\mathbf { N } \\right)$ using gradient flow. " + ], + "image_footnote": [], + "bbox": [ + 241, + 465, + 761, + 547 + ], + "page_idx": 55 + }, + { + "type": "image", + "img_path": "images/322fdc9b1f1799546e45643baf826573ce3bc5d4b767f0efbcb8c3be28c4e87e.jpg", + "image_caption": [ + "Figure 75: We train a ReLU CNN $\\left( + \\mathbf { B } \\mathbf { N } \\right)$ using gradient descent. " + ], + "image_footnote": [], + "bbox": [ + 238, + 604, + 754, + 771 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "K.1 RELATION TO SANTURKAR ET AL. (2018) ", + "text_level": 1, + "bbox": [ + 174, + 839, + 509, + 853 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "We have demonstrated that the sharpness hovers right at (or just above) the value $2 / \\eta$ when both BN and non-BN networks are trained using gradient descent at reasonable step sizes. Therefore, at least in the case of full-batch gradient descent, it cannot be said that batch normalization decreases the sharpness (i.e. improves the local $L$ -smoothness) along the optimization trajectory. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Santurkar et al. (2018) argued that batch normalization improves the effective smoothness along the optimization trajectory, where effective smoothness is defined as the Lipschitz constant of the gradient in the update direction (i.e. the negative gradient direction, for full-batch GD). That is, given an objective function $f$ , an iterate $\\theta$ , and a distance $\\alpha$ , the effective smoothness of $f$ at parameter $\\theta$ and distance $\\alpha$ is defined in Santurkar et al. (2018) as ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/4bb8fbc3125ed90eca39e65f098540103626303b22e374262062ab02fbd04554.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\gamma \\in [ 0 , \\alpha ] } \\frac { \\| \\nabla f ( \\theta ) - \\nabla f ( \\theta - \\gamma \\nabla f ( \\theta ) ) \\| _ { 2 } } { \\| \\gamma \\nabla f ( \\theta ) \\| _ { 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 180, + 629, + 218 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "where the sup can be numerically approximated by evaluating the given ratio at several values $\\gamma$ spaced uniformly between 0 and $\\alpha$ . ", + "bbox": [ + 173, + 227, + 821, + 255 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "In Figure 76, we train two ReLU CNNs — one with BN, one without — at the same set of step sizes, and we monitor both the sharpness (i.e. the $L$ -smoothness) and the effective smoothness. When computing the effective smoothness, we use a distance $\\alpha = \\eta$ . Observe that for both the BN and the non-BN network, the effective smoothness initially hovers around zero, but once gradient descent enters the Edge of Stability, the effective smoothness jumps to the value $2 / \\eta$ and then remains there. Thus, at least for full-batch gradient descent on this particular architecture, batch normalization does not improve the effective smoothness along the optimization trajectory. (Despite this, note that for each step size, the BN network trains faster than the non-BN network, confirming that BN does accelerate training.) ", + "bbox": [ + 173, + 261, + 825, + 387 + ], + "page_idx": 56 + }, + { + "type": "image", + "img_path": "images/eae8d2d243df8c1253b39e48a1ddcded8efeab7593d115d544011a9caeda745d.jpg", + "image_caption": [ + "Figure 76: On a 5,000-size subset of CIFAR-10, we train a ReLU CNN both with BN (top row) and without BN (bottom row) at the same grid of step sizes. We plot the sharpness/smoothness (center column) as well as the effective smoothness (Santurkar et al., 2018) (right column). Observe that for both networks, the effective smoothness hovers around zero initially, and jumps up to $2 / \\eta$ once gradient descent enters the Edge of Stability. " + ], + "image_footnote": [], + "bbox": [ + 183, + 405, + 825, + 612 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "Note that this finding is actually consistent with Figure 4(c) in Santurkar et al. (2018), which is meant to show that BN improves effective smoothness when training a VGG network using SGD. Their Figure 4(c) shows that during SGD with step size $\\eta = 0 . 1$ , the effective smoothness hovers around the value 20 for both the BN and the non-BN network. Since $2 0 = 2 / ( 0 . 1 )$ , this is fully consistent with our findings (though they use SGD rather than full-batch GD). Figure 4(c) does show that the effective smoothness behaves more regularly for the BN network than for the non-BN network. But we disagree with their interpretation of this figure as demonstrating that BN improves the effective smoothness during training. ", + "bbox": [ + 173, + 720, + 825, + 833 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "The other piece of evidence in Santurkar et al. (2018) in support of the argument that batch normalization improves the effective smoothness during training is their Figure 9(c). This figure shows that a deep linear network (DLN) trained without BN has a much larger (i.e. worse) effective smoothness during training than a DLN trained with BN. However, for this figure, the distance $\\alpha$ used to compute effective smoothness was larger than the training step size $\\eta$ by a factor of 30. The effective smoothness at distances larger than the step size does not affect training. We have verified that when effective smoothness is computed at a distance equal to the training step size (i.e. $\\alpha = \\eta$ ), the effective smoothness for the DLN with BN and for the DLN without BN both hover right at $2 / \\eta$ . ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "Specifically, in Figure 77 and Figure 78, we train a DLN both with and without BN (respectively), and we measure the effective smoothness at a distance $\\alpha = 3 0 \\eta$ , as done in Figure 9(c) of Santurkar et al. (2018). We use the same experimental setup and the same step size of $\\eta = 1 \\mathrm { e } { - 6 }$ as they do, and we repeat the experiment across four random seeds. Observe that when training the BN network, the effective smoothness hovers right at $2 / \\eta$ (marked by the horizontal black line), whereas when training the non-BN network, the effective smoothness is much larger. This is consistent with Figure 9(c) in Santurkar et al. (2018). However, in Figure 79 and 80, we measure the effective smoothness at the actual step size $\\alpha = \\eta$ . When effective smoothness is computed in this way, we observe that for both the network with BN and the network without BN, the effective smoothness hovers right at $2 / \\eta$ . Therefore, we conclude that there is no evidence that the use of batch normalization improves either the smoothness or the effective smoothness along the optimization trajectory. (That said, this experiment possibly explains why the batch-normalized network permits training with larger step sizes.) ", + "bbox": [ + 173, + 138, + 825, + 319 + ], + "page_idx": 57 + }, + { + "type": "image", + "img_path": "images/711d45e6c38cb8db57825974e6d118c638c192753afb1ad27d8b8f11db673d6c.jpg", + "image_caption": [ + "Figure 77: When training a deep linear network without BN, we measure effective smoothness at a distance $\\alpha = 3 0 \\eta$ that is $\\mathbf { 3 0 x }$ larger than the step size. " + ], + "image_footnote": [], + "bbox": [ + 217, + 337, + 779, + 417 + ], + "page_idx": 57 + }, + { + "type": "image", + "img_path": "images/3bc9cf62b9fb4ee1881a277840f0bd332c0e1c1c324b2af253089f6e9e89439b.jpg", + "image_caption": [ + "Figure 78: When training a deep linear network with BN, we measure effective smoothness at a distance $\\alpha = 3 0 \\eta$ that is $\\mathbf { 3 0 x }$ larger than the step size. " + ], + "image_footnote": [], + "bbox": [ + 217, + 493, + 779, + 574 + ], + "page_idx": 57 + }, + { + "type": "image", + "img_path": "images/d626344fdf7f9ec1855efc2601d1c5c489599e0a6d439755681a405dc577f6f8.jpg", + "image_caption": [ + "Figure 79: When training a deep linear network without BN, we measure effective smoothness at a distance $\\alpha = \\eta$ that is equal to the step size. " + ], + "image_footnote": [], + "bbox": [ + 217, + 651, + 779, + 732 + ], + "page_idx": 57 + }, + { + "type": "image", + "img_path": "images/7b6737fa587e9662dcb6397b65243ac836bc908a5ac2bd03931e816debfd9868.jpg", + "image_caption": [ + "Figure 80: When training a deep linear network with BN, we measure effective smoothness at a distance $\\alpha = \\eta$ that is equal to the step size. " + ], + "image_footnote": [], + "bbox": [ + 217, + 446, + 779, + 527 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "L ADDITIONAL TASKS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 377, + 118 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "So far, we have verified our findings on image classification and language modeling. In this appendix, we verify our findings on three additional tasks: training a Transformer on the WikiText-2 language modeling dataset (L.1), training a one-hidden-layer network on a one-dimensional toy regression task (L.2), and training a deep linear network (L.3). ", + "bbox": [ + 174, + 133, + 825, + 189 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "L.1 TRANSFORMER ON WIKITEXT-2 ", + "text_level": 1, + "bbox": [ + 176, + 205, + 441, + 220 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "We consider the problem of training a Transformer on the WikText-2 word-level language modeling dataset (Merity et al., 2016). See $\\ S 1 . 4$ for full experimental details. In Figure 81, we train using gradient flow (only partially, not to completion). Observe that the sharpness continually rises. In Figure 82, we train using gradient descent at a range of step sizes. Consistent with our general findings, for each step size $\\eta$ , we observe that the sharpness rises to $2 / \\eta$ and then hovers right at, or just above, that value. However, for this Transformer, we do not observe that gradient descent closely tracks the gradient flow trajectory at the beginning of training. ", + "bbox": [ + 173, + 232, + 826, + 330 + ], + "page_idx": 59 + }, + { + "type": "image", + "img_path": "images/12cad1f71acba6d23954d772c7367242e0c306c559e7b0d15a5a6edc67b72da6.jpg", + "image_caption": [ + "Figure 81: Training a Transformer using gradient flow. " + ], + "image_footnote": [], + "bbox": [ + 287, + 348, + 710, + 444 + ], + "page_idx": 59 + }, + { + "type": "image", + "img_path": "images/bffbb195c94c724c5b9c7619674a20d6367ab03d9eb252d20206a69b8297a98d.jpg", + "image_caption": [ + "Figure 82: Training a Transformer using gradient descent. We train a Transformer for 15,000 iterations on the WikiText-2 language modeling dataset. (For this problem, training to completion would not be computationally practical.) Top left: we plot the train loss curves, with a vertical dotted line marking the iteration where the sharpness first crosses $2 / \\eta$ . The train loss decreases monotonically before this line, but behaves non-monotonically afterwards. Top right: we plot the evolution of the sharpness, with a horizontal dashed line marking the value $2 / \\eta$ . Observe that for each step size, the sharpness rises to $2 / \\eta$ and then hovers right at, or just above, that value. Bottom left: for the initial phase of training, we plot the distance between the gradient descent trajectory and the gradient flow trajectory, with a vertical dotted line marking the gradient descent iteration where the sharpness first crosses $2 / \\eta$ . Observe that the distance begins to rise from the start of training, indicating that for this architecture, gradient descent does not track the gradient flow trajectory initially. Bottom right: for the initial phase of training, we plot the evolution of the sharpness by “time” $=$ iteration $\\times \\eta$ rather than iteration. The black dots are the sharpness during gradient flow. " + ], + "image_footnote": [], + "bbox": [ + 287, + 507, + 710, + 708 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "Task Our toy regression problem is to approximate a Chebyshev polynomial using a neural network. To generate a toy dataset, we take 20 points spaced uniformly on the interval $[ - 1 , 1 ]$ , and we label them noiselessly using the Chebyshev polynomial of some degree $k$ . Note that the Chebyshev polynomial of degree $k$ is a polynomial with $k$ zeros that maps the domain $[ - 1 , 1 ]$ to the range $[ - 1 , 1 ]$ . Figure 83 shows the Chebyshev datasets for degree 3, 4, and 5. ", + "bbox": [ + 173, + 128, + 825, + 200 + ], + "page_idx": 60 + }, + { + "type": "image", + "img_path": "images/924b6cd0121397ed2391aa23bcdb5462952dedd01348ef21d88eadd4015708fe.jpg", + "image_caption": [ + "Figure 83: The datasets for the toy one-dimensional regression task. " + ], + "image_footnote": [], + "bbox": [ + 184, + 218, + 825, + 338 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Network For the network, we use a tanh architecture with one hidden layer of $h = 1 0 0$ units, initialized using Xavier initialization. We train using the MSE loss until the loss reaches 0.05. ", + "bbox": [ + 173, + 397, + 825, + 426 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Results In Figure 84, we fit the Chebyshev degree 3, 4, and 5 datasets using gradient flow. Empirically, the higher the degree, the more the sharpness rises during the course of training: on the degree 3 polynomial, the sharpness rises by a factor of 1.2; on the degree 4 polynomial, by a factor of 3.2; and on the degree 5 polynomial, by a factor of 63.5. ", + "bbox": [ + 173, + 441, + 825, + 497 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "In Figure 85 and Figure 86, we fit the degree 4 and 5 datasets using gradient descent at a range of step sizes. We observe mostly the same Edge of Stability behavior as elsewhere in the paper. The only difference is that for the degree 5 dataset, after the sharpness hits $2 / \\eta$ , the training loss first undergoes a temporary period of non-monotonicity in which no progress is made, and then it decreases monotonically until training is finished (in contrast to our other experiments where we observe that training loss behaves non-monotonically at the same time as it is consistently decreasing). ", + "bbox": [ + 174, + 503, + 825, + 588 + ], + "page_idx": 60 + }, + { + "type": "image", + "img_path": "images/402e98d48907bb48bb8cd932791b00422fdd9bb17a24b93290a6f5868d68c871.jpg", + "image_caption": [ + "Figure 84: Fitting Chebyshev polynomials using gradient flow. We use gradient flow to fit a one-hidden-layer tanh network to the Chebyshev polynomials of degrees 3, 4, and 5. We observe that the sharpness rises more when fitting a higher-degree polynomial, likely because the dataset is more complex. " + ], + "image_footnote": [], + "bbox": [ + 263, + 103, + 740, + 453 + ], + "page_idx": 61 + }, + { + "type": "image", + "img_path": "images/1fb3bdb91531e90387d4d5bbeec87d783a7f112d402f03037e318faf47f13d84.jpg", + "image_caption": [ + "Figure 85: Chebyshev degree 4 (gradient descent). We fit the Chebyshev polynomial of degree 4 using gradient descent at a range of step sizes (see legend). " + ], + "image_footnote": [], + "bbox": [ + 184, + 555, + 821, + 827 + ], + "page_idx": 61 + }, + { + "type": "image", + "img_path": "images/f863a93dc042fd5c3a02028c53f622ee65d3283ade9cfd4badfb4234ea590310.jpg", + "image_caption": [ + "Figure 86: Chebyshev degree 5 (gradient descent). We fit the Chebyshev polynomial of degree 5 using gradient descent at a range of step sizes (see legend). " + ], + "image_footnote": [], + "bbox": [ + 184, + 108, + 820, + 378 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "L.3 DEEP LINEAR NETWORK ", + "text_level": 1, + "bbox": [ + 174, + 460, + 385, + 474 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "is to map rror is me $n$ inputs ured u $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { n } \\subseteq \\mathbb { R } ^ { d }$ to ss, $n$ te. $\\mathbf { y } _ { 1 } , \\ldots , \\mathbf { y } _ { n } \\subseteq \\mathbb { R } ^ { d }$ $f : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ $\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| f ( \\mathbf { { \\bar { x } } } _ { i } ) - \\mathbf { y } _ { i } \\| _ { 2 } ^ { 2 } } \\end{array}$ $\\mathbf { X } \\in \\mathbb { R } ^ { n \\times d }$ $\\mathbf { Y } \\in \\mathbb { R } ^ { n \\times d }$ targets. We first generate $\\mathbf { X }$ as a random whitened matrix (i.e. $\\textstyle { \\frac { 1 } { n } } \\mathbf { X } ^ { T } \\mathbf { X } = \\mathbf { I } )$ . To generate $\\mathbf { X }$ as a random whitened matrix, we sample a $n \\times d$ matrix of standard Gaussians, and then set $\\mathbf { X }$ to be $\\sqrt { n }$ times the Q factor in the QR factorization of that matrix. We then generate $\\mathbf { Y }$ via $\\mathbf { Y } = \\mathbf { X } \\mathbf { A } ^ { T }$ , where $\\mathbf { A } \\in \\mathbb { R } ^ { d \\times d }$ is a random matrix whose entries are sampled i.i.d from the standard normal distribution. ", + "bbox": [ + 173, + 486, + 825, + 588 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "We use $n = 5 0$ datapoints with a dimension of $d = 5 0$ . ", + "bbox": [ + 176, + 595, + 534, + 609 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "Network The function $f : \\mathbb { R } ^ { d } \\mathbb { R } ^ { d }$ is implemented as a $L$ -layer deep linear network: $f ( \\mathbf { x } ) =$ $\\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 2 } \\mathbf { W } _ { 1 } \\mathbf { x }$ , with $\\mathbf { W } _ { \\ell } \\in \\mathbb { R } ^ { d \\times d }$ . We initialize all layers of the deep linear network using Xavier initialization: all entries of each $\\mathbf { W } _ { \\ell }$ are drawn i.i.d from $\\textstyle { \\mathcal { N } } ( 0 , { \\frac { 1 } { d } } )$ . ", + "bbox": [ + 174, + 623, + 825, + 667 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "We use a network with $L = 2 0$ layers. ", + "bbox": [ + 176, + 674, + 423, + 688 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "Results In Figure 87, we train the network using gradient flow. (Since it is unclear whether the network can be trained to zero loss, and how long this would take, we arbitrarily chose to stop training at time 100.) In Figure 88, we train the network using gradient descent at a range of step sizes. We observe mostly the same Edge of Stability behavior as elsewhere in the paper. The only difference is that in Figure 88, the train loss does not really behave non-monotonically — for each step size $\\eta$ , there is a brief blip at some point, but otherwise, the train loss decreases monotonically. ", + "bbox": [ + 173, + 704, + 825, + 787 + ], + "page_idx": 62 + }, + { + "type": "image", + "img_path": "images/e326bc0690fd1050067b147d39b711c643bccb982cd01f9353f4e60775546f3a.jpg", + "image_caption": [ + "Figure 87: Training a deep linear network using gradient flow. We use gradient flow to train a deep linear network. Since it is unclear whether this network can be trained to zero loss (or how long that would take), we arbitrarily chose to stop training at time 100. " + ], + "image_footnote": [], + "bbox": [ + 259, + 190, + 740, + 299 + ], + "page_idx": 63 + }, + { + "type": "image", + "img_path": "images/2fe7e0caa74be9dfbde7de428e59112895a019374e9f9a1754cddfbe6cc6c1d9.jpg", + "image_caption": [ + "Figure 88: Training a deep linear network using gradient descent. We train a deep linear network using gradient descent at a range of step sizes (see legend). (Note that in the top left pane, the blue, orange and green dotted lines are directly on top of one another.) " + ], + "image_footnote": [], + "bbox": [ + 240, + 549, + 758, + 768 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "M EXPERIMENTS: STANDARD ARCHITECTURES ON CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 707, + 117 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "In this appendix, we demonstrate that our findings hold for three standard architectures on the standard dataset CIFAR-10. The three architectures are: a VGG with batch normalization (Figures 89-90), a VGG without batch normalization (Figures 91-92), and a ResNet with batch normalization (Figures 93 -94). See $\\ S 1 . 2$ for full experimental details. ", + "bbox": [ + 174, + 138, + 821, + 194 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "For each of these three architectures, we confirm our main points: (1) so long as the sharpness is less than $2 / \\eta$ , the sharpness tends to increase; and (2) if the sharpness reaches $2 / \\eta$ , gradient descent enters a regime (the Edge of Stability) in which (a) the training loss behaves non-monotonically, yet consistently decreases over long timescales, and (b) the sharpness hovers right at, or just above, the value $2 / \\eta$ . Moreover, we observe that even though these architectures use the ReLU activation function (which is not continuously differentiable), gradient descent closely tracks the Runge-Kutta trajectory until reaching the point on that trajectory where the sharpness hits $2 / \\eta$ . ", + "bbox": [ + 174, + 202, + 825, + 299 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Furthermore, we observe that for these three standard architectures, the following additional points hold: (1) progressive sharpening occurs to a dramatic degree, and (2) stable step sizes are so small as to be completely unreasonable. ", + "bbox": [ + 174, + 306, + 823, + 348 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Progressive sharpening occurs to a dramatic degree To assess the degree of progressive sharpening, we train these networks using Runge-Kutta / gradient flow, which can be viewed as gradient descent with an infinitesimally small step size. (In practice, the Runge-Kutta algorithm does have a step size parameter, and throughout training, we periodically adjust this step size in order to ensure that the algorithm remains stable.) Intuitively, training with gradient flow tell us how far the sharpness would rise “if it didn’t have to worry about” instability caused by nonzero step sizes. In Figure 89, we train the VGG with BN to completion $9 9 \\%$ training accuracy) using Runge-Kutta / gradient flow, and find that the sharpness rises from its initial value of 6.38 to a peak value of 2227.6. For the other two architectures, progressive sharpening occurs to such a degree that it is not computationally feasible for to train using Runge-Kutta / gradient flow all the way to completion. (The reason is that in regions where the sharpness is high, the Runge-Kutta step size must be made small, so Runge-Kutta requires very many iterations.) Therefore, we instead train these two networks only partially. In Figure 91, for the VGG without BN, we find that the sharpness rises from its initial value of 0.64 to the value 2461.78 at $3 7 . 1 \\%$ accuracy, when we stop training. In Figure 93, for the ResNet, we find that the sharpness rises from its initial value of 1.07 to the value 760.6 at $4 3 . 2 \\%$ accuracy, when we stop training. Thus, even though we observed in Appendix D that progressive sharpening attenuates as the width of fully-connected networks is made larger, it appears that either: (1) this does not happen for modern families of architectures such as ResNet and VGG, or (2) this does happen for modern families of architectures, but practical network widths lie on the narrow end of the scale. ", + "bbox": [ + 173, + 371, + 825, + 648 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Stable step sizes are so small as to be unreasonable Recall from 3.3 that if $\\lambda _ { \\mathrm { m a x } }$ is the maximum sharpness along the gradient flow trajectory, then any stable step size must be less than $2 / \\lambda _ { \\mathrm { m a x } }$ . Therefore, for these three architectures, because progressive sharpening occurs to a dramatic degree (i.e. $\\lambda _ { \\mathrm { m a x } }$ is extraordinarily large), any stable step size must be extraordinarily small, which means that training will require many iterations. Yet, at the same time, we find that these three networks can be successfully trained in far fewer iterations by using larger step step size. This means that training at a stable step size is extremely suboptimal. We now elaborate on this point: ", + "bbox": [ + 173, + 672, + 825, + 770 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "VGG with BN. For this network, gradient flow terminates at time 15.66, and the maximum sharpness along the gradient flow trajectory is 2227.6. Therefore, the largest stable step size is $2 / 2 2 2 7 . 5 9 = 0 . 0 0 0 8 9 7$ , and training to completion at this step size would take $1 4 . 9 1 / 0 . 0 0 0 8 9 7 =$ 16622 iterations. Meanwhile, we empirically observe that the network can also be trained to completion at the much larger step size of $\\eta = 0 . 1 6$ in just 329 iterations. Therefore, using a stable step size is suboptimal by a factor of at least $1 6 6 2 2 / 3 2 9 = 5 0 . 5 $ . ", + "bbox": [ + 174, + 777, + 823, + 861 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "For the other two architectures, since we are unable to train to completion using gradient flow, we are unable to obtain a tight lower bound for the number of iterations required to run gradient descent to completion at a stable step size. Therefore, by extension, we are unable to compute a tight lower bound for the suboptimality factor of stable step sizes. As a substitute, we will instead compute both: (1) a tight lower bound on the suboptimality of training partially at a stable step size, and (2) a very loose lower bound on the suboptimality of training to completion at a stable step size. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 821, + 132 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "VGG without BN. For this network, gradient flow reaches $3 7 . 1 \\%$ accuracy at time 8, and the maximum sharpness up through this point on the gradient flow trajectory is 2461.8. Therefore, the largest stable step size is $2 / 2 4 6 1 . 8 = 0 . 0 0 0 8 1$ , and training to $3 7 . 1 \\%$ accuracy at this step size would require $8 / 0 . 0 0 0 8 1 = 9 , \\quad$ 876 iterations. Meanwhile, we empirically observe that the network can also be trained to $3 7 . 1 \\%$ accuracy at the larger step size of $\\eta = 0 . 1 6$ in just 355 iterations. Therefore, when training this network to $37 \\%$ accuracy, stable step sizes are suboptimal by a factor of at least $9 8 7 6 / 3 5 5 = \\mathrm { \\bar { 2 7 } } . 8 .$ . This is a tight lower bound on the suboptimality of training to $3 7 . 1 \\%$ accuracy at a stable step size. ", + "bbox": [ + 174, + 138, + 825, + 251 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "To obtain a loose lower bound on the suboptimality of training the VGG without BN to completion at a stable step size, we note that (a) since the maximum sharpness up through time 8 is 2461.8, the maximum sharpness along the entire gradient flow trajectory must be at least 2461.8; and (b) since by time 8 gradient flow has only attained $3 7 . 1 \\%$ training accuracy, the time to reach $9 9 \\%$ accuracy (i.e. completion) must be at least 8. (Note that both of these lower bounds are extremely loose.) Therefore, training this network to completion at a stable step size would require at least $8 / ( 2 / 2 4 6 1 . 8 ) \\ = \\ 9 , 8 7 6$ iterations (which is the same number of iterations as training to $3 7 . 1 \\%$ accuracy at a stable step size). Meanwhile, we find that the network can be trained to completion at the larger step size of $\\eta = 0 . 1 6$ in just 1782 iterations. Therefore, training to completion at a stable step size is suboptimal by a factor of at least $9 , 8 7 6 / 1 7 8 2 = 5 . 5 4$ . ", + "bbox": [ + 173, + 257, + 825, + 397 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "ResNet. For this network, gradient flow reaches $4 3 . 2 \\%$ accuracy at time 70, and the maximum sharpness up through this point on the gradient flow trajectory is 760.6. Therefore, the largest stable step size is $2 / 7 6 0 . 6 = 0 . 0 0 2 6$ , and training to $4 3 . 2 \\%$ accuracy at this step size would require $7 0 / 0 . 0 0 2 6 = 2 6 ,$ 923 iterations. Meanwhile, we empirically observe that the network can also be trained to $4 3 . 2 \\%$ accuracy at the larger step size of $\\eta = 2 . 0$ in just 99 iterations. Therefore, when training this network to $4 3 . 2 \\%$ accuracy, stable step sizes are suboptimal by a factor of at least $2 6 , 9 2 { \\bar { 3 } } / 9 9 = 2 7 1 . 9 .$ . This is a tight lower bound on the suboptimality of training to $4 3 . 2 \\%$ accuracy at a stable step size. For the loose lower bound on the suboptimality of training to completion at a stable step size, note that (by similar reasoning as the VGG-without-BN above), training to completion at a stable step size must require at least 26, 923 iterations. Meanwhile, we find that the network can be trained to completion at the larger step size of $\\eta = 2 . 0$ in just 807 iterations. Therefore, training to completion at a stable step size is suboptimal by a factor of at least $2 6 , 9 2 3 / 8 0 7 = 3 3 . 3$ . ", + "bbox": [ + 173, + 404, + 825, + 570 + ], + "page_idx": 65 + }, + { + "type": "image", + "img_path": "images/1aea1ce5c7e9dddceab313af8fc6f861b105d3164143d97c0beb229fb91197d9.jpg", + "image_caption": [ + "Figure 89: We train a VGG with BN to completion using gradient flow (Runge-Kutta). Observe that the sharpness rises dramatically from 6.38 at initialization to a peak of 2227.59. " + ], + "image_footnote": [], + "bbox": [ + 179, + 106, + 825, + 204 + ], + "page_idx": 66 + }, + { + "type": "image", + "img_path": "images/4b37034eb812e5a6f0e13a18abb8eed6b97188c126c689072faf59d018799676.jpg", + "image_caption": [ + "Figure 90: We train a VGG with BN to completion using gradient descent at different step sizes (see legend in the top right pane). Top left: we plot the train loss, with a vertical dotted line marking the iteration where the sharpness first crosses $2 / \\eta$ . Top center: we plot the evolution of the sharpness, with a horizontal dashed line (of the appropriate color) marking the value $2 / \\eta$ . Top right: we plot the train accuracy. Bottom left: for the initial phase of training, we monitor the distance between the gradient descent iterate at iteration $t / \\eta$ , and the gradient flow solution at time $t$ , with a vertical dotted line (of the appropriate color) marking the time when the sharpness crosses $2 / \\eta$ . Observe that the distance between gradient descent and gradient flow is almost zero before this instant, but starts to rise shortly afterwards. Bottom center: we plot the sharpness by time $\\left( = \\right.$ iteration $\\times \\eta$ ) rather than iteration. The black dots are the sharpness of the gradient flow trajectory, which shoots up immediately. Bottom right: we plot the test accuracy. " + ], + "image_footnote": [], + "bbox": [ + 179, + 279, + 825, + 484 + ], + "page_idx": 66 + }, + { + "type": "image", + "img_path": "images/2cba675712614b25e4b0e73c288854542afdda1705bd41eca425139b8bfcf0e0.jpg", + "image_caption": [ + "Figure 91: We train a VGG without BN to $3 7 . 1 \\%$ accuracy using gradient flow (Runge-Kutta). Observe that the sharpness rises dramatically from 0.64 at initialization to 2461.78 at $3 7 . 1 \\%$ accuracy. We train this network only partway because training this network to completion would be too computationally expensive: Runge-Kutta runs very slowly when the sharpness is high (because it is forced to take small steps) and for this network the sharpness is extremely high when the train accuracy is only $37 \\%$ (which means that there is a long way to go). " + ], + "image_footnote": [], + "bbox": [ + 179, + 127, + 825, + 224 + ], + "page_idx": 67 + }, + { + "type": "image", + "img_path": "images/d4bfe9fcbdd1335d834762612e090edacd2eb089dd52c241375224a57890a8ba.jpg", + "image_caption": [ + "Figure 92: We train a VGG without BN to completion using gradient descent at a range of step sizes (see legend in the top right pane). Refer to the Figure 90 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 178, + 385, + 825, + 588 + ], + "page_idx": 67 + }, + { + "type": "image", + "img_path": "images/8fd5dfa00a6140106d5a5b7860d9738fe09f879d8763d36414f68080d6a1f13a.jpg", + "image_caption": [ + "Figure 93: We train a ResNet to $4 3 . 2 \\%$ accuracy using gradient flow (Runge-Kutta). Observe that the sharpness rises dramatically from 1.07 at initialization to 760.63 at $4 3 . 2 \\%$ accuracy. We train this network only partway because training this network to completion would be too computationally expensive: Runge-Kutta runs very slowly when the sharpness is high (because it is forced to take small steps) and for this network the sharpness is extremely high when the train accuracy is only $42 \\%$ (which means that there is a long way to go). " + ], + "image_footnote": [], + "bbox": [ + 179, + 695, + 825, + 792 + ], + "page_idx": 67 + }, + { + "type": "image", + "img_path": "images/130d034a56276190c7b4f98f4f4c55a94d894e02d8caf6718923f78cc85394ad.jpg", + "image_caption": [ + "Figure 94: We train a ResNet to completion using gradient descent at a range of step sizes (see legend in the top right pane). Refer to the Figure 90 caption for more information. " + ], + "image_footnote": [], + "bbox": [ + 179, + 387, + 825, + 588 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "N EXPERIMENTS: MOMENTUM ", + "text_level": 1, + "bbox": [ + 176, + 102, + 449, + 118 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "This appendix contains systematic experiments for gradient descent with Polyak momentum and Nesterov momentum. Our aim is to demonstrate that the sharpness rises until reaching the maximum stable sharpness (MSS) given by equation 1, and then either plateaus just above that value, or oscillates around that value. ", + "bbox": [ + 174, + 133, + 825, + 189 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "We experiment on a $5 \\mathrm { k }$ -sized subset of CIFAR-10, using four architectures: a tanh fully-connected network (section N.1), a ReLU fully-connected network (section N.2), a tanh convolutional network (section N.3), and a ReLU convolutional network (section N.4). For each of these four architectures, we experiment with both the square loss (for classification) and cross-entropy loss. For each architecture and loss function, we experiment with both Polyak momentum at $\\beta = 0 . 9$ , and gradient descent with Nesterov momentum at $\\beta = 0 . 9$ . We run gradient descent at a range of several step sizes which were chosen by hand so that the MSS’s are approximately spaced evenly. Note that for Polyak momentum with step size $\\eta$ and momentum parameter $\\beta = 0 . 9$ , the MSS is $\\begin{array} { r } { \\frac { \\dot { 2 } + 2 \\beta } { \\eta } = \\frac { 3 . 8 } { \\eta } } \\end{array}$ . For Netsterov momentum with step size $\\eta$ and momentum parameter $\\beta = 0 . 9$ , the MSS is η2+2 β $\\begin{array} { r } { \\frac { 2 + 2 \\beta } { \\eta ( 1 + 2 \\beta ) } \\approx \\frac { 1 . 3 5 7 1 4 } { \\eta } } \\end{array}$ . We run gradient descent until reaching $9 9 \\%$ accuracy. ", + "bbox": [ + 173, + 195, + 825, + 345 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "We find that the sharpness rises until reaching the maximum stable sharpness (MSS) given by equation 1, and then either plateaus just above that value, or oscillates around that value. Sometimes these oscillations are rapid (e.g. Figure 96), sometimes they are a bit slower (e.g. Figure 103), and sometimes they are slow (e.g. Figure 109). ", + "bbox": [ + 174, + 352, + 825, + 409 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "N.1 FULLY-CONNECTED TANH NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 472, + 117 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) In the middle plot, the horizontal dashed line marks the MSS. ", + "bbox": [ + 173, + 128, + 825, + 185 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "N.1.1 SQUARE LOSS", + "text_level": 1, + "bbox": [ + 174, + 200, + 330, + 215 + ], + "page_idx": 70 + }, + { + "type": "image", + "img_path": "images/c635e15ac8da04def67411e9c66fea250f5acccf22c2f0c8445aac14ff7b7c11.jpg", + "image_caption": [ + "Figure 95: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 233, + 857, + 337 + ], + "page_idx": 70 + }, + { + "type": "image", + "img_path": "images/18b1d4e8ddd5dfcd4ac260780cfdb5e61e128a05bb4d176d635cf0dd95b00dc8.jpg", + "image_caption": [ + "Figure 96: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 393, + 857, + 497 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "N.1.2 CROSS-ENTROPY LOSS ", + "text_level": 1, + "bbox": [ + 174, + 554, + 390, + 566 + ], + "page_idx": 70 + }, + { + "type": "image", + "img_path": "images/577b27608f5c5362d2236e7a2aeb96fe6ab7600bf3fd5b2a5e0cb51aca438af6.jpg", + "image_caption": [ + "Figure 97: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 584, + 857, + 688 + ], + "page_idx": 70 + }, + { + "type": "image", + "img_path": "images/9c8eaa747084c6fd0eee24a15f13fe7cab787990986b26ec928a5677672ddaf9.jpg", + "image_caption": [ + "Figure 98: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 743, + 857, + 847 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "N.2 FULLY-CONNECTED RELU NETWORK", + "text_level": 1, + "bbox": [ + 174, + 103, + 478, + 118 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) ", + "bbox": [ + 174, + 128, + 828, + 171 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "In the middle plot, the horizontal dashed line marks the MSS. ", + "bbox": [ + 173, + 179, + 575, + 193 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "N.2.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 208, + 330, + 223 + ], + "page_idx": 71 + }, + { + "type": "image", + "img_path": "images/58f01faf3df568b2a611cf0d8ce60484e8dcc031f4f236b7649e0ac6bab8e324.jpg", + "image_caption": [ + "Figure 99: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 241, + 856, + 344 + ], + "page_idx": 71 + }, + { + "type": "image", + "img_path": "images/01212b19f8d82ee634814f18695512cc31c0c763ee6db34ef2cfee597613003f.jpg", + "image_caption": [ + "Figure 100: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 400, + 857, + 505 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "N.2.2 CROSS-ENTROPY LOSS ", + "text_level": 1, + "bbox": [ + 174, + 560, + 392, + 574 + ], + "page_idx": 71 + }, + { + "type": "image", + "img_path": "images/5a0256dead0f65338f9bdc82b061f281d9cac406d9ae79ce3d539c957956a303.jpg", + "image_caption": [ + "Figure 101: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 592, + 857, + 695 + ], + "page_idx": 71 + }, + { + "type": "image", + "img_path": "images/63f0f077747f117a1df6884844d6a4a3b1a3276917f710cfb44b76989bb04095.jpg", + "image_caption": [ + "Figure 102: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 751, + 857, + 854 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "N.3 CONVOLUTIONAL TANH NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 459, + 117 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) ", + "bbox": [ + 174, + 128, + 828, + 171 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "In the middle plot, the horizontal dashed line marks the MSS. ", + "bbox": [ + 173, + 179, + 575, + 193 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "N.3.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 208, + 330, + 223 + ], + "page_idx": 72 + }, + { + "type": "image", + "img_path": "images/16667ae2c68eb53ffd87e79e6fe4d3cde565358efaa0b64ad1256a29c7af8fbe.jpg", + "image_caption": [ + "Figure 103: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 241, + 856, + 344 + ], + "page_idx": 72 + }, + { + "type": "image", + "img_path": "images/c8bfb03ffd79b7881dc6f4c9ad5f6524d8e445c98d75c00bad920245ddf8a405.jpg", + "image_caption": [ + "Figure 104: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 400, + 857, + 505 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "N.3.2 CROSS-ENTROPY LOSS ", + "text_level": 1, + "bbox": [ + 174, + 560, + 390, + 574 + ], + "page_idx": 72 + }, + { + "type": "image", + "img_path": "images/e921c61207bbe83724e6a142d823673404c7c0134a5e904bcd11dabbb95b70a7.jpg", + "image_caption": [ + "Figure 105: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 592, + 857, + 695 + ], + "page_idx": 72 + }, + { + "type": "image", + "img_path": "images/9fc7427c21335831099aaf8091c3c0268425939a5a01e4ff54dc4a93183cde98.jpg", + "image_caption": [ + "Figure 106: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 751, + 857, + 854 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "N.4 CONVOLUTIONAL RELU NETWORK ", + "text_level": 1, + "bbox": [ + 176, + 103, + 464, + 117 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "In the leftmost plot, the vertical dotted line marks the iteration where the sharpness first crosses the MSS. (Note that unlike vanilla gradient descent, momentum gradient descent can sometimes cause the train loss to increase even when the algorithm is stable (Goh, 2017).) In the middle plot, the horizontal dashed line marks the MSS. ", + "bbox": [ + 173, + 128, + 825, + 185 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "N.4.1 SQUARE LOSS ", + "text_level": 1, + "bbox": [ + 174, + 200, + 330, + 215 + ], + "page_idx": 73 + }, + { + "type": "image", + "img_path": "images/6ab9b3351c47aa3ed6d83b7c6bb70ba04a789ccdb2cce6d05f735cb944472443.jpg", + "image_caption": [ + "Figure 107: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 234, + 856, + 337 + ], + "page_idx": 73 + }, + { + "type": "image", + "img_path": "images/86c7cdbe8905f56f022877678effcf54958de7fa8d763725e681bc7244609100.jpg", + "image_caption": [ + "Figure 108: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 393, + 857, + 497 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "N.4.2 CROSS-ENTROPY LOSS ", + "text_level": 1, + "bbox": [ + 174, + 553, + 390, + 566 + ], + "page_idx": 73 + }, + { + "type": "image", + "img_path": "images/73f8de1c537a70383970b3f9589de4d80a14e524268e65eaafa99b19a2f620d2.jpg", + "image_caption": [ + "Figure 109: Gradient descent with Polyak momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 584, + 857, + 688 + ], + "page_idx": 73 + }, + { + "type": "image", + "img_path": "images/bc70ca76e68f98d0817877e5c550174a8635e44bdf5fc5e1b97b862a0368b914.jpg", + "image_caption": [ + "Figure 110: Gradient descent with Nesterov momentum, $\\beta = 0 . 9$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 743, + 857, + 847 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "O EXPERIMENTS: LEARNING RATE DROP ", + "text_level": 1, + "bbox": [ + 174, + 102, + 532, + 117 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "In this appendix, we run gradient descent until reaching the Edge of Stability, and then we cut the step size. We will see that the sharpness starts increasing as soon as the step size is cut, and only stops increasing once gradient descent is back at the Edge of Stability (or training is finished). As a consequence of this experiment, one can interpret the Edge of Stability as a regime in which gradient descent is constantly “trying” to increase the sharpness beyond $2 / \\eta$ , but is constantly being blocked from doing so. Our experiments focus on image classification on a $5 \\mathrm { k }$ -sized subset of CIFAR-10. We study two architectures (a fully-connected tanh network and a convolutional ReLU network) and two loss functions (squared loss and cross-entropy loss). ", + "bbox": [ + 174, + 133, + 825, + 246 + ], + "page_idx": 74 + }, + { + "type": "image", + "img_path": "images/63079ee2bf16f1303274a111a15b75f6e37ada1fe66fe1fef66f1cb5d153e95a.jpg", + "image_caption": [ + "Figure 111: Top row: we train a network using gradient descent at a range of step sizes $\\eta$ (see the legend in the right pane). Bottom three rows: Once gradient descent has reached the Edge of Stability, we cut the step size. The black vertical dotted line marks the iteration where the step size is cut. This iteration was chosen by hand, but was not cherry-picked. We use different colors (consistent with the legend) to plot the train loss, sharpness, and train accuracy before and after the learning rate drop. In the middle sharpness plot, the two horizontal lines mark the maximum stable sharpness $2 / \\eta$ for both the old and new step size. Takeaway: once we decrease the step size, the sharpness immediately starts to increase until it reaches the maximum stable sharpness $2 / \\eta$ for the new step size $\\eta$ (or training finishes). " + ], + "image_footnote": [], + "bbox": [ + 179, + 136, + 857, + 618 + ], + "page_idx": 75 + }, + { + "type": "image", + "img_path": "images/e08ca739281869a73b1c196159c8d03258ad2755b0bdd25baec5ac29e060ff67.jpg", + "image_caption": [ + "Figure 112: Refer to the caption for Figure 111. " + ], + "image_footnote": [], + "bbox": [ + 179, + 142, + 857, + 744 + ], + "page_idx": 76 + }, + { + "type": "image", + "img_path": "images/2aa1ff8f8343899eb878ab9e97c7a19948fb4248c1d7567ef426779380863791.jpg", + "image_caption": [ + "Figure 113: Refer to the caption for Figure 111. " + ], + "image_footnote": [], + "bbox": [ + 179, + 136, + 857, + 618 + ], + "page_idx": 77 + }, + { + "type": "image", + "img_path": "images/d2a98f1537dc53f080bbd9d04ff78a1b399991fb816ab1e705225c67befad5fd.jpg", + "image_caption": [ + "Figure 114: Refer to the caption for Figure 111. " + ], + "image_footnote": [], + "bbox": [ + 179, + 135, + 857, + 617 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "P OTHER EIGENVALUES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 387, + 117 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Throughout most of this paper, we have studied the evolution of the maximum Hessian eigenvalue during gradient descent. In this appendix, we examine the evolution of the top six eigenvalues. While training the network from $\\ S 3$ , we monitor the evolution of the top six Hessian eigenvalues. For each of cross-entropy loss and MSE loss, we train at four different step sizes. The results are shown in Figure 115. Observe that each of the top six eigenvalues rises and then plateaus. The precise details differ between MSE loss and cross-entropy loss. For MSE loss, each eigenvalue rises past $2 / \\eta$ , and then plateaus just above that value. In contrast, for cross-entropy loss, some of the lesser eigenvalues plateau below $2 / \\eta$ . ", + "bbox": [ + 173, + 133, + 826, + 246 + ], + "page_idx": 79 + }, + { + "type": "image", + "img_path": "images/b4f593a216377f4dd8239fb942354819c685c31f0dfada89c725a888501f40d1.jpg", + "image_caption": [ + "Figure 115: Evolution of top six eigenvalues. For both cross-entropy loss (top) and MSE loss (bottom), we run gradient descent at four different step sizes (columns), and plot the evolution of the top six eigenvalues (different colors; see legend). " + ], + "image_footnote": [], + "bbox": [ + 171, + 266, + 831, + 465 + ], + "page_idx": 79 + } +] \ No newline at end of file diff --git a/parse/train/jh-rTtvkGeM/jh-rTtvkGeM_middle.json b/parse/train/jh-rTtvkGeM/jh-rTtvkGeM_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..dcb306b36f7997bcca5ca46f9e81e1e6d02436ab --- /dev/null +++ b/parse/train/jh-rTtvkGeM/jh-rTtvkGeM_middle.json @@ -0,0 +1,141604 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 502, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 504, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 504, + 97 + ], + "score": 1.0, + "content": "GRADIENT DESCENT ON NEURAL NETWORKS TYPI-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 99, + 435, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 435, + 118 + ], + "score": 1.0, + "content": "CALLY OCCURS AT THE EDGE OF STABILITY", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 134, + 474, + 169 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 474, + 147 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 474, + 147 + ], + "score": 1.0, + "content": "Jeremy Cohen Simran Kaur Yuanzhi Li J. Zico Kolter1 and Ameet Talwalkar2", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 365, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 365, + 158 + ], + "score": 1.0, + "content": "Carnegie Mellon University and: 1Bosch AI 2 Determined AI", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 113, + 158, + 309, + 170 + ], + "spans": [ + { + "bbox": [ + 113, + 158, + 309, + 170 + ], + "score": 1.0, + "content": "Correspondence to: jeremycohen@cmu.edu", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 210 + ], + "lines": [ + { + "bbox": [ + 276, + 197, + 335, + 211 + ], + "spans": [ + { + "bbox": [ + 276, + 197, + 335, + 211 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 223, + 468, + 322 + ], + "lines": [ + { + "bbox": [ + 142, + 223, + 469, + 235 + ], + "spans": [ + { + "bbox": [ + 142, + 223, + 469, + 235 + ], + "score": 1.0, + "content": "We empirically demonstrate that full-batch gradient descent on neural network", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 469, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 469, + 247 + ], + "score": 1.0, + "content": "training objectives typically operates in a regime we call the Edge of Stability.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 470, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 470, + 257 + ], + "score": 1.0, + "content": "In this regime, the maximum eigenvalue of the training loss Hessian hovers just", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "score": 1.0, + "content": "above the value 2/(step size), and the training loss behaves non-monotonically", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "over short timescales, yet consistently decreases over long timescales. Since this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 469, + 290 + ], + "score": 1.0, + "content": "behavior is inconsistent with several widespread presumptions in the field of op-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "score": 1.0, + "content": "timization, our findings raise questions as to whether these presumptions are rel-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 469, + 313 + ], + "score": 1.0, + "content": "evant to neural network training. We hope that our findings will inspire future", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 459, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 459, + 324 + ], + "score": 1.0, + "content": "efforts aimed at rigorously understanding optimization at the Edge of Stability.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 109, + 343, + 206, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 208, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 208, + 358 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "Neural networks are almost never trained using (full-batch) gradient descent, even though gradient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "descent is the conceptual basis for popular optimization algorithms such as SGD. In this paper, we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "train neural networks using gradient descent, and find two surprises. 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Second, this characterization is strongly at odds with prevailing beliefs in optimization.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 429, + 452 + ], + "score": 1.0, + "content": "In more detail, as we train neural networks using gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 430, + 442, + 436, + 451 + ], + "score": 0.68, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 440, + 505, + 452 + ], + "score": 1.0, + "content": ", we measure the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "evolution of the sharpness — the maximum eigenvalue of the training loss Hessian. Empirically,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "the behavior of the sharpness is consistent across architectures and tasks: so long as the sharpness", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 192, + 485 + ], + "score": 1.0, + "content": "is less than the value", + "type": "text" + }, + { + "bbox": [ + 193, + 473, + 209, + 484 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 472, + 505, + 485 + ], + "score": 1.0, + "content": ", it tends to continually rise (§3.1). We call this phenomenon progressive", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 281, + 496 + ], + "score": 1.0, + "content": "sharpening. The significance of the value", + "type": "text" + }, + { + "bbox": [ + 282, + 484, + 299, + 495 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "is that gradient descent on quadratic objectives is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "unstable if the sharpness exceeds this threshold (§2). Indeed, in neural network training, if the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 200, + 519 + ], + "score": 1.0, + "content": "sharpness ever crosses", + "type": "text" + }, + { + "bbox": [ + 201, + 505, + 217, + 517 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 505, + 506, + 519 + ], + "score": 1.0, + "content": ", gradient descent quickly becomes destabilized — that is, the iterates", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "start to oscillate with ever-increasing magnitude along the direction of greatest curvature. 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On three separate architec-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 317, + 703 + ], + "score": 1.0, + "content": "tures, we run gradient descent at a range of step sizes", + "type": "text" + }, + { + "bbox": [ + 318, + 693, + 324, + 702 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 690, + 506, + 703 + ], + "score": 1.0, + "content": ", and plot both the train loss (top row) and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 278, + 714 + ], + "score": 1.0, + "content": "sharpness (bottom row). 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Zico Kolter1 and Ameet Talwalkar2", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 146, + 365, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 365, + 158 + ], + "score": 1.0, + "content": "Carnegie Mellon University and: 1Bosch AI 2 Determined AI", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 113, + 158, + 309, + 170 + ], + "spans": [ + { + "bbox": [ + 113, + 158, + 309, + 170 + ], + "score": 1.0, + "content": "Correspondence to: jeremycohen@cmu.edu", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + } + ], + "index": 3, + "bbox_fs": [ + 111, + 134, + 474, + 170 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 198, + 333, + 210 + ], + "lines": [ + { + "bbox": [ + 276, + 197, + 335, + 211 + ], + "spans": [ + { + "bbox": [ + 276, + 197, + 335, + 211 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 223, + 468, + 322 + ], + "lines": [ + { + "bbox": [ + 142, + 223, + 469, + 235 + ], + "spans": [ + { + "bbox": [ + 142, + 223, + 469, + 235 + ], + "score": 1.0, + "content": "We empirically demonstrate that full-batch gradient descent on neural network", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 469, + 247 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 469, + 247 + ], + "score": 1.0, + "content": "training objectives typically operates in a regime we call the Edge of Stability.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 244, + 470, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 470, + 257 + ], + "score": 1.0, + "content": "In this regime, the maximum eigenvalue of the training loss Hessian hovers just", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "score": 1.0, + "content": "above the value 2/(step size), and the training loss behaves non-monotonically", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "over short timescales, yet consistently decreases over long timescales. Since this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 469, + 290 + ], + "score": 1.0, + "content": "behavior is inconsistent with several widespread presumptions in the field of op-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "score": 1.0, + "content": "timization, our findings raise questions as to whether these presumptions are rel-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 469, + 313 + ], + "score": 1.0, + "content": "evant to neural network training. We hope that our findings will inspire future", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 459, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 459, + 324 + ], + "score": 1.0, + "content": "efforts aimed at rigorously understanding optimization at the Edge of Stability.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 141, + 223, + 470, + 324 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 343, + 206, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 208, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 208, + 358 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "Neural networks are almost never trained using (full-batch) gradient descent, even though gradient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "descent is the conceptual basis for popular optimization algorithms such as SGD. In this paper, we", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "train neural networks using gradient descent, and find two surprises. First, while little is known", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "about the dynamics of neural network training in general, we find that in the special case of gradient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "descent, there is a simple characterization that holds across a broad range of network architectures", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 422, + 498, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 498, + 435 + ], + "score": 1.0, + "content": "and tasks. Second, this characterization is strongly at odds with prevailing beliefs in optimization.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 367, + 506, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 429, + 452 + ], + "score": 1.0, + "content": "In more detail, as we train neural networks using gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 430, + 442, + 436, + 451 + ], + "score": 0.68, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 440, + 505, + 452 + ], + "score": 1.0, + "content": ", we measure the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "evolution of the sharpness — the maximum eigenvalue of the training loss Hessian. Empirically,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "the behavior of the sharpness is consistent across architectures and tasks: so long as the sharpness", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 192, + 485 + ], + "score": 1.0, + "content": "is less than the value", + "type": "text" + }, + { + "bbox": [ + 193, + 473, + 209, + 484 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 472, + 505, + 485 + ], + "score": 1.0, + "content": ", it tends to continually rise (§3.1). We call this phenomenon progressive", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 281, + 496 + ], + "score": 1.0, + "content": "sharpening. The significance of the value", + "type": "text" + }, + { + "bbox": [ + 282, + 484, + 299, + 495 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "is that gradient descent on quadratic objectives is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "unstable if the sharpness exceeds this threshold (§2). Indeed, in neural network training, if the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 200, + 519 + ], + "score": 1.0, + "content": "sharpness ever crosses", + "type": "text" + }, + { + "bbox": [ + 201, + 505, + 217, + 517 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 505, + 506, + 519 + ], + "score": 1.0, + "content": ", gradient descent quickly becomes destabilized — that is, the iterates", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "start to oscillate with ever-increasing magnitude along the direction of greatest curvature. Yet once", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "this happens, gradient descent does not diverge entirely or stall. Instead, it enters a new regime", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "we call the Edge of Stability1 (§3.2), in which (1) the sharpness hovers right at, or just above, the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 131, + 118 + ], + "score": 1.0, + "content": "value", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 131, + 104, + 148, + 117 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 148, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "; and (2) the train loss behaves non-monotonically, yet consistently decreases over long", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "timescales. 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On three separate architec-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 317, + 703 + ], + "score": 1.0, + "content": "tures, we run gradient descent at a range of step sizes", + "type": "text" + }, + { + "bbox": [ + 318, + 693, + 324, + 702 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 690, + 506, + 703 + ], + "score": 1.0, + "content": ", and plot both the train loss (top row) and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 278, + 714 + ], + "score": 1.0, + "content": "sharpness (bottom row). For each step size", + "type": "text" + }, + { + "bbox": [ + 278, + 703, + 285, + 713 + ], + "score": 0.75, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 702, + 425, + 714 + ], + "score": 1.0, + "content": ", observe that the sharpness rises to", + "type": "text" + }, + { + "bbox": [ + 425, + 702, + 442, + 713 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 702, + 506, + 714 + ], + "score": 1.0, + "content": "(marked by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 711, + 494, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 494, + 725 + ], + "score": 1.0, + "content": "horizontal dashed line of the appropriate color) and then hovers right at, or just above, this value.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + } + ], + "index": 32.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "this happens, gradient descent does not diverge entirely or stall. Instead, it enters a new regime", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "we call the Edge of Stability1 (§3.2), in which (1) the sharpness hovers right at, or just above, the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 131, + 118 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 131, + 104, + 148, + 117 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "; and (2) the train loss behaves non-monotonically, yet consistently decreases over long", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "timescales. In this regime, gradient descent is constantly “trying” to increase the sharpness, but is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "constantly restrained from doing so. The net effect is that gradient descent continues to successfully", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 486, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 486, + 150 + ], + "score": 1.0, + "content": "optimize the training objective, but in such a way as to avoid further increasing the sharpness.2", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 347, + 167 + ], + "score": 1.0, + "content": "In principle, it is possible to run gradient descent at step sizes", + "type": "text" + }, + { + "bbox": [ + 347, + 156, + 354, + 166 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "so small that the sharpness never rises", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 116, + 178 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 165, + 133, + 177 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 165, + 505, + 178 + ], + "score": 1.0, + "content": ". However, these step sizes are suboptimal from the point of view of training speed, sometimes", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "score": 1.0, + "content": "dramatically so. In particular, for standard architectures on the standard dataset CIFAR-10, such step", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "sizes are so small as to be completely unreasonable — at all reasonable step sizes, gradient descent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 280, + 211 + ], + "score": 1.0, + "content": "eventually enters the Edge of Stability (see", + "type": "text" + }, + { + "bbox": [ + 281, + 199, + 293, + 209 + ], + "score": 0.76, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "). Thus, at least for standard networks on CIFAR-10,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 428, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 428, + 222 + ], + "score": 1.0, + "content": "the Edge of Stability regime should be viewed as the “rule,” not the “exception.”", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 178, + 238 + ], + "score": 1.0, + "content": "As we describe in", + "type": "text" + }, + { + "bbox": [ + 179, + 226, + 190, + 237 + ], + "score": 0.63, + "content": "\\ S 5", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 226, + 505, + 238 + ], + "score": 1.0, + "content": ", the Edge of Stability regime is inconsistent with several pieces of conventional", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 358, + 250 + ], + "score": 1.0, + "content": "wisdom in optimization theory: convergence analyses based on", + "type": "text" + }, + { + "bbox": [ + 359, + 237, + 367, + 247 + ], + "score": 0.71, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "-smoothness or monotone descent,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "quadratic Taylor approximations as a model for local progress, and certain heuristics for step size", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "selection. We hope that our empirical findings will both nudge the optimization community away", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "from widespread presumptions that appear to be untrue in the case of neural network training, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 496, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 496, + 294 + ], + "score": 1.0, + "content": "also point the way forward by identifying precise empirical phenomena suitable for further study.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "Certain aspects of the Edge of Stability have been observed in previous empirical studies of full-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "batch gradient descent (Xing et al., 2018; Wu et al., 2018); our paper provides a unified explanation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "for these observations. Furthermore, Jastrz˛ebski et al. (2020) proposed a simplified model for the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 331, + 504, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 504, + 343 + ], + "score": 1.0, + "content": "evolution of the sharpness during stochastic gradient descent which matches our empirical observa-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "tions in the special case of full-batch SGD (i.e. gradient descent). However, outside the full-batch", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "special case, there is no evidence that their model matches experiments with any degree of quanti-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 362, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 378 + ], + "score": 1.0, + "content": "tative precision, although their model does successfully predict the directional trend that large step", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 388 + ], + "score": 1.0, + "content": "sizes and/or small batch sizes steer SGD into regions of low sharpness. We discuss SGD at greater", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 145, + 398 + ], + "score": 1.0, + "content": "length in", + "type": "text" + }, + { + "bbox": [ + 145, + 385, + 157, + 397 + ], + "score": 0.74, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 385, + 504, + 398 + ], + "score": 1.0, + "content": ". To summarize, while the sharpness does not obey simple dynamics during SGD (as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "it does during GD), there are indications that the “Edge of Stability” intuition might generalize", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 408, + 403, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 403, + 420 + ], + "score": 1.0, + "content": "somehow to SGD, just in a way that does not center around the sharpness.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 106, + 439, + 478, + 453 + ], + "lines": [ + { + "bbox": [ + 104, + 437, + 480, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 480, + 456 + ], + "score": 1.0, + "content": "2 BACKGROUND: STABILITY OF GRADIENT DESCENT ON QUADRATICS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "In this section, we review the stability properties of gradient descent on quadratic functions. Later,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "we will see that the stability of gradient descent on neural training objectives is partly well-modeled", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 490, + 405, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 405, + 503 + ], + "score": 1.0, + "content": "by the stability of gradient descent on the quadratic Taylor approximation.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 504, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 245, + 521 + ], + "score": 1.0, + "content": "On a quadratic objective function", + "type": "text" + }, + { + "bbox": [ + 246, + 506, + 366, + 520 + ], + "score": 0.95, + "content": "\\begin{array} { r } { f ( \\mathbf { x } ) = \\frac { 1 } { 2 } \\mathbf { x } ^ { T } \\mathbf { A } \\mathbf { x } + \\mathbf { b } ^ { T } \\mathbf { x } + c , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 506, + 497, + 521 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 497, + 510, + 504, + 519 + ], + "score": 0.73, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 343, + 533 + ], + "score": 1.0, + "content": "will diverge if3 any eigenvalue of A exceeds the threshold", + "type": "text" + }, + { + "bbox": [ + 343, + 520, + 360, + 532 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 519, + 505, + 533 + ], + "score": 1.0, + "content": ". To see why, consider first the one-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 197, + 544 + ], + "score": 1.0, + "content": "dimensional quadratic", + "type": "text" + }, + { + "bbox": [ + 198, + 530, + 291, + 544 + ], + "score": 0.92, + "content": "\\textstyle f ( x ) = { \\frac { 1 } { 2 } } a x ^ { 2 } + b x + c", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 530, + 315, + 544 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 315, + 532, + 340, + 541 + ], + "score": 0.89, + "content": "a > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 530, + 454, + 544 + ], + "score": 1.0, + "content": ". This function has optimum", + "type": "text" + }, + { + "bbox": [ + 454, + 531, + 501, + 543 + ], + "score": 0.91, + "content": "x ^ { * } = - b / a", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 530, + 505, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 309, + 555 + ], + "score": 1.0, + "content": "Consider running gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 309, + 544, + 316, + 553 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 541, + 374, + 555 + ], + "score": 1.0, + "content": "starting from", + "type": "text" + }, + { + "bbox": [ + 374, + 543, + 385, + 553 + ], + "score": 0.83, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 541, + 471, + 555 + ], + "score": 1.0, + "content": ". The update rule is", + "type": "text" + }, + { + "bbox": [ + 471, + 544, + 505, + 553 + ], + "score": 0.86, + "content": "x _ { t + 1 } =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 107, + 553, + 173, + 565 + ], + "score": 0.92, + "content": "x _ { t } - \\eta ( a x _ { t } + b )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 552, + 285, + 566 + ], + "score": 1.0, + "content": ", which means that the error", + "type": "text" + }, + { + "bbox": [ + 285, + 554, + 318, + 564 + ], + "score": 0.91, + "content": "x _ { t } - x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 552, + 363, + 566 + ], + "score": 1.0, + "content": "evolves as", + "type": "text" + }, + { + "bbox": [ + 363, + 553, + 501, + 565 + ], + "score": 0.92, + "content": "( x _ { t + 1 } - x ^ { * } ) = ( 1 - \\eta a ) ( x _ { t } - x ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 552, + 506, + 566 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 220, + 577 + ], + "score": 1.0, + "content": "Therefore, the error at step", + "type": "text" + }, + { + "bbox": [ + 221, + 565, + 226, + 574 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 563, + 238, + 577 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 239, + 564, + 379, + 575 + ], + "score": 0.87, + "content": "( x _ { t } - x ^ { * } ) = ( 1 - \\eta a ) ^ { t } ( x _ { 0 } - \\dot { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 563, + 488, + 577 + ], + "score": 1.0, + "content": ", and so the iterate at step", + "type": "text" + }, + { + "bbox": [ + 489, + 566, + 494, + 574 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 228, + 586 + ], + "score": 0.91, + "content": "x _ { t } = ( 1 - \\eta a ) ^ { t } ( x _ { 0 } - x ^ { * } ) \\bar { + } x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 574, + 242, + 588 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 242, + 575, + 277, + 587 + ], + "score": 0.88, + "content": "a > 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 574, + 300, + 588 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 301, + 576, + 361, + 587 + ], + "score": 0.88, + "content": "( 1 - \\eta a ) < - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 574, + 430, + 588 + ], + "score": 1.0, + "content": ", so the sequence", + "type": "text" + }, + { + "bbox": [ + 430, + 574, + 450, + 587 + ], + "score": 0.92, + "content": "\\{ x _ { t } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "will oscillate", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 585, + 331, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 136, + 600 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 137, + 587, + 148, + 596 + ], + "score": 0.85, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 585, + 331, + 600 + ], + "score": 1.0, + "content": "with ever-increasing magnitude, and diverge.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 504, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 210, + 616 + ], + "score": 1.0, + "content": "Now consider the general", + "type": "text" + }, + { + "bbox": [ + 210, + 603, + 217, + 613 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 602, + 308, + 616 + ], + "score": 1.0, + "content": "-dimensional case. Let", + "type": "text" + }, + { + "bbox": [ + 308, + 603, + 339, + 614 + ], + "score": 0.92, + "content": "\\left( a _ { i } , \\mathbf { q } _ { i } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 602, + 366, + 616 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 366, + 604, + 370, + 613 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 602, + 506, + 616 + ], + "score": 1.0, + "content": "-th largest eigenvalue/eigenvector", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 118, + 627 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 614, + 127, + 624 + ], + "score": 0.31, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 612, + 228, + 627 + ], + "score": 1.0, + "content": ". As shown in Appendix", + "type": "text" + }, + { + "bbox": [ + 228, + 614, + 236, + 624 + ], + "score": 0.27, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 612, + 380, + 627 + ], + "score": 1.0, + "content": ", when the gradient descent iterates", + "type": "text" + }, + { + "bbox": [ + 381, + 613, + 401, + 626 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 612, + 505, + 627 + ], + "score": 1.0, + "content": "are expressed in the spe-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "score": 1.0, + "content": "cial coordinate system whose axes are the eigenvectors of A, each coordinate evolves separately.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 635, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 304, + 649 + ], + "score": 1.0, + "content": "In particular, the coordinate for each eigenvector", + "type": "text" + }, + { + "bbox": [ + 305, + 637, + 316, + 647 + ], + "score": 0.86, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 635, + 352, + 649 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + }, + { + "bbox": [ + 353, + 636, + 384, + 648 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { i } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 635, + 505, + 649 + ], + "score": 1.0, + "content": ", evolves according to the dy-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 491, + 660 + ], + "score": 1.0, + "content": "namics of gradient descent on a one-dimensional quadratic objective with second derivative", + "type": "text" + }, + { + "bbox": [ + 491, + 649, + 501, + 658 + ], + "score": 0.83, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 646, + 506, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 669, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 668, + 362, + 681 + ], + "spans": [ + { + "bbox": [ + 119, + 668, + 362, + 681 + ], + "score": 1.0, + "content": "1This nomenclature was inspired by the title of Giladi et al. (2020).", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 119, + 680, + 132, + 690 + ], + "score": 0.28, + "content": "^ 2 \\mathrm { I n }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "the literature, the term “sharpness” has been used to refer to a variety of quantities, often connected to", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "generalization (e.g. Keskar et al. (2016)). In this paper, “sharpness” strictly means the maximum eigenvalue of", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 700, + 459, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 459, + 712 + ], + "score": 1.0, + "content": "the training loss Hessian. We do not claim that this quantity has any connection to generalization.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "3For convex quadratics, this is “if and only if.” However, if A has a negative eigenvalue, then gradient", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 417, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 417, + 734 + ], + "score": 1.0, + "content": "descent with any (positive) step size will diverge along the corresponding eigenvector.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 150 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 347, + 167 + ], + "score": 1.0, + "content": "In principle, it is possible to run gradient descent at step sizes", + "type": "text" + }, + { + "bbox": [ + 347, + 156, + 354, + 166 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "so small that the sharpness never rises", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 116, + 178 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 165, + 133, + 177 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 165, + 505, + 178 + ], + "score": 1.0, + "content": ". However, these step sizes are suboptimal from the point of view of training speed, sometimes", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 190 + ], + "score": 1.0, + "content": "dramatically so. In particular, for standard architectures on the standard dataset CIFAR-10, such step", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "sizes are so small as to be completely unreasonable — at all reasonable step sizes, gradient descent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 280, + 211 + ], + "score": 1.0, + "content": "eventually enters the Edge of Stability (see", + "type": "text" + }, + { + "bbox": [ + 281, + 199, + 293, + 209 + ], + "score": 0.76, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "). Thus, at least for standard networks on CIFAR-10,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 428, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 428, + 222 + ], + "score": 1.0, + "content": "the Edge of Stability regime should be viewed as the “rule,” not the “exception.”", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 154, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 178, + 238 + ], + "score": 1.0, + "content": "As we describe in", + "type": "text" + }, + { + "bbox": [ + 179, + 226, + 190, + 237 + ], + "score": 0.63, + "content": "\\ S 5", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 226, + 505, + 238 + ], + "score": 1.0, + "content": ", the Edge of Stability regime is inconsistent with several pieces of conventional", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 358, + 250 + ], + "score": 1.0, + "content": "wisdom in optimization theory: convergence analyses based on", + "type": "text" + }, + { + "bbox": [ + 359, + 237, + 367, + 247 + ], + "score": 0.71, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "-smoothness or monotone descent,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "quadratic Taylor approximations as a model for local progress, and certain heuristics for step size", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "score": 1.0, + "content": "selection. We hope that our empirical findings will both nudge the optimization community away", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "from widespread presumptions that appear to be untrue in the case of neural network training, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 496, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 496, + 294 + ], + "score": 1.0, + "content": "also point the way forward by identifying precise empirical phenomena suitable for further study.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 226, + 506, + 294 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "Certain aspects of the Edge of Stability have been observed in previous empirical studies of full-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "batch gradient descent (Xing et al., 2018; Wu et al., 2018); our paper provides a unified explanation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "for these observations. Furthermore, Jastrz˛ebski et al. (2020) proposed a simplified model for the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 331, + 504, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 504, + 343 + ], + "score": 1.0, + "content": "evolution of the sharpness during stochastic gradient descent which matches our empirical observa-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "tions in the special case of full-batch SGD (i.e. gradient descent). However, outside the full-batch", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "special case, there is no evidence that their model matches experiments with any degree of quanti-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 362, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 378 + ], + "score": 1.0, + "content": "tative precision, although their model does successfully predict the directional trend that large step", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 373, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 388 + ], + "score": 1.0, + "content": "sizes and/or small batch sizes steer SGD into regions of low sharpness. We discuss SGD at greater", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 145, + 398 + ], + "score": 1.0, + "content": "length in", + "type": "text" + }, + { + "bbox": [ + 145, + 385, + 157, + 397 + ], + "score": 0.74, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 385, + 504, + 398 + ], + "score": 1.0, + "content": ". To summarize, while the sharpness does not obey simple dynamics during SGD (as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "it does during GD), there are indications that the “Edge of Stability” intuition might generalize", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 408, + 403, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 403, + 420 + ], + "score": 1.0, + "content": "somehow to SGD, just in a way that does not center around the sharpness.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 298, + 506, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 439, + 478, + 453 + ], + "lines": [ + { + "bbox": [ + 104, + 437, + 480, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 480, + 456 + ], + "score": 1.0, + "content": "2 BACKGROUND: STABILITY OF GRADIENT DESCENT ON QUADRATICS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "In this section, we review the stability properties of gradient descent on quadratic functions. Later,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "we will see that the stability of gradient descent on neural training objectives is partly well-modeled", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 490, + 405, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 405, + 503 + ], + "score": 1.0, + "content": "by the stability of gradient descent on the quadratic Taylor approximation.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 468, + 505, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 504, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 245, + 521 + ], + "score": 1.0, + "content": "On a quadratic objective function", + "type": "text" + }, + { + "bbox": [ + 246, + 506, + 366, + 520 + ], + "score": 0.95, + "content": "\\begin{array} { r } { f ( \\mathbf { x } ) = \\frac { 1 } { 2 } \\mathbf { x } ^ { T } \\mathbf { A } \\mathbf { x } + \\mathbf { b } ^ { T } \\mathbf { x } + c , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 506, + 497, + 521 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 497, + 510, + 504, + 519 + ], + "score": 0.73, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 519, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 343, + 533 + ], + "score": 1.0, + "content": "will diverge if3 any eigenvalue of A exceeds the threshold", + "type": "text" + }, + { + "bbox": [ + 343, + 520, + 360, + 532 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 519, + 505, + 533 + ], + "score": 1.0, + "content": ". To see why, consider first the one-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 530, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 197, + 544 + ], + "score": 1.0, + "content": "dimensional quadratic", + "type": "text" + }, + { + "bbox": [ + 198, + 530, + 291, + 544 + ], + "score": 0.92, + "content": "\\textstyle f ( x ) = { \\frac { 1 } { 2 } } a x ^ { 2 } + b x + c", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 530, + 315, + 544 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 315, + 532, + 340, + 541 + ], + "score": 0.89, + "content": "a > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 530, + 454, + 544 + ], + "score": 1.0, + "content": ". This function has optimum", + "type": "text" + }, + { + "bbox": [ + 454, + 531, + 501, + 543 + ], + "score": 0.91, + "content": "x ^ { * } = - b / a", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 530, + 505, + 544 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 309, + 555 + ], + "score": 1.0, + "content": "Consider running gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 309, + 544, + 316, + 553 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 541, + 374, + 555 + ], + "score": 1.0, + "content": "starting from", + "type": "text" + }, + { + "bbox": [ + 374, + 543, + 385, + 553 + ], + "score": 0.83, + "content": "x _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 541, + 471, + 555 + ], + "score": 1.0, + "content": ". The update rule is", + "type": "text" + }, + { + "bbox": [ + 471, + 544, + 505, + 553 + ], + "score": 0.86, + "content": "x _ { t + 1 } =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 107, + 553, + 173, + 565 + ], + "score": 0.92, + "content": "x _ { t } - \\eta ( a x _ { t } + b )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 552, + 285, + 566 + ], + "score": 1.0, + "content": ", which means that the error", + "type": "text" + }, + { + "bbox": [ + 285, + 554, + 318, + 564 + ], + "score": 0.91, + "content": "x _ { t } - x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 552, + 363, + 566 + ], + "score": 1.0, + "content": "evolves as", + "type": "text" + }, + { + "bbox": [ + 363, + 553, + 501, + 565 + ], + "score": 0.92, + "content": "( x _ { t + 1 } - x ^ { * } ) = ( 1 - \\eta a ) ( x _ { t } - x ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 552, + 506, + 566 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 220, + 577 + ], + "score": 1.0, + "content": "Therefore, the error at step", + "type": "text" + }, + { + "bbox": [ + 221, + 565, + 226, + 574 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 563, + 238, + 577 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 239, + 564, + 379, + 575 + ], + "score": 0.87, + "content": "( x _ { t } - x ^ { * } ) = ( 1 - \\eta a ) ^ { t } ( x _ { 0 } - \\dot { x } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 563, + 488, + 577 + ], + "score": 1.0, + "content": ", and so the iterate at step", + "type": "text" + }, + { + "bbox": [ + 489, + 566, + 494, + 574 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 228, + 586 + ], + "score": 0.91, + "content": "x _ { t } = ( 1 - \\eta a ) ^ { t } ( x _ { 0 } - x ^ { * } ) \\bar { + } x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 574, + 242, + 588 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 308, + 603, + 339, + 614 + ], + "score": 0.92, + "content": "\\left( a _ { i } , \\mathbf { q } _ { i } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 602, + 366, + 616 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 366, + 604, + 370, + 613 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 602, + 506, + 616 + ], + "score": 1.0, + "content": "-th largest eigenvalue/eigenvector", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 118, + 627 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 614, + 127, + 624 + ], + "score": 0.31, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 612, + 228, + 627 + ], + "score": 1.0, + "content": ". As shown in Appendix", + "type": "text" + }, + { + "bbox": [ + 228, + 614, + 236, + 624 + ], + "score": 0.27, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 612, + 380, + 627 + ], + "score": 1.0, + "content": ", when the gradient descent iterates", + "type": "text" + }, + { + "bbox": [ + 381, + 613, + 401, + 626 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 612, + 505, + 627 + ], + "score": 1.0, + "content": "are expressed in the spe-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 638 + ], + "score": 1.0, + "content": "cial coordinate system whose axes are the eigenvectors of A, each coordinate evolves separately.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 635, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 304, + 649 + ], + "score": 1.0, + "content": "In particular, the coordinate for each eigenvector", + "type": "text" + }, + { + "bbox": [ + 305, + 637, + 316, + 647 + ], + "score": 0.86, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 635, + 352, + 649 + ], + "score": 1.0, + "content": ", namely", + "type": "text" + }, + { + "bbox": [ + 353, + 636, + 384, + 648 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { i } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 635, + 505, + 649 + ], + "score": 1.0, + "content": ", evolves according to the dy-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 491, + 660 + ], + "score": 1.0, + "content": "namics of gradient descent on a one-dimensional quadratic objective with second derivative", + "type": "text" + }, + { + "bbox": [ + 491, + 649, + 501, + 658 + ], + "score": 0.83, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 646, + 506, + 660 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 602, + 506, + 660 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 316, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 316, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 159, + 96 + ], + "score": 1.0, + "content": "Therefore, if", + "type": "text" + }, + { + "bbox": [ + 159, + 82, + 197, + 95 + ], + "score": 0.93, + "content": "a _ { i } > 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 80, + 273, + 96 + ], + "score": 1.0, + "content": ", then the sequence", + "type": "text" + }, + { + "bbox": [ + 273, + 82, + 316, + 95 + ], + "score": 0.93, + "content": "\\left\\{ \\left. \\mathbf { q } _ { i } , \\mathbf { x } _ { t } \\right. \\right\\}", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 317, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 317, + 105 + ], + "score": 1.0, + "content": "will oscillate with ever-increasing magnitude; in this", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 317, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 234, + 118 + ], + "score": 1.0, + "content": "case, we say that the iterates", + "type": "text" + }, + { + "bbox": [ + 234, + 104, + 255, + 117 + ], + "score": 0.92, + "content": "\\left\\{ { \\bf x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 103, + 317, + 118 + ], + "score": 1.0, + "content": "diverge along", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 318, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 162, + 127 + ], + "score": 1.0, + "content": "the direction", + "type": "text" + }, + { + "bbox": [ + 163, + 117, + 173, + 127 + ], + "score": 0.85, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 115, + 318, + 127 + ], + "score": 1.0, + "content": ". 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Sutskever et al., 2013)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "are notable variants of gradient descent which often improve the convergence speed. On quadratic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "functions, these two algorithms also diverge if the sharpness exceeds a certain threshold, which we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "call the “maximum stable sharpness,” or MSS. In particular, we prove in Appendix B that gradient", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 253, + 455, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 195, + 265 + ], + "score": 1.0, + "content": "descent with step size", + "type": "text" + }, + { + "bbox": [ + 195, + 255, + 202, + 264 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 253, + 310, + 265 + ], + "score": 1.0, + "content": "and momentum parameter", + "type": "text" + }, + { + "bbox": [ + 311, + 253, + 318, + 264 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 253, + 455, + 265 + ], + "score": 1.0, + "content": "diverges if the sharpness exceeds:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 270, + 447, + 297 + ], + "lines": [ + { + "bbox": [ + 163, + 270, + 447, + 297 + ], + "spans": [ + { + "bbox": [ + 163, + 270, + 447, + 297 + ], + "score": 0.92, + "content": "\\mathbf { M S S } _ { \\mathrm { P o l y a k } } ( \\eta , \\beta ) = \\frac { 1 } { \\eta } \\left( 2 + 2 \\beta \\right) , \\quad \\mathbf { M S S } _ { \\mathrm { N e s t e r o v } } ( \\eta , \\beta ) = \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) .", + "type": "interline_equation", + "image_path": "191d8add630829e8afec6fec3d67a720312e04238cbdb8abeba3bf3daaa9b752.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 163, + 270, + 447, + 279.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 163, + 279.0, + 447, + 288.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 163, + 288.0, + 447, + 297.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "The Polyak result previously appeared in Goh (2017); the Nesterov one seems to be new. Note that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 405, + 325 + ], + "score": 1.0, + "content": "this discussion only applies to full-batch gradient descent. As we discuss in", + "type": "text" + }, + { + "bbox": [ + 406, + 313, + 417, + 324 + ], + "score": 0.8, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 313, + 505, + 325 + ], + "score": 1.0, + "content": ", several recent papers", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 324, + 441, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 441, + 336 + ], + "score": 1.0, + "content": "have proposed stability analyses for SGD (Wu et al., 2018; Jastrz˛ebski et al., 2020).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "Neural network training objectives are not globally quadratic. However, the second-order Taylor", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 353, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 237, + 364 + ], + "score": 1.0, + "content": "approximation around any point", + "type": "text" + }, + { + "bbox": [ + 237, + 353, + 249, + 363 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 353, + 506, + 364 + ], + "score": 1.0, + "content": "in parameter space is a quadratic function whose “A” matrix is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 362, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 165, + 377 + ], + "score": 1.0, + "content": "the Hessian at", + "type": "text" + }, + { + "bbox": [ + 165, + 364, + 177, + 374 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 362, + 351, + 377 + ], + "score": 1.0, + "content": ". If any eigenvalue of this Hessian exceeds", + "type": "text" + }, + { + "bbox": [ + 351, + 363, + 367, + 375 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 362, + 497, + 377 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 497, + 366, + 504, + 375 + ], + "score": 0.68, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "score": 1.0, + "content": "would diverge if run on this quadratic function — the iterates would oscillate with ever-increasing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 390, + 398 + ], + "score": 1.0, + "content": "magnitude along the corresponding eigenvector. Therefore, at any point", + "type": "text" + }, + { + "bbox": [ + 390, + 386, + 402, + 396 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "in parameter space where", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 197, + 409 + ], + "score": 1.0, + "content": "the sharpness exceeds", + "type": "text" + }, + { + "bbox": [ + 198, + 396, + 214, + 408 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 395, + 344, + 409 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 344, + 398, + 351, + 407 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "would diverge if run on the quadratic", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 406, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 322, + 420 + ], + "score": 1.0, + "content": "Taylor approximation to the training objective around", + "type": "text" + }, + { + "bbox": [ + 322, + 408, + 334, + 418 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 406, + 338, + 420 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 361, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 433, + 362, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 362, + 449 + ], + "score": 1.0, + "content": "3 GRADIENT DESCENT ON NEURAL NETWORKS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 503, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "In this section, we empirically characterize the behavior of gradient descent on neural network train-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 470, + 407, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 407, + 483 + ], + "score": 1.0, + "content": "ing objectives. Section 4 will show that this characterization holds broadly.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 108, + 496, + 249, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 250, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 250, + 507 + ], + "score": 1.0, + "content": "3.1 PROGRESSIVE SHARPENING", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "When training neural networks, it seems to be a general rule that so long as the sharpness is small", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 280, + 540 + ], + "score": 1.0, + "content": "enough for gradient descent to be stable", + "type": "text" + }, + { + "bbox": [ + 280, + 527, + 311, + 539 + ], + "score": 0.9, + "content": "( < 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 527, + 505, + 540 + ], + "score": 1.0, + "content": ", for vanilla gradient descent), gradient descent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "has an overwhelming tendency to continually increase the sharpness. We call this phenomenon", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "progressive sharpening. By “overwhelming tendency,” we mean that gradient descent can occasion-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "ally decrease the sharpness (especially at the beginning of training), but these brief decreases always", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "seem be followed by a return to continual increase. Jastrz˛ebski et al. 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We consider both MSE loss (left) and cross-entropy loss (right).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53 + } + ], + "index": 51.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 316, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 316, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 159, + 96 + ], + "score": 1.0, + "content": "Therefore, if", + "type": "text" + }, + { + "bbox": [ + 159, + 82, + 197, + 95 + ], + "score": 0.93, + "content": "a _ { i } > 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 80, + 273, + 96 + ], + "score": 1.0, + "content": ", then the sequence", + "type": "text" + }, + { + "bbox": [ + 273, + 82, + 316, + 95 + ], + "score": 0.93, + "content": "\\left\\{ \\left. \\mathbf { q } _ { i } , \\mathbf { x } _ { t } \\right. \\right\\}", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 317, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 317, + 105 + ], + "score": 1.0, + "content": "will oscillate with ever-increasing magnitude; in this", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 317, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 234, + 118 + ], + "score": 1.0, + "content": "case, we say that the iterates", + "type": "text" + }, + { + "bbox": [ + 234, + 104, + 255, + 117 + ], + "score": 0.92, + "content": "\\left\\{ { \\bf x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 103, + 317, + 118 + ], + "score": 1.0, + "content": "diverge along", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 318, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 162, + 127 + ], + "score": 1.0, + "content": "the direction", + "type": "text" + }, + { + "bbox": [ + 163, + 117, + 173, + 127 + ], + "score": 0.85, + "content": "\\mathbf { q } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 115, + 318, + 127 + ], + "score": 1.0, + "content": ". To illustrate, Figure 2 shows a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 317, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 257, + 138 + ], + "score": 1.0, + "content": "quadratic function with eigenvalues", + "type": "text" + }, + { + "bbox": [ + 257, + 127, + 298, + 137 + ], + "score": 0.9, + "content": "a _ { 1 } ~ = ~ 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 127, + 317, + 138 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 318, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 141, + 149 + ], + "score": 0.9, + "content": "a _ { 2 } ~ = ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 137, + 318, + 149 + ], + "score": 1.0, + "content": ". 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Sutskever et al., 2013)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "are notable variants of gradient descent which often improve the convergence speed. On quadratic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "functions, these two algorithms also diverge if the sharpness exceeds a certain threshold, which we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "call the “maximum stable sharpness,” or MSS. In particular, we prove in Appendix B that gradient", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 253, + 455, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 195, + 265 + ], + "score": 1.0, + "content": "descent with step size", + "type": "text" + }, + { + "bbox": [ + 195, + 255, + 202, + 264 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 253, + 310, + 265 + ], + "score": 1.0, + "content": "and momentum parameter", + "type": "text" + }, + { + "bbox": [ + 311, + 253, + 318, + 264 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 253, + 455, + 265 + ], + "score": 1.0, + "content": "diverges if the sharpness exceeds:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 208, + 505, + 265 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 270, + 447, + 297 + ], + "lines": [ + { + "bbox": [ + 163, + 270, + 447, + 297 + ], + "spans": [ + { + "bbox": [ + 163, + 270, + 447, + 297 + ], + "score": 0.92, + "content": "\\mathbf { M S S } _ { \\mathrm { P o l y a k } } ( \\eta , \\beta ) = \\frac { 1 } { \\eta } \\left( 2 + 2 \\beta \\right) , \\quad \\mathbf { M S S } _ { \\mathrm { N e s t e r o v } } ( \\eta , \\beta ) = \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) .", + "type": "interline_equation", + "image_path": "191d8add630829e8afec6fec3d67a720312e04238cbdb8abeba3bf3daaa9b752.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 163, + 270, + 447, + 279.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 163, + 279.0, + 447, + 288.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 163, + 288.0, + 447, + 297.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "The Polyak result previously appeared in Goh (2017); the Nesterov one seems to be new. Note that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 405, + 325 + ], + "score": 1.0, + "content": "this discussion only applies to full-batch gradient descent. As we discuss in", + "type": "text" + }, + { + "bbox": [ + 406, + 313, + 417, + 324 + ], + "score": 0.8, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 313, + 505, + 325 + ], + "score": 1.0, + "content": ", several recent papers", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 324, + 441, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 441, + 336 + ], + "score": 1.0, + "content": "have proposed stability analyses for SGD (Wu et al., 2018; Jastrz˛ebski et al., 2020).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 302, + 506, + 336 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "Neural network training objectives are not globally quadratic. However, the second-order Taylor", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 353, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 237, + 364 + ], + "score": 1.0, + "content": "approximation around any point", + "type": "text" + }, + { + "bbox": [ + 237, + 353, + 249, + 363 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 353, + 506, + 364 + ], + "score": 1.0, + "content": "in parameter space is a quadratic function whose “A” matrix is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 362, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 165, + 377 + ], + "score": 1.0, + "content": "the Hessian at", + "type": "text" + }, + { + "bbox": [ + 165, + 364, + 177, + 374 + ], + "score": 0.84, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 362, + 351, + 377 + ], + "score": 1.0, + "content": ". If any eigenvalue of this Hessian exceeds", + "type": "text" + }, + { + "bbox": [ + 351, + 363, + 367, + 375 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 362, + 497, + 377 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 497, + 366, + 504, + 375 + ], + "score": 0.68, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "score": 1.0, + "content": "would diverge if run on this quadratic function — the iterates would oscillate with ever-increasing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 390, + 398 + ], + "score": 1.0, + "content": "magnitude along the corresponding eigenvector. Therefore, at any point", + "type": "text" + }, + { + "bbox": [ + 390, + 386, + 402, + 396 + ], + "score": 0.86, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "in parameter space where", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 197, + 409 + ], + "score": 1.0, + "content": "the sharpness exceeds", + "type": "text" + }, + { + "bbox": [ + 198, + 396, + 214, + 408 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 395, + 344, + 409 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 344, + 398, + 351, + 407 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "would diverge if run on the quadratic", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 406, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 322, + 420 + ], + "score": 1.0, + "content": "Taylor approximation to the training objective around", + "type": "text" + }, + { + "bbox": [ + 322, + 408, + 334, + 418 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 406, + 338, + 420 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 341, + 506, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 361, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 433, + 362, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 362, + 449 + ], + "score": 1.0, + "content": "3 GRADIENT DESCENT ON NEURAL NETWORKS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 503, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "In this section, we empirically characterize the behavior of gradient descent on neural network train-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 470, + 407, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 407, + 483 + ], + "score": 1.0, + "content": "ing objectives. Section 4 will show that this characterization holds broadly.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 459, + 505, + 483 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 496, + 249, + 506 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 250, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 250, + 507 + ], + "score": 1.0, + "content": "3.1 PROGRESSIVE SHARPENING", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "When training neural networks, it seems to be a general rule that so long as the sharpness is small", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 280, + 540 + ], + "score": 1.0, + "content": "enough for gradient descent to be stable", + "type": "text" + }, + { + "bbox": [ + 280, + 527, + 311, + 539 + ], + "score": 0.9, + "content": "( < 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 527, + 505, + 540 + ], + "score": 1.0, + "content": ", for vanilla gradient descent), gradient descent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "has an overwhelming tendency to continually increase the sharpness. We call this phenomenon", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "progressive sharpening. By “overwhelming tendency,” we mean that gradient descent can occasion-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "ally decrease the sharpness (especially at the beginning of training), but these brief decreases always", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "seem be followed by a return to continual increase. Jastrz˛ebski et al. (2020) previously hypothesized", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "(in their Assumption 4) that a similar phenomenon may hold for SGD, but the evidence for, and the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 593, + 473, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 473, + 605 + ], + "score": 1.0, + "content": "precise scope of, this effect are both currently far clearer for gradient descent than for SGD.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 516, + 506, + 605 + ] + }, + { + "type": "image", + "bbox": [ + 110, + 622, + 507, + 684 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 622, + 507, + 684 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 622, + 507, + 684 + ], + "spans": [ + { + "bbox": [ + 110, + 622, + 507, + 684 + ], + "score": 0.96, + "type": "image", + "image_path": "56cc0c19b654465ab0c58a070a13f44506ef5b48cb3127451a26a57a6f217e3d.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 110, + 622, + 507, + 642.6666666666666 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 110, + 642.6666666666666, + 507, + 663.3333333333333 + ], + "spans": [], + "index": 50 + }, + { + "bbox": [ + 110, + 663.3333333333333, + 507, + 683.9999999999999 + ], + "spans": [], + "index": 51 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 695, + 505, + 729 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 694, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 313, + 709 + ], + "score": 1.0, + "content": "Figure 3: So long as the sharpness is less than", + "type": "text" + }, + { + "bbox": [ + 314, + 695, + 330, + 707 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 694, + 505, + 709 + ], + "score": 1.0, + "content": ", it tends to continually increase during", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 707, + 506, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 323, + 719 + ], + "score": 1.0, + "content": "gradient descent. We train a network to completion", + "type": "text" + }, + { + "bbox": [ + 323, + 707, + 343, + 717 + ], + "score": 0.86, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 707, + 506, + 719 + ], + "score": 1.0, + "content": "accuracy) using gradient descent with a", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 717, + 450, + 730 + ], + "spans": [ + { + "bbox": [ + 106, + 717, + 450, + 730 + ], + "score": 1.0, + "content": "very small step size. We consider both MSE loss (left) and cross-entropy loss (right).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53 + } + ], + "index": 51.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Progressive sharpening is illustrated in Figure 3. Here, we use (full-batch) gradient descent to train", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "a network on a subset of 5,000 examples from CIFAR-10, and we monitor the evolution of the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "sharpness during training. The network is a fully-connected architecture with two hidden layers", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "of width 200, and tanh activations. In Figure 3(a), we train using the mean squared error loss for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "classification (Hui & Belkin, 2020), encoding the correct class with 1 and the other classes with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 244, + 150 + ], + "score": 1.0, + "content": "0. We use the small step size of", + "type": "text" + }, + { + "bbox": [ + 244, + 137, + 292, + 149 + ], + "score": 0.92, + "content": "\\eta \\ : = \\ : 2 / 6 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 136, + 481, + 150 + ], + "score": 1.0, + "content": ", and stop when the training accuracy reaches", + "type": "text" + }, + { + "bbox": [ + 482, + 137, + 501, + 148 + ], + "score": 0.89, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 136, + 505, + 150 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "We plot both the train loss and the sharpness, with a horizontal dashed line marking the stability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 146, + 172 + ], + "score": 1.0, + "content": "threshold", + "type": "text" + }, + { + "bbox": [ + 146, + 159, + 163, + 171 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 158, + 506, + 172 + ], + "score": 1.0, + "content": ". Observe that the sharpness continually rises during training (except for a brief dip at", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "score": 1.0, + "content": "the beginning). This is progressive sharpening. For this experiment, we intentionally chose a step", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 488, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 124, + 194 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 183, + 131, + 193 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 181, + 334, + 194 + ], + "score": 1.0, + "content": "small enough that the sharpness remained beneath", + "type": "text" + }, + { + "bbox": [ + 335, + 181, + 351, + 193 + ], + "score": 0.89, + "content": "2 / \\bar { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 181, + 488, + 194 + ], + "score": 1.0, + "content": "for the entire duration of training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Cross-entropy. When training with cross-entropy loss, there is an exception to the rule that the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "sharpness tends to continually increase: with cross-entropy loss, the sharpness typically drops at the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "end of training. This behavior can be seen in Figure 3(b), where we train the same network using the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "cross-entropy loss rather than MSE. This drop occurs because once most data points are classified", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "correctly, gradient descent tries to drive the cross-entropy loss to zero by scaling up the margins, as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 494, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 494, + 266 + ], + "score": 1.0, + "content": "detailed in Soudry et al. (2018). As we explain in Appendix C, this causes the sharpness to drop.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "The effect of width. It is known that when networks parameterized in a certain way (the “NTK", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "parameterization”) are made infinitely wide, the Hessian moves a vanishingly small amount during", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "training (Jacot et al., 2018; Lee et al., 2019; Li & Liang, 2018), which implies that no progressive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "sharpening occurs. In Appendix D, we experiment with networks of varying width, under both NTK", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "score": 1.0, + "content": "and standard parameterizations. We find that progressive sharpening occurs to a lesser degree as net-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 376, + 338 + ], + "score": 1.0, + "content": "works become increasingly wide. Nevertheless, our experiments in", + "type": "text" + }, + { + "bbox": [ + 376, + 325, + 388, + 336 + ], + "score": 0.74, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "demonstrate that progressive", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "sharpening occurs to a dramatic degree for standard architectures on the standard dataset CIFAR-10.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 353, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "We do not know why progressive sharpening occurs, or whether “sharp” solutions differ in any", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "important way from “not sharp” solutions. These are important questions for future work. Note that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 483, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 483, + 387 + ], + "score": 1.0, + "content": "Mulayoff & Michaeli (2020) studied the latter question in the context of deep linear networks.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 402, + 238, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 240, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 240, + 415 + ], + "score": 1.0, + "content": "3.2 THE EDGE OF STABILITY", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 382, + 438 + ], + "score": 1.0, + "content": "In the preceding section, we ran gradient descent using step sizes", + "type": "text" + }, + { + "bbox": [ + 382, + 427, + 389, + 436 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "so small that the sharpness", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 255, + 448 + ], + "score": 1.0, + "content": "never reached the stability threshold", + "type": "text" + }, + { + "bbox": [ + 255, + 435, + 272, + 448 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 436, + 505, + 448 + ], + "score": 1.0, + "content": ". In Figure 4(a), we start to train the same network at the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 183, + 460 + ], + "score": 1.0, + "content": "larger step size of", + "type": "text" + }, + { + "bbox": [ + 183, + 447, + 224, + 458 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 446, + 421, + 460 + ], + "score": 1.0, + "content": ", and pause training once the sharpness rises to", + "type": "text" + }, + { + "bbox": [ + 421, + 446, + 470, + 459 + ], + "score": 0.91, + "content": "2 / \\eta = 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 446, + 506, + 460 + ], + "score": 1.0, + "content": ". Recall", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 456, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 129, + 471 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 130, + 458, + 141, + 468 + ], + "score": 0.8, + "content": "\\ S 2", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 456, + 343, + 471 + ], + "score": 1.0, + "content": "that in any region where the sharpness exceeds", + "type": "text" + }, + { + "bbox": [ + 344, + 457, + 360, + 469 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 456, + 497, + 471 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 497, + 459, + 504, + 469 + ], + "score": 0.69, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "would be unstable if run on the quadratic Taylor approximation to the training objective — the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "gradient descent iterates would oscillate with ever-increasing magnitude along the leading Hessian", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "eigenvector. Empirically, we find that gradient descent on the real neural training objective behaves", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 241, + 514 + ], + "score": 1.0, + "content": "similarly — at first. Namely, let", + "type": "text" + }, + { + "bbox": [ + 242, + 502, + 254, + 513 + ], + "score": 0.81, + "content": "{ \\bf q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "be the leading Hessian eigenvector at the iteration where the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 181, + 526 + ], + "score": 1.0, + "content": "sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 181, + 512, + 198, + 524 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 512, + 506, + 526 + ], + "score": 1.0, + "content": ". In Figure 4(b), we resume training the network, and we monitor both the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 219, + 536 + ], + "score": 1.0, + "content": "train loss and the quantity", + "type": "text" + }, + { + "bbox": [ + 219, + 523, + 252, + 535 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 522, + 429, + 536 + ], + "score": 1.0, + "content": "for the next 215 iterations. Observe that", + "type": "text" + }, + { + "bbox": [ + 430, + 523, + 463, + 535 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "oscillates", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "with ever-increasing magnitude, similar to the divergent quadratic example in Figure 2(b). At first,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "these oscillations are too small to affect the objective appreciably, and so the train loss continues to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "monotonically decrease. But eventually, these oscillations grow big enough that the train loss spikes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33 + }, + { + "type": "image", + "bbox": [ + 110, + 591, + 505, + 672 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 591, + 505, + 672 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 591, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 110, + 591, + 505, + 672 + ], + "score": 0.961, + "type": "image", + "image_path": "d167549c8548e4247c63c0dafe3ea4b11804fc67c00c956313b55282c60c6c65.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 110, + 591, + 505, + 618.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 110, + 618.0, + 505, + 645.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 110, + 645.0, + 505, + 672.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 684, + 505, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 261, + 697 + ], + "score": 1.0, + "content": "Figure 4: Once the sharpness crosses", + "type": "text" + }, + { + "bbox": [ + 262, + 684, + 278, + 696 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 684, + 505, + 697 + ], + "score": 1.0, + "content": ", gradient descent becomes destabilized. We run gradi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 695, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 164, + 708 + ], + "score": 1.0, + "content": "ent descent at", + "type": "text" + }, + { + "bbox": [ + 164, + 696, + 202, + 707 + ], + "score": 0.93, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 695, + 357, + 708 + ], + "score": 1.0, + "content": ". (a) The sharpness eventually reaches", + "type": "text" + }, + { + "bbox": [ + 357, + 695, + 374, + 707 + ], + "score": 0.84, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 695, + 505, + 708 + ], + "score": 1.0, + "content": ". (b) Once the sharpness crosses", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 706, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 706, + 123, + 718 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 706, + 264, + 720 + ], + "score": 1.0, + "content": ", the iterates start to oscillate along", + "type": "text" + }, + { + "bbox": [ + 264, + 708, + 276, + 718 + ], + "score": 0.75, + "content": "{ \\bf q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 706, + 506, + 720 + ], + "score": 1.0, + "content": "with ever-increasing magnitude. (c) Somehow, GD does", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 717, + 471, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 471, + 730 + ], + "score": 1.0, + "content": "not diverge entirely; instead, the train loss continues to decrease, albeit non-monotonically.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "index": 42.75 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Progressive sharpening is illustrated in Figure 3. Here, we use (full-batch) gradient descent to train", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "a network on a subset of 5,000 examples from CIFAR-10, and we monitor the evolution of the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "sharpness during training. The network is a fully-connected architecture with two hidden layers", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "of width 200, and tanh activations. In Figure 3(a), we train using the mean squared error loss for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "classification (Hui & Belkin, 2020), encoding the correct class with 1 and the other classes with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 244, + 150 + ], + "score": 1.0, + "content": "0. We use the small step size of", + "type": "text" + }, + { + "bbox": [ + 244, + 137, + 292, + 149 + ], + "score": 0.92, + "content": "\\eta \\ : = \\ : 2 / 6 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 136, + 481, + 150 + ], + "score": 1.0, + "content": ", and stop when the training accuracy reaches", + "type": "text" + }, + { + "bbox": [ + 482, + 137, + 501, + 148 + ], + "score": 0.89, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 136, + 505, + 150 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "We plot both the train loss and the sharpness, with a horizontal dashed line marking the stability", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 146, + 172 + ], + "score": 1.0, + "content": "threshold", + "type": "text" + }, + { + "bbox": [ + 146, + 159, + 163, + 171 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 158, + 506, + 172 + ], + "score": 1.0, + "content": ". Observe that the sharpness continually rises during training (except for a brief dip at", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 185 + ], + "score": 1.0, + "content": "the beginning). This is progressive sharpening. For this experiment, we intentionally chose a step", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 488, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 124, + 194 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 124, + 183, + 131, + 193 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 181, + 334, + 194 + ], + "score": 1.0, + "content": "small enough that the sharpness remained beneath", + "type": "text" + }, + { + "bbox": [ + 335, + 181, + 351, + 193 + ], + "score": 0.89, + "content": "2 / \\bar { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 181, + 488, + 194 + ], + "score": 1.0, + "content": "for the entire duration of training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Cross-entropy. When training with cross-entropy loss, there is an exception to the rule that the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "sharpness tends to continually increase: with cross-entropy loss, the sharpness typically drops at the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "end of training. This behavior can be seen in Figure 3(b), where we train the same network using the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "cross-entropy loss rather than MSE. This drop occurs because once most data points are classified", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "correctly, gradient descent tries to drive the cross-entropy loss to zero by scaling up the margins, as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 494, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 494, + 266 + ], + "score": 1.0, + "content": "detailed in Soudry et al. (2018). As we explain in Appendix C, this causes the sharpness to drop.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 198, + 505, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 504, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "The effect of width. It is known that when networks parameterized in a certain way (the “NTK", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "parameterization”) are made infinitely wide, the Hessian moves a vanishingly small amount during", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "training (Jacot et al., 2018; Lee et al., 2019; Li & Liang, 2018), which implies that no progressive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "sharpening occurs. In Appendix D, we experiment with networks of varying width, under both NTK", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "score": 1.0, + "content": "and standard parameterizations. We find that progressive sharpening occurs to a lesser degree as net-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 376, + 338 + ], + "score": 1.0, + "content": "works become increasingly wide. Nevertheless, our experiments in", + "type": "text" + }, + { + "bbox": [ + 376, + 325, + 388, + 336 + ], + "score": 0.74, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "demonstrate that progressive", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "sharpening occurs to a dramatic degree for standard architectures on the standard dataset CIFAR-10.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 269, + 506, + 348 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 353, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "We do not know why progressive sharpening occurs, or whether “sharp” solutions differ in any", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "important way from “not sharp” solutions. These are important questions for future work. Note that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 483, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 483, + 387 + ], + "score": 1.0, + "content": "Mulayoff & Michaeli (2020) studied the latter question in the context of deep linear networks.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 352, + 505, + 387 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 402, + 238, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 240, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 240, + 415 + ], + "score": 1.0, + "content": "3.2 THE EDGE OF STABILITY", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 382, + 438 + ], + "score": 1.0, + "content": "In the preceding section, we ran gradient descent using step sizes", + "type": "text" + }, + { + "bbox": [ + 382, + 427, + 389, + 436 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "so small that the sharpness", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 255, + 448 + ], + "score": 1.0, + "content": "never reached the stability threshold", + "type": "text" + }, + { + "bbox": [ + 255, + 435, + 272, + 448 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 436, + 505, + 448 + ], + "score": 1.0, + "content": ". In Figure 4(a), we start to train the same network at the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 446, + 183, + 460 + ], + "score": 1.0, + "content": "larger step size of", + "type": "text" + }, + { + "bbox": [ + 183, + 447, + 224, + 458 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 446, + 421, + 460 + ], + "score": 1.0, + "content": ", and pause training once the sharpness rises to", + "type": "text" + }, + { + "bbox": [ + 421, + 446, + 470, + 459 + ], + "score": 0.91, + "content": "2 / \\eta = 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 446, + 506, + 460 + ], + "score": 1.0, + "content": ". Recall", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 456, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 129, + 471 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 130, + 458, + 141, + 468 + ], + "score": 0.8, + "content": "\\ S 2", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 456, + 343, + 471 + ], + "score": 1.0, + "content": "that in any region where the sharpness exceeds", + "type": "text" + }, + { + "bbox": [ + 344, + 457, + 360, + 469 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 456, + 497, + 471 + ], + "score": 1.0, + "content": ", gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 497, + 459, + 504, + 469 + ], + "score": 0.69, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "would be unstable if run on the quadratic Taylor approximation to the training objective — the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "gradient descent iterates would oscillate with ever-increasing magnitude along the leading Hessian", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "eigenvector. Empirically, we find that gradient descent on the real neural training objective behaves", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 241, + 514 + ], + "score": 1.0, + "content": "similarly — at first. Namely, let", + "type": "text" + }, + { + "bbox": [ + 242, + 502, + 254, + 513 + ], + "score": 0.81, + "content": "{ \\bf q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "be the leading Hessian eigenvector at the iteration where the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 181, + 526 + ], + "score": 1.0, + "content": "sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 181, + 512, + 198, + 524 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 512, + 506, + 526 + ], + "score": 1.0, + "content": ". In Figure 4(b), we resume training the network, and we monitor both the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 219, + 536 + ], + "score": 1.0, + "content": "train loss and the quantity", + "type": "text" + }, + { + "bbox": [ + 219, + 523, + 252, + 535 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 522, + 429, + 536 + ], + "score": 1.0, + "content": "for the next 215 iterations. Observe that", + "type": "text" + }, + { + "bbox": [ + 430, + 523, + 463, + 535 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "oscillates", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "with ever-increasing magnitude, similar to the divergent quadratic example in Figure 2(b). At first,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "these oscillations are too small to affect the objective appreciably, and so the train loss continues to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "monotonically decrease. But eventually, these oscillations grow big enough that the train loss spikes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 424, + 506, + 569 + ] + }, + { + "type": "image", + "bbox": [ + 110, + 591, + 505, + 672 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 591, + 505, + 672 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 591, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 110, + 591, + 505, + 672 + ], + "score": 0.961, + "type": "image", + "image_path": "d167549c8548e4247c63c0dafe3ea4b11804fc67c00c956313b55282c60c6c65.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 110, + 591, + 505, + 618.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 110, + 618.0, + 505, + 645.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 110, + 645.0, + 505, + 672.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 684, + 505, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 261, + 697 + ], + "score": 1.0, + "content": "Figure 4: Once the sharpness crosses", + "type": "text" + }, + { + "bbox": [ + 262, + 684, + 278, + 696 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 684, + 505, + 697 + ], + "score": 1.0, + "content": ", gradient descent becomes destabilized. We run gradi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 695, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 164, + 708 + ], + "score": 1.0, + "content": "ent descent at", + "type": "text" + }, + { + "bbox": [ + 164, + 696, + 202, + 707 + ], + "score": 0.93, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 695, + 357, + 708 + ], + "score": 1.0, + "content": ". (a) The sharpness eventually reaches", + "type": "text" + }, + { + "bbox": [ + 357, + 695, + 374, + 707 + ], + "score": 0.84, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 695, + 505, + 708 + ], + "score": 1.0, + "content": ". (b) Once the sharpness crosses", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 706, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 706, + 123, + 718 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 706, + 264, + 720 + ], + "score": 1.0, + "content": ", the iterates start to oscillate along", + "type": "text" + }, + { + "bbox": [ + 264, + 708, + 276, + 718 + ], + "score": 0.75, + "content": "{ \\bf q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 706, + 506, + 720 + ], + "score": 1.0, + "content": "with ever-increasing magnitude. (c) Somehow, GD does", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 717, + 471, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 471, + 730 + ], + "score": 1.0, + "content": "not diverge entirely; instead, the train loss continues to decrease, albeit non-monotonically.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "index": 42.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Once gradient descent becomes destabilized in this manner, classical optimization theory gives no", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "clues as to what will happen next. One might imagine that perhaps gradient descent might diverge", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "entirely, or that gradient descent might stall while failing to make progress, or that gradient descent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "might jump to a flatter region and remain there. In reality, none of these outcomes occurs. In Figure", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 249, + 140 + ], + "score": 1.0, + "content": "4(c), we plot both the train loss and", + "type": "text" + }, + { + "bbox": [ + 249, + 127, + 282, + 138 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 126, + 484, + 140 + ], + "score": 1.0, + "content": "for 1000 iterations after the sharpness first crossed", + "type": "text" + }, + { + "bbox": [ + 485, + 126, + 501, + 138 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 126, + 505, + 140 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "Observe that gradient descent somehow avoids diverging entirely. Instead, after initially spiking", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 440, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 440, + 161 + ], + "score": 1.0, + "content": "around iteration 215, the train loss continues to decrease, albeit non-monotonically.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 459, + 178 + ], + "score": 1.0, + "content": "This numerical example is representative. In general, after the sharpness initially crosses", + "type": "text" + }, + { + "bbox": [ + 459, + 165, + 475, + 177 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 165, + 504, + 178 + ], + "score": 1.0, + "content": ", gradi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "ent descent enters a regime we call the Edge of Stability, in which (1) the sharpness hovers right at,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 200, + 200 + ], + "score": 1.0, + "content": "or just above, the value", + "type": "text" + }, + { + "bbox": [ + 200, + 187, + 217, + 199 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "; and (2) the train loss behaves non-monotonically over short timescales,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "yet decreases consistently over long timescales. Indeed, in Figure 5, we run gradient descent at a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "range of step sizes using both MSE and cross-entropy loss. The left plane plots the train loss curves,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "with a vertical dotted line (of the appropriate color) marking the iteration where the sharpness first", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 138, + 243 + ], + "score": 1.0, + "content": "crosses", + "type": "text" + }, + { + "bbox": [ + 139, + 231, + 155, + 243 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 231, + 506, + 243 + ], + "score": 1.0, + "content": ". Observe that the train loss decreases monotonically before this dotted line, but be-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "haves non-monotonically afterwards. The middle plane plots the evolution of the sharpness, with a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 351, + 266 + ], + "score": 1.0, + "content": "horizontal dashed line (of the appropriate color) at the value", + "type": "text" + }, + { + "bbox": [ + 352, + 253, + 368, + 265 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 253, + 505, + 266 + ], + "score": 1.0, + "content": ". Observe that once the sharpness", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 139, + 277 + ], + "score": 1.0, + "content": "reaches", + "type": "text" + }, + { + "bbox": [ + 139, + 264, + 156, + 276 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 264, + 487, + 277 + ], + "score": 1.0, + "content": ", it ceases to increase further, and instead hovers right at, or just above, the value", + "type": "text" + }, + { + "bbox": [ + 487, + 264, + 504, + 276 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "for the remainder of training. (The precise meaning of “just above” varies: in Figure 5, for MSE", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 284, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 337, + 300 + ], + "score": 1.0, + "content": "loss, the sharpness hovers just a minuscule amount above", + "type": "text" + }, + { + "bbox": [ + 337, + 286, + 354, + 298 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 284, + 505, + 300 + ], + "score": 1.0, + "content": ", while for cross-entropy loss, the gap", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 346, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 215, + 310 + ], + "score": 1.0, + "content": "between the sharpness and", + "type": "text" + }, + { + "bbox": [ + 215, + 297, + 232, + 309 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 296, + 346, + 310 + ], + "score": 1.0, + "content": "is small yet non-miniscule.)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "At the Edge of Stability, gradient descent is “trying” to increase the sharpness further, but is being", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 420, + 338 + ], + "score": 1.0, + "content": "restrained from doing so. To demonstrate this, in Figure 7, we train at step size", + "type": "text" + }, + { + "bbox": [ + 420, + 325, + 447, + 336 + ], + "score": 0.52, + "content": "2 / 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "until reaching", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 349 + ], + "score": 1.0, + "content": "the Edge of Stability, and then at iteration 6,000 (marked by the vertical black line), we drop the step", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 135, + 358 + ], + "score": 1.0, + "content": "size to", + "type": "text" + }, + { + "bbox": [ + 135, + 347, + 180, + 358 + ], + "score": 0.92, + "content": "\\eta = 2 / 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 347, + 505, + 358 + ], + "score": 1.0, + "content": ". Observe that after the learning rate drop, the sharpness immediately starts to in-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "crease, and only stops increasing once gradient descent is back at the Edge of Stability. Appendix O", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "repeats this experiment on more architectures. Intuitively, gradient descent with fixed step sizes acts", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 356, + 392 + ], + "score": 1.0, + "content": "like a constrained optimization algorithm: the use of step size", + "type": "text" + }, + { + "bbox": [ + 356, + 381, + 363, + 391 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 379, + 445, + 392 + ], + "score": 1.0, + "content": "imposes an implicit", + "type": "text" + }, + { + "bbox": [ + 446, + 379, + 462, + 392 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "constraint", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 479, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 479, + 402 + ], + "score": 1.0, + "content": "on the sharpness (Nar & Sastry, 2018), and at the Edge of Stability this constraint is “active.”", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 319, + 420 + ], + "score": 1.0, + "content": "Observe from Figure 5 that there do exist step sizes", + "type": "text" + }, + { + "bbox": [ + 319, + 409, + 326, + 419 + ], + "score": 0.75, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "(in purple) small enough that the sharpness", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 161, + 430 + ], + "score": 1.0, + "content": "never rises to", + "type": "text" + }, + { + "bbox": [ + 162, + 418, + 178, + 430 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 419, + 505, + 430 + ], + "score": 1.0, + "content": ". We call such a step size stable. However, observe that with cross-entropy loss, it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "takes 3700 iterations to train at the stable step size in purple, but only 1000 iterations to train at the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "larger step size in blue. In general, we always observe that stable step sizes are suboptimal in terms", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "image", + "bbox": [ + 105, + 490, + 525, + 660 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 490, + 525, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 490, + 525, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 525, + 660 + ], + "score": 0.974, + "type": "image", + "image_path": "1d1a6cde8e4d00e092499a010db67cddcc397824f725bfbf3fd1dc2519962684.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 105, + 490, + 525, + 546.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 105, + 546.6666666666666, + 525, + 603.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 105, + 603.3333333333333, + 525, + 659.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 673, + 506, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 272, + 686 + ], + "score": 1.0, + "content": "Figure 5: After the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 272, + 673, + 289, + 685 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 673, + 505, + 686 + ], + "score": 1.0, + "content": ", gradient descent enters the Edge of Stability. A", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "score": 1.0, + "content": "network is trained with gradient descent at a range of step sizes (see legend), using both MSE loss", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "score": 1.0, + "content": "(top row) and cross-entropy (bottom row). Left: the train loss curves, with a vertical dotted line at", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 706, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 289, + 720 + ], + "score": 1.0, + "content": "the iteration where the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 289, + 706, + 306, + 718 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 706, + 506, + 720 + ], + "score": 1.0, + "content": ". Center: the sharpness, with a horizontal dashed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 717, + 478, + 730 + ], + "spans": [ + { + "bbox": [ + 106, + 718, + 172, + 730 + ], + "score": 1.0, + "content": "line at the value", + "type": "text" + }, + { + "bbox": [ + 172, + 717, + 189, + 729 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 718, + 329, + 730 + ], + "score": 1.0, + "content": ". Right: sharpness plotted by time", + "type": "text" + }, + { + "bbox": [ + 330, + 718, + 338, + 727 + ], + "score": 0.64, + "content": "\\mathit { \\Psi } = \\mathit { \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 718, + 374, + 730 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 374, + 718, + 391, + 729 + ], + "score": 0.77, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 718, + 478, + 730 + ], + "score": 1.0, + "content": ") rather than iteration.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + } + ], + "index": 35.0 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Once gradient descent becomes destabilized in this manner, classical optimization theory gives no", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "clues as to what will happen next. One might imagine that perhaps gradient descent might diverge", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "entirely, or that gradient descent might stall while failing to make progress, or that gradient descent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "might jump to a flatter region and remain there. In reality, none of these outcomes occurs. In Figure", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 249, + 140 + ], + "score": 1.0, + "content": "4(c), we plot both the train loss and", + "type": "text" + }, + { + "bbox": [ + 249, + 127, + 282, + 138 + ], + "score": 0.93, + "content": "\\left. \\mathbf { q } _ { 1 } , \\mathbf { x } _ { t } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 126, + 484, + 140 + ], + "score": 1.0, + "content": "for 1000 iterations after the sharpness first crossed", + "type": "text" + }, + { + "bbox": [ + 485, + 126, + 501, + 138 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 126, + 505, + 140 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "Observe that gradient descent somehow avoids diverging entirely. Instead, after initially spiking", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 440, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 440, + 161 + ], + "score": 1.0, + "content": "around iteration 215, the train loss continues to decrease, albeit non-monotonically.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 81, + 506, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 459, + 178 + ], + "score": 1.0, + "content": "This numerical example is representative. In general, after the sharpness initially crosses", + "type": "text" + }, + { + "bbox": [ + 459, + 165, + 475, + 177 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 165, + 504, + 178 + ], + "score": 1.0, + "content": ", gradi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "ent descent enters a regime we call the Edge of Stability, in which (1) the sharpness hovers right at,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 200, + 200 + ], + "score": 1.0, + "content": "or just above, the value", + "type": "text" + }, + { + "bbox": [ + 200, + 187, + 217, + 199 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "; and (2) the train loss behaves non-monotonically over short timescales,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "yet decreases consistently over long timescales. Indeed, in Figure 5, we run gradient descent at a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "range of step sizes using both MSE and cross-entropy loss. The left plane plots the train loss curves,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "with a vertical dotted line (of the appropriate color) marking the iteration where the sharpness first", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 138, + 243 + ], + "score": 1.0, + "content": "crosses", + "type": "text" + }, + { + "bbox": [ + 139, + 231, + 155, + 243 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 231, + 506, + 243 + ], + "score": 1.0, + "content": ". Observe that the train loss decreases monotonically before this dotted line, but be-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "haves non-monotonically afterwards. The middle plane plots the evolution of the sharpness, with a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 351, + 266 + ], + "score": 1.0, + "content": "horizontal dashed line (of the appropriate color) at the value", + "type": "text" + }, + { + "bbox": [ + 352, + 253, + 368, + 265 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 253, + 505, + 266 + ], + "score": 1.0, + "content": ". Observe that once the sharpness", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 504, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 139, + 277 + ], + "score": 1.0, + "content": "reaches", + "type": "text" + }, + { + "bbox": [ + 139, + 264, + 156, + 276 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 264, + 487, + 277 + ], + "score": 1.0, + "content": ", it ceases to increase further, and instead hovers right at, or just above, the value", + "type": "text" + }, + { + "bbox": [ + 487, + 264, + 504, + 276 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "for the remainder of training. (The precise meaning of “just above” varies: in Figure 5, for MSE", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 284, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 337, + 300 + ], + "score": 1.0, + "content": "loss, the sharpness hovers just a minuscule amount above", + "type": "text" + }, + { + "bbox": [ + 337, + 286, + 354, + 298 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 284, + 505, + 300 + ], + "score": 1.0, + "content": ", while for cross-entropy loss, the gap", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 346, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 215, + 310 + ], + "score": 1.0, + "content": "between the sharpness and", + "type": "text" + }, + { + "bbox": [ + 215, + 297, + 232, + 309 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 296, + 346, + 310 + ], + "score": 1.0, + "content": "is small yet non-miniscule.)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 165, + 506, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 327 + ], + "score": 1.0, + "content": "At the Edge of Stability, gradient descent is “trying” to increase the sharpness further, but is being", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 420, + 338 + ], + "score": 1.0, + "content": "restrained from doing so. To demonstrate this, in Figure 7, we train at step size", + "type": "text" + }, + { + "bbox": [ + 420, + 325, + 447, + 336 + ], + "score": 0.52, + "content": "2 / 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "until reaching", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 349 + ], + "score": 1.0, + "content": "the Edge of Stability, and then at iteration 6,000 (marked by the vertical black line), we drop the step", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 135, + 358 + ], + "score": 1.0, + "content": "size to", + "type": "text" + }, + { + "bbox": [ + 135, + 347, + 180, + 358 + ], + "score": 0.92, + "content": "\\eta = 2 / 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 347, + 505, + 358 + ], + "score": 1.0, + "content": ". Observe that after the learning rate drop, the sharpness immediately starts to in-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "crease, and only stops increasing once gradient descent is back at the Edge of Stability. Appendix O", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 104, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "repeats this experiment on more architectures. Intuitively, gradient descent with fixed step sizes acts", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 356, + 392 + ], + "score": 1.0, + "content": "like a constrained optimization algorithm: the use of step size", + "type": "text" + }, + { + "bbox": [ + 356, + 381, + 363, + 391 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 379, + 445, + 392 + ], + "score": 1.0, + "content": "imposes an implicit", + "type": "text" + }, + { + "bbox": [ + 446, + 379, + 462, + 392 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "constraint", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 479, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 479, + 402 + ], + "score": 1.0, + "content": "on the sharpness (Nar & Sastry, 2018), and at the Edge of Stability this constraint is “active.”", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 312, + 506, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 319, + 420 + ], + "score": 1.0, + "content": "Observe from Figure 5 that there do exist step sizes", + "type": "text" + }, + { + "bbox": [ + 319, + 409, + 326, + 419 + ], + "score": 0.75, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "(in purple) small enough that the sharpness", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 161, + 430 + ], + "score": 1.0, + "content": "never rises to", + "type": "text" + }, + { + "bbox": [ + 162, + 418, + 178, + 430 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 419, + 505, + 430 + ], + "score": 1.0, + "content": ". We call such a step size stable. However, observe that with cross-entropy loss, it", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "takes 3700 iterations to train at the stable step size in purple, but only 1000 iterations to train at the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "larger step size in blue. In general, we always observe that stable step sizes are suboptimal in terms", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 246, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 237, + 262 + ], + "score": 1.0, + "content": "of convergence speed. In fact, in", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 237, + 249, + 249, + 259 + ], + "score": 0.81, + "content": "\\ S 4", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 249, + 246, + 505, + 262 + ], + "score": 1.0, + "content": "we will see that for standard networks on CIFAR-10, stable step", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 259, + 388, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 388, + 271 + ], + "score": 1.0, + "content": "sizes are so suboptimally small that they are completely unreasonable.", + "type": "text", + "cross_page": true + } + ], + "index": 18 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 406, + 506, + 453 + ] + }, + { + "type": "image", + "bbox": [ + 105, + 490, + 525, + 660 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 490, + 525, + 660 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 490, + 525, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 525, + 660 + ], + "score": 0.974, + "type": "image", + "image_path": "1d1a6cde8e4d00e092499a010db67cddcc397824f725bfbf3fd1dc2519962684.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 105, + 490, + 525, + 546.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 105, + 546.6666666666666, + 525, + 603.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 105, + 603.3333333333333, + 525, + 659.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 673, + 506, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 272, + 686 + ], + "score": 1.0, + "content": "Figure 5: After the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 272, + 673, + 289, + 685 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 673, + 505, + 686 + ], + "score": 1.0, + "content": ", gradient descent enters the Edge of Stability. A", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 505, + 697 + ], + "score": 1.0, + "content": "network is trained with gradient descent at a range of step sizes (see legend), using both MSE loss", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 506, + 708 + ], + "score": 1.0, + "content": "(top row) and cross-entropy (bottom row). Left: the train loss curves, with a vertical dotted line at", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 706, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 289, + 720 + ], + "score": 1.0, + "content": "the iteration where the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 289, + 706, + 306, + 718 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 706, + 506, + 720 + ], + "score": 1.0, + "content": ". Center: the sharpness, with a horizontal dashed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 717, + 478, + 730 + ], + "spans": [ + { + "bbox": [ + 106, + 718, + 172, + 730 + ], + "score": 1.0, + "content": "line at the value", + "type": "text" + }, + { + "bbox": [ + 172, + 717, + 189, + 729 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 718, + 329, + 730 + ], + "score": 1.0, + "content": ". Right: sharpness plotted by time", + "type": "text" + }, + { + "bbox": [ + 330, + 718, + 338, + 727 + ], + "score": 0.64, + "content": "\\mathit { \\Psi } = \\mathit { \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 718, + 374, + 730 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 374, + 718, + 391, + 729 + ], + "score": 0.77, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 718, + 478, + 730 + ], + "score": 1.0, + "content": ") rather than iteration.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + } + ], + "index": 35.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 85, + 258, + 155 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 85, + 258, + 155 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 85, + 258, + 155 + ], + "spans": [ + { + "bbox": [ + 113, + 85, + 258, + 155 + ], + "score": 0.952, + "type": "image", + "image_path": "1f252f03c147233621dfa8b75b997a2e37764b54085eddf968800144261a3aa1.jpg" + } + ] + } + ], + "index": 0.5, + "virtual_lines": [ + { + "bbox": [ + 113, + 85, + 258, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 120.0, + 258, + 155.0 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 109, + 167, + 261, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 167, + 262, + 178 + ], + "spans": [ + { + "bbox": [ + 108, + 167, + 262, + 178 + ], + "score": 1.0, + "content": "Figure 6: Momentum. We run GD", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 109, + 178, + 262, + 190 + ], + "spans": [ + { + "bbox": [ + 109, + 178, + 165, + 190 + ], + "score": 1.0, + "content": "with step size", + "type": "text" + }, + { + "bbox": [ + 165, + 179, + 203, + 190 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 178, + 262, + 190 + ], + "score": 1.0, + "content": "and Polyak or", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 109, + 189, + 263, + 200 + ], + "spans": [ + { + "bbox": [ + 109, + 189, + 234, + 200 + ], + "score": 1.0, + "content": "Nesterov momentum at various", + "type": "text" + }, + { + "bbox": [ + 235, + 190, + 242, + 200 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 189, + 263, + 200 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 108, + 200, + 262, + 211 + ], + "spans": [ + { + "bbox": [ + 108, + 200, + 262, + 211 + ], + "score": 1.0, + "content": "each algorithm, the horizontal dashed", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 108, + 210, + 261, + 223 + ], + "spans": [ + { + "bbox": [ + 108, + 210, + 261, + 223 + ], + "score": 1.0, + "content": "line marks the MSS from Equation 1.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + } + ], + "index": 4.75 + }, + { + "type": "image", + "bbox": [ + 275, + 85, + 516, + 156 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 275, + 85, + 516, + 156 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 275, + 85, + 516, + 156 + ], + "spans": [ + { + "bbox": [ + 275, + 85, + 516, + 156 + ], + "score": 0.966, + "type": "image", + "image_path": "46bb801641419ab649df218983ffaa1aa8aacf166e8d5d6ec575bf2ff10b7ccd.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 275, + 85, + 516, + 99.2 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 275, + 99.2, + 516, + 113.4 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 275, + 113.4, + 516, + 127.60000000000001 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 275, + 127.60000000000001, + 516, + 141.8 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 275, + 141.8, + 516, + 156.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 270, + 169, + 502, + 224 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 269, + 168, + 502, + 182 + ], + "spans": [ + { + "bbox": [ + 269, + 168, + 502, + 182 + ], + "score": 1.0, + "content": "Figure 7: After a learning rate drop, progressive sharp-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 269, + 180, + 502, + 193 + ], + "spans": [ + { + "bbox": [ + 269, + 180, + 417, + 193 + ], + "score": 1.0, + "content": "ening resumes. We start training at", + "type": "text" + }, + { + "bbox": [ + 418, + 181, + 465, + 192 + ], + "score": 0.94, + "content": "\\eta = 2 / 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 180, + 502, + 193 + ], + "score": 1.0, + "content": "(orange)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 270, + 191, + 502, + 203 + ], + "spans": [ + { + "bbox": [ + 270, + 191, + 502, + 203 + ], + "score": 1.0, + "content": "and then after 6000 iterations (dotted vertical black line),", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 270, + 202, + 503, + 214 + ], + "spans": [ + { + "bbox": [ + 270, + 202, + 359, + 214 + ], + "score": 1.0, + "content": "we cut the step size to", + "type": "text" + }, + { + "bbox": [ + 359, + 202, + 404, + 214 + ], + "score": 0.92, + "content": "\\eta = 2 / 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 202, + 503, + 214 + ], + "score": 1.0, + "content": "(green). Observe that as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 270, + 214, + 488, + 226 + ], + "spans": [ + { + "bbox": [ + 270, + 214, + 488, + 226 + ], + "score": 1.0, + "content": "soon as the step size is cut, the sharpness starts to rise.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + } + ], + "index": 9.0 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 503, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 237, + 262 + ], + "score": 1.0, + "content": "of convergence speed. In fact, in", + "type": "text" + }, + { + "bbox": [ + 237, + 249, + 249, + 259 + ], + "score": 0.81, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 246, + 505, + 262 + ], + "score": 1.0, + "content": "we will see that for standard networks on CIFAR-10, stable step", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 259, + 388, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 388, + 271 + ], + "score": 1.0, + "content": "sizes are so suboptimally small that they are completely unreasonable.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "The “Edge of Stability” effect generalizes to gradient descent with momentum. In Figure 6, we train", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 257, + 299 + ], + "score": 1.0, + "content": "using gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 257, + 288, + 296, + 299 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 288, + 505, + 299 + ], + "score": 1.0, + "content": ", and varying amounts of either Polyak or Nesterov", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "momentum. Observe that in each case, the sharpness rises until reaching the MSS given by Equation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 309, + 405, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 255, + 322 + ], + "score": 1.0, + "content": "1, and then plateaus there. Appendix", + "type": "text" + }, + { + "bbox": [ + 255, + 310, + 264, + 320 + ], + "score": 0.26, + "content": "\\mathbf { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 309, + 405, + 322 + ], + "score": 1.0, + "content": "has more momentum experiments.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 503, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "In Appendix P, we briefly examine the evolution of the next few Hessian eigenvalues during gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 336, + 421, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 401, + 351 + ], + "score": 1.0, + "content": "descent. We find that each of these eigenvalues rises until plateauing near", + "type": "text" + }, + { + "bbox": [ + 402, + 337, + 418, + 349 + ], + "score": 0.89, + "content": "\\bar { 2 } / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 336, + 421, + 351 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "Prior work. Aspects of the Edge of Stability have been observed previously in the literature. Wu", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "score": 1.0, + "content": "et al. (2018) noted that the sharpness at the solution reached by full-batch gradient descent was not", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 163, + 389 + ], + "score": 1.0, + "content": "just less than", + "type": "text" + }, + { + "bbox": [ + 164, + 376, + 180, + 388 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 375, + 505, + 389 + ], + "score": 1.0, + "content": ", as was expected due to stability considerations, but was mysteriously approx-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 176, + 400 + ], + "score": 1.0, + "content": "imately equal to", + "type": "text" + }, + { + "bbox": [ + 177, + 387, + 193, + 398 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 386, + 505, + 400 + ], + "score": 1.0, + "content": ". In retrospect, we can attribute this observation to progressive sharpening.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "Xing et al. (2018) observed that full-batch gradient descent eventually enters a regime (the Edge of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "Stability) in which the training loss behaves non-monotonically, and the iterates oscillate along the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "direction of largest curvature; however, they did not relate this regime to the sharpness. Lewkowycz", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "et al. (2020) found that in neural network training, if the sharpness at initialization is larger than", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 123, + 454 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 442, + 505, + 455 + ], + "score": 1.0, + "content": ", then after becoming initially destabilized, gradient descent does not always diverge entirely (as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "the quadratic Taylor approximation would suggest), but rather sometimes “catapults” into a flatter", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "region that is flat enough to stably accommodate the step size. It seems plausible that whichever", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "properties of neural training objectives permit this so-called “catapult” behavior may also be the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "same properties that permit successful optimization at the Edge of Stability. Indeed, optimization", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "at the Edge of Stability can conceivably be viewed as a never-ending series of micro-catapults. As", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 229, + 520 + ], + "score": 1.0, + "content": "we discuss at greater length in", + "type": "text" + }, + { + "bbox": [ + 230, + 507, + 241, + 518 + ], + "score": 0.79, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 507, + 506, + 520 + ], + "score": 1.0, + "content": ", several papers (Jastrz˛ebski et al., 2017; 2019) have observed that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "large step sizes steer stochastic gradient descent into less sharp regions of the loss landscape, and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Jastrz˛ebski et al. (2020) attributed this effect to the stability properties of SGD. Finally, our pre-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "cise characterization of the behavior of the sharpness during full-batch gradient descent adds to a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "growing body of work that empirically investigates the Hessian spectrum of neural networks (Sagun", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 563, + 423, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 423, + 574 + ], + "score": 1.0, + "content": "et al., 2017; Ghorbani et al., 2019; Li et al., 2020a; Papyan, 2018; 2019; 2020).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 589, + 280, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 282, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 282, + 601 + ], + "score": 1.0, + "content": "3.3 THE GRADIENT FLOW TRAJECTORY", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "In the right pane of Figure 5, we plot the evolution of the sharpness during gradient descent, with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 135, + 635 + ], + "score": 1.0, + "content": "“time”", + "type": "text" + }, + { + "bbox": [ + 135, + 623, + 144, + 632 + ], + "score": 0.79, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 621, + 181, + 635 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 181, + 623, + 197, + 633 + ], + "score": 0.86, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 621, + 315, + 635 + ], + "score": 1.0, + "content": ", rather than iteration, on the", + "type": "text" + }, + { + "bbox": [ + 315, + 623, + 322, + 632 + ], + "score": 0.39, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "-axis. This allows us to directly compare the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 261, + 645 + ], + "score": 1.0, + "content": "sharpness after, say, 100 iterations at", + "type": "text" + }, + { + "bbox": [ + 261, + 633, + 302, + 644 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 633, + 460, + 645 + ], + "score": 1.0, + "content": "to the sharpness after 50 iterations at", + "type": "text" + }, + { + "bbox": [ + 460, + 633, + 501, + 644 + ], + "score": 0.9, + "content": "\\eta = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 633, + 504, + 645 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "both are time 1. Observe that when plotted by time, the sharpnesses for gradient descent at different", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 317, + 668 + ], + "score": 1.0, + "content": "step sizes coincide until the time where each reaches", + "type": "text" + }, + { + "bbox": [ + 318, + 655, + 334, + 667 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 655, + 506, + 668 + ], + "score": 1.0, + "content": ". This is because for this network, gradient", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 147, + 679 + ], + "score": 1.0, + "content": "descent at", + "type": "text" + }, + { + "bbox": [ + 147, + 666, + 185, + 677 + ], + "score": 0.9, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 666, + 277, + 679 + ], + "score": 1.0, + "content": "and gradient descent at", + "type": "text" + }, + { + "bbox": [ + 278, + 666, + 315, + 677 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "initially travel the same path (moving at a speed", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 169, + 690 + ], + "score": 1.0, + "content": "proportional to", + "type": "text" + }, + { + "bbox": [ + 169, + 679, + 176, + 688 + ], + "score": 0.71, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 676, + 443, + 690 + ], + "score": 1.0, + "content": ") until each reaches the point on that path where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 443, + 677, + 460, + 689 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ". This path", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 370, + 700 + ], + "score": 1.0, + "content": "is the gradient flow trajectory. The gradient flow solution at time", + "type": "text" + }, + { + "bbox": [ + 370, + 689, + 375, + 698 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 687, + 476, + 700 + ], + "score": 1.0, + "content": "is defined as the limit as", + "type": "text" + }, + { + "bbox": [ + 476, + 688, + 504, + 699 + ], + "score": 0.9, + "content": "\\eta 0", + "type": "inline_equation" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 273, + 712 + ], + "score": 1.0, + "content": "of the gradient descent iterate at iteration", + "type": "text" + }, + { + "bbox": [ + 274, + 699, + 289, + 711 + ], + "score": 0.9, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "(if this limit exists). The empirical finding of interest", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 104, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "is that for this particular network, gradient descent does not only track the gradient flow trajectory", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 720, + 489, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 489, + 733 + ], + "score": 1.0, + "content": "in the limit of infinitesimally small step sizes, but for any step size that is less than 2/sharpness.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 51 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 85, + 258, + 155 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 85, + 258, + 155 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 85, + 258, + 155 + ], + "spans": [ + { + "bbox": [ + 113, + 85, + 258, + 155 + ], + "score": 0.952, + "type": "image", + "image_path": "1f252f03c147233621dfa8b75b997a2e37764b54085eddf968800144261a3aa1.jpg" + } + ] + } + ], + "index": 0.5, + "virtual_lines": [ + { + "bbox": [ + 113, + 85, + 258, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 120.0, + 258, + 155.0 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 109, + 167, + 261, + 222 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 167, + 262, + 178 + ], + "spans": [ + { + "bbox": [ + 108, + 167, + 262, + 178 + ], + "score": 1.0, + "content": "Figure 6: Momentum. We run GD", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 109, + 178, + 262, + 190 + ], + "spans": [ + { + "bbox": [ + 109, + 178, + 165, + 190 + ], + "score": 1.0, + "content": "with step size", + "type": "text" + }, + { + "bbox": [ + 165, + 179, + 203, + 190 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 178, + 262, + 190 + ], + "score": 1.0, + "content": "and Polyak or", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 109, + 189, + 263, + 200 + ], + "spans": [ + { + "bbox": [ + 109, + 189, + 234, + 200 + ], + "score": 1.0, + "content": "Nesterov momentum at various", + "type": "text" + }, + { + "bbox": [ + 235, + 190, + 242, + 200 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 189, + 263, + 200 + ], + "score": 1.0, + "content": ". 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We start training at", + "type": "text" + }, + { + "bbox": [ + 418, + 181, + 465, + 192 + ], + "score": 0.94, + "content": "\\eta = 2 / 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 180, + 502, + 193 + ], + "score": 1.0, + "content": "(orange)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 270, + 191, + 502, + 203 + ], + "spans": [ + { + "bbox": [ + 270, + 191, + 502, + 203 + ], + "score": 1.0, + "content": "and then after 6000 iterations (dotted vertical black line),", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 270, + 202, + 503, + 214 + ], + "spans": [ + { + "bbox": [ + 270, + 202, + 359, + 214 + ], + "score": 1.0, + "content": "we cut the step size to", + "type": "text" + }, + { + "bbox": [ + 359, + 202, + 404, + 214 + ], + "score": 0.92, + "content": "\\eta = 2 / 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 202, + 503, + 214 + ], + "score": 1.0, + "content": "(green). Observe that as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 270, + 214, + 488, + 226 + ], + "spans": [ + { + "bbox": [ + 270, + 214, + 488, + 226 + ], + "score": 1.0, + "content": "soon as the step size is cut, the sharpness starts to rise.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + } + ], + "index": 9.0 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 503, + 270 + ], + "lines": [], + "index": 17.5, + "bbox_fs": [ + 105, + 246, + 505, + 271 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "The “Edge of Stability” effect generalizes to gradient descent with momentum. In Figure 6, we train", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 257, + 299 + ], + "score": 1.0, + "content": "using gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 257, + 288, + 296, + 299 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 288, + 505, + 299 + ], + "score": 1.0, + "content": ", and varying amounts of either Polyak or Nesterov", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "momentum. Observe that in each case, the sharpness rises until reaching the MSS given by Equation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 309, + 405, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 255, + 322 + ], + "score": 1.0, + "content": "1, and then plateaus there. Appendix", + "type": "text" + }, + { + "bbox": [ + 255, + 310, + 264, + 320 + ], + "score": 0.26, + "content": "\\mathbf { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 309, + 405, + 322 + ], + "score": 1.0, + "content": "has more momentum experiments.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 275, + 505, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 503, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "In Appendix P, we briefly examine the evolution of the next few Hessian eigenvalues during gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 336, + 421, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 401, + 351 + ], + "score": 1.0, + "content": "descent. We find that each of these eigenvalues rises until plateauing near", + "type": "text" + }, + { + "bbox": [ + 402, + 337, + 418, + 349 + ], + "score": 0.89, + "content": "\\bar { 2 } / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 336, + 421, + 351 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 326, + 505, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "Prior work. Aspects of the Edge of Stability have been observed previously in the literature. Wu", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "score": 1.0, + "content": "et al. (2018) noted that the sharpness at the solution reached by full-batch gradient descent was not", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 163, + 389 + ], + "score": 1.0, + "content": "just less than", + "type": "text" + }, + { + "bbox": [ + 164, + 376, + 180, + 388 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 375, + 505, + 389 + ], + "score": 1.0, + "content": ", as was expected due to stability considerations, but was mysteriously approx-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 176, + 400 + ], + "score": 1.0, + "content": "imately equal to", + "type": "text" + }, + { + "bbox": [ + 177, + 387, + 193, + 398 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 386, + 505, + 400 + ], + "score": 1.0, + "content": ". In retrospect, we can attribute this observation to progressive sharpening.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "Xing et al. (2018) observed that full-batch gradient descent eventually enters a regime (the Edge of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "Stability) in which the training loss behaves non-monotonically, and the iterates oscillate along the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "direction of largest curvature; however, they did not relate this regime to the sharpness. Lewkowycz", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "et al. (2020) found that in neural network training, if the sharpness at initialization is larger than", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 123, + 454 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 442, + 505, + 455 + ], + "score": 1.0, + "content": ", then after becoming initially destabilized, gradient descent does not always diverge entirely (as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "the quadratic Taylor approximation would suggest), but rather sometimes “catapults” into a flatter", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "region that is flat enough to stably accommodate the step size. It seems plausible that whichever", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "properties of neural training objectives permit this so-called “catapult” behavior may also be the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "same properties that permit successful optimization at the Edge of Stability. Indeed, optimization", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "at the Edge of Stability can conceivably be viewed as a never-ending series of micro-catapults. As", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 229, + 520 + ], + "score": 1.0, + "content": "we discuss at greater length in", + "type": "text" + }, + { + "bbox": [ + 230, + 507, + 241, + 518 + ], + "score": 0.79, + "content": "\\ S 6", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 507, + 506, + 520 + ], + "score": 1.0, + "content": ", several papers (Jastrz˛ebski et al., 2017; 2019) have observed that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "large step sizes steer stochastic gradient descent into less sharp regions of the loss landscape, and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Jastrz˛ebski et al. (2020) attributed this effect to the stability properties of SGD. Finally, our pre-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "cise characterization of the behavior of the sharpness during full-batch gradient descent adds to a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "growing body of work that empirically investigates the Hessian spectrum of neural networks (Sagun", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 563, + 423, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 423, + 574 + ], + "score": 1.0, + "content": "et al., 2017; Ghorbani et al., 2019; Li et al., 2020a; Papyan, 2018; 2019; 2020).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 354, + 506, + 574 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 589, + 280, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 282, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 282, + 601 + ], + "score": 1.0, + "content": "3.3 THE GRADIENT FLOW TRAJECTORY", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "In the right pane of Figure 5, we plot the evolution of the sharpness during gradient descent, with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 135, + 635 + ], + "score": 1.0, + "content": "“time”", + "type": "text" + }, + { + "bbox": [ + 135, + 623, + 144, + 632 + ], + "score": 0.79, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 621, + 181, + 635 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 181, + 623, + 197, + 633 + ], + "score": 0.86, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 621, + 315, + 635 + ], + "score": 1.0, + "content": ", rather than iteration, on the", + "type": "text" + }, + { + "bbox": [ + 315, + 623, + 322, + 632 + ], + "score": 0.39, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "-axis. This allows us to directly compare the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 633, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 261, + 645 + ], + "score": 1.0, + "content": "sharpness after, say, 100 iterations at", + "type": "text" + }, + { + "bbox": [ + 261, + 633, + 302, + 644 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 633, + 460, + 645 + ], + "score": 1.0, + "content": "to the sharpness after 50 iterations at", + "type": "text" + }, + { + "bbox": [ + 460, + 633, + 501, + 644 + ], + "score": 0.9, + "content": "\\eta = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 633, + 504, + 645 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "both are time 1. Observe that when plotted by time, the sharpnesses for gradient descent at different", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 317, + 668 + ], + "score": 1.0, + "content": "step sizes coincide until the time where each reaches", + "type": "text" + }, + { + "bbox": [ + 318, + 655, + 334, + 667 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 655, + 506, + 668 + ], + "score": 1.0, + "content": ". This is because for this network, gradient", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 147, + 679 + ], + "score": 1.0, + "content": "descent at", + "type": "text" + }, + { + "bbox": [ + 147, + 666, + 185, + 677 + ], + "score": 0.9, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 666, + 277, + 679 + ], + "score": 1.0, + "content": "and gradient descent at", + "type": "text" + }, + { + "bbox": [ + 278, + 666, + 315, + 677 + ], + "score": 0.91, + "content": "\\eta = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "initially travel the same path (moving at a speed", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 169, + 690 + ], + "score": 1.0, + "content": "proportional to", + "type": "text" + }, + { + "bbox": [ + 169, + 679, + 176, + 688 + ], + "score": 0.71, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 676, + 443, + 690 + ], + "score": 1.0, + "content": ") until each reaches the point on that path where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 443, + 677, + 460, + 689 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ". This path", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 370, + 700 + ], + "score": 1.0, + "content": "is the gradient flow trajectory. The gradient flow solution at time", + "type": "text" + }, + { + "bbox": [ + 370, + 689, + 375, + 698 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 687, + 476, + 700 + ], + "score": 1.0, + "content": "is defined as the limit as", + "type": "text" + }, + { + "bbox": [ + 476, + 688, + 504, + 699 + ], + "score": 0.9, + "content": "\\eta 0", + "type": "inline_equation" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 273, + 712 + ], + "score": 1.0, + "content": "of the gradient descent iterate at iteration", + "type": "text" + }, + { + "bbox": [ + 274, + 699, + 289, + 711 + ], + "score": 0.9, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "(if this limit exists). The empirical finding of interest", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 104, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "is that for this particular network, gradient descent does not only track the gradient flow trajectory", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 720, + 489, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 489, + 733 + ], + "score": 1.0, + "content": "in the limit of infinitesimally small step sizes, but for any step size that is less than 2/sharpness.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 51, + "bbox_fs": [ + 104, + 611, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We can numerically approximate gradient flow trajectories by using the Runge-Kutta RK4 algorithm", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "(Press et al., 1992) to numerically integrate the gradient flow ODE. Empirically, for many but not all", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 423, + 117 + ], + "score": 1.0, + "content": "networks studied in this paper, we find that gradient descent at any step size", + "type": "text" + }, + { + "bbox": [ + 423, + 106, + 430, + 116 + ], + "score": 0.73, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "closely tracks the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 484, + 129 + ], + "score": 1.0, + "content": "Runge-Kutta trajectory until reaching the point on that trajectory where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 485, + 115, + 501, + 127 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 114, + 505, + 129 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "(This sometimes occurs even for networks with ReLU activations or max-pooling, which give rise", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "to training objectives that are not continuously differentiable, which means that the gradient flow", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "trajectory is not necessarily guaranteed to exist.) For such networks, the gradient flow trajectory", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "provides a coherent framework for reasoning about which step sizes will eventually enter the Edge", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 171, + 184 + ], + "score": 1.0, + "content": "of Stability. Let", + "type": "text" + }, + { + "bbox": [ + 172, + 171, + 183, + 181 + ], + "score": 0.88, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 169, + 346, + 184 + ], + "score": 1.0, + "content": "be the sharpness at initialization, and let", + "type": "text" + }, + { + "bbox": [ + 346, + 171, + 369, + 182 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 169, + 505, + 184 + ], + "score": 1.0, + "content": "be the maximum sharpness along", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 228, + 195 + ], + "score": 1.0, + "content": "the gradient flow trajectory. If", + "type": "text" + }, + { + "bbox": [ + 228, + 181, + 279, + 193 + ], + "score": 0.92, + "content": "\\eta < 2 / \\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 181, + 506, + 195 + ], + "score": 1.0, + "content": ", then gradient descent will stably track the gradient flow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "trajectory for the entire duration of training, and will never enter the Edge of Stability. On the other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 139, + 218 + ], + "score": 1.0, + "content": "hand, if", + "type": "text" + }, + { + "bbox": [ + 139, + 203, + 219, + 215 + ], + "score": 0.93, + "content": "\\eta \\in [ 2 / \\lambda _ { \\operatorname* { m a x } } , 2 / \\lambda _ { 0 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 201, + 506, + 218 + ], + "score": 1.0, + "content": ", then gradient descent will stably track the gradient flow trajectory only", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 372, + 227 + ], + "score": 1.0, + "content": "until reaching the point on that trajectory where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 372, + 214, + 389, + 226 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "; shortly afterwards, gradient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "descent will become destabilized, depart the gradient flow trajectory, and enter the Edge of Stability.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 257, + 249, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 251, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 251, + 271 + ], + "score": 1.0, + "content": "4 FURTHER EXPERIMENTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 504, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "Section 3 focused for exposition on a single architecture and task. In this section, we show that our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 297, + 504, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 504, + 307 + ], + "score": 1.0, + "content": "characterization of gradient descent holds broadly across a wide range of architectures and tasks.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 375, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 375, + 319 + ], + "score": 1.0, + "content": "We detail several known caveats and qualifications in Appendix A.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 504, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 338, + 335 + ], + "score": 1.0, + "content": "Architectures. In Appendix J, we fix the task of training a", + "type": "text" + }, + { + "bbox": [ + 338, + 325, + 350, + 335 + ], + "score": 0.47, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "subset of CIFAR-10, and we systemat-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "ically vary the network architecture. We consider fully-connected networks, as well as convolutional", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "networks with both max-pooling and average pooling. For all of these architectures, we consider", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "tanh, ReLU, and ELU activations, and for fully-connected networks we moreover consider softplus", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "and hardtanh —- eleven networks in total. We train each network with both cross-entropy and MSE", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 328, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 328, + 391 + ], + "score": 1.0, + "content": "loss. In each case, we successfully reproduce Figure 5.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 504, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "Since batch normalization (Ioffe & Szegedy, 2015) is known to have unusual optimization properties", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "(Li & Arora, 2019), it is natural to wonder whether our findings still hold with batch normalization.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 418, + 495, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 495, + 430 + ], + "score": 1.0, + "content": "In Appendix K, we confirm that they do, and we reconcile this point with Santurkar et al. (2018).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 504, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "Tasks. In Appendix L, we verify our findings on: (1) a Transformer trained on the WikiText-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "2 language modeling task; (2) fully-connected tanh networks with one hidden layer, trained on a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "score": 1.0, + "content": "synthetic one-dimensional toy regression task; and (3) deep linear networks trained on Gaussian", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 466, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 445, + 480 + ], + "score": 1.0, + "content": "data. In each case, the sharpness rises until hovering right at, or just above, the value", + "type": "text" + }, + { + "bbox": [ + 446, + 467, + 462, + 479 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 466, + 466, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Standard networks on CIFAR-10. In Appendix M, we verify our findings on three standard ar-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "chitectures trained on the full CIFAR-10 dataset: a ResNet with BN, a VGG with BN, and a VGG", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "without BN. For all three architectures, we find that progressive sharpening occurs to a dramatic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "degree, and, relatedly, that stable step sizes are dramatically suboptimal. For example, when we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 194, + 542 + ], + "score": 1.0, + "content": "train the VGG-BN to", + "type": "text" + }, + { + "bbox": [ + 195, + 528, + 215, + 539 + ], + "score": 0.86, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "accuracy using gradient flow / Runge-Kutta, we find that the sharpness", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "rises from 6.3 at initialization to a peak sharpness of 2227.6. Since this is an architecture for which", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "gradient descent closely hews to the gradient flow trajectory, we can conclude that any stable step", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 312, + 574 + ], + "score": 1.0, + "content": "size for gradient descent would need to be less than", + "type": "text" + }, + { + "bbox": [ + 313, + 561, + 403, + 573 + ], + "score": 0.87, + "content": "2 / 2 2 2 7 . 6 = 0 . 0 0 0 8 9 7 .", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 561, + 506, + 574 + ], + "score": 1.0, + "content": ". Training finishes at time", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 393, + 585 + ], + "score": 1.0, + "content": "14.91, so gradient descent at any stable step size would require at least", + "type": "text" + }, + { + "bbox": [ + 394, + 572, + 502, + 584 + ], + "score": 0.7, + "content": "1 4 . 9 1 / 0 . 0 \\bar { 0 } 0 8 9 7 = 1 6 , 6 2 2", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 571, + 505, + 585 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "iterations. Yet empirically, this network can be trained to completion at the larger, “Edge of Stabil-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 170, + 606 + ], + "score": 1.0, + "content": "ity” step size of", + "type": "text" + }, + { + "bbox": [ + 170, + 595, + 208, + 605 + ], + "score": 0.89, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "in just 329 iterations. Therefore, training at a stable step size is suboptimal", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 197, + 618 + ], + "score": 1.0, + "content": "by a factor of at least", + "type": "text" + }, + { + "bbox": [ + 197, + 605, + 277, + 617 + ], + "score": 0.87, + "content": "1 6 6 2 2 / 3 2 9 = 5 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 604, + 505, + 618 + ], + "score": 1.0, + "content": ". The situation is similar for the other two architectures", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "we consider. In short, for standard architectures on the standard dataset CIFAR-10, stable step sizes", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "are not just suboptimally small, they are so suboptimal as to be completely unreasonable. For these", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "networks, gradient descent at any reasonable step size eventually enters the Edge of Stability regime.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "Tracking gradient flow. Recall that for some (but not all) architectures, gradient descent closely", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 407, + 678 + ], + "score": 1.0, + "content": "hews to the gradient flow trajectory so long as the sharpness is less than", + "type": "text" + }, + { + "bbox": [ + 407, + 666, + 424, + 678 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 666, + 505, + 678 + ], + "score": 1.0, + "content": ". Among the archi-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "tectures considered in Appendix J, we found this to be true for the architectures with continuously", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "differentiable components, as well as some, but not all, with ReLU, hardtanh, and max-pooling.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Among the architectures in Appendix L, we found this to be true for the tanh network, but not for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "the deep linear network or the Transformer. Finally, we did find this to be true for the three standard", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 399, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 399, + 732 + ], + "score": 1.0, + "content": "architectures in Appendix M, even though those architectures use ReLU.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 7 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We can numerically approximate gradient flow trajectories by using the Runge-Kutta RK4 algorithm", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "(Press et al., 1992) to numerically integrate the gradient flow ODE. Empirically, for many but not all", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 423, + 117 + ], + "score": 1.0, + "content": "networks studied in this paper, we find that gradient descent at any step size", + "type": "text" + }, + { + "bbox": [ + 423, + 106, + 430, + 116 + ], + "score": 0.73, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "closely tracks the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 484, + 129 + ], + "score": 1.0, + "content": "Runge-Kutta trajectory until reaching the point on that trajectory where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 485, + 115, + 501, + 127 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 114, + 505, + 129 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "(This sometimes occurs even for networks with ReLU activations or max-pooling, which give rise", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "to training objectives that are not continuously differentiable, which means that the gradient flow", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "trajectory is not necessarily guaranteed to exist.) For such networks, the gradient flow trajectory", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "provides a coherent framework for reasoning about which step sizes will eventually enter the Edge", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 171, + 184 + ], + "score": 1.0, + "content": "of Stability. Let", + "type": "text" + }, + { + "bbox": [ + 172, + 171, + 183, + 181 + ], + "score": 0.88, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 169, + 346, + 184 + ], + "score": 1.0, + "content": "be the sharpness at initialization, and let", + "type": "text" + }, + { + "bbox": [ + 346, + 171, + 369, + 182 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 169, + 505, + 184 + ], + "score": 1.0, + "content": "be the maximum sharpness along", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 228, + 195 + ], + "score": 1.0, + "content": "the gradient flow trajectory. If", + "type": "text" + }, + { + "bbox": [ + 228, + 181, + 279, + 193 + ], + "score": 0.92, + "content": "\\eta < 2 / \\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 181, + 506, + 195 + ], + "score": 1.0, + "content": ", then gradient descent will stably track the gradient flow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "trajectory for the entire duration of training, and will never enter the Edge of Stability. On the other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 201, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 139, + 218 + ], + "score": 1.0, + "content": "hand, if", + "type": "text" + }, + { + "bbox": [ + 139, + 203, + 219, + 215 + ], + "score": 0.93, + "content": "\\eta \\in [ 2 / \\lambda _ { \\operatorname* { m a x } } , 2 / \\lambda _ { 0 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 201, + 506, + 218 + ], + "score": 1.0, + "content": ", then gradient descent will stably track the gradient flow trajectory only", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 372, + 227 + ], + "score": 1.0, + "content": "until reaching the point on that trajectory where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 372, + 214, + 389, + 226 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "; shortly afterwards, gradient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "descent will become destabilized, depart the gradient flow trajectory, and enter the Edge of Stability.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 82, + 506, + 239 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 257, + 249, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 251, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 251, + 271 + ], + "score": 1.0, + "content": "4 FURTHER EXPERIMENTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 504, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "Section 3 focused for exposition on a single architecture and task. In this section, we show that our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 297, + 504, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 504, + 307 + ], + "score": 1.0, + "content": "characterization of gradient descent holds broadly across a wide range of architectures and tasks.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 375, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 375, + 319 + ], + "score": 1.0, + "content": "We detail several known caveats and qualifications in Appendix A.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 285, + 506, + 319 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 504, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 338, + 335 + ], + "score": 1.0, + "content": "Architectures. In Appendix J, we fix the task of training a", + "type": "text" + }, + { + "bbox": [ + 338, + 325, + 350, + 335 + ], + "score": 0.47, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "subset of CIFAR-10, and we systemat-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "ically vary the network architecture. We consider fully-connected networks, as well as convolutional", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "networks with both max-pooling and average pooling. For all of these architectures, we consider", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "tanh, ReLU, and ELU activations, and for fully-connected networks we moreover consider softplus", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "and hardtanh —- eleven networks in total. We train each network with both cross-entropy and MSE", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 328, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 328, + 391 + ], + "score": 1.0, + "content": "loss. In each case, we successfully reproduce Figure 5.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 324, + 505, + 391 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 504, + 429 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "Since batch normalization (Ioffe & Szegedy, 2015) is known to have unusual optimization properties", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "(Li & Arora, 2019), it is natural to wonder whether our findings still hold with batch normalization.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 418, + 495, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 495, + 430 + ], + "score": 1.0, + "content": "In Appendix K, we confirm that they do, and we reconcile this point with Santurkar et al. (2018).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 395, + 506, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 504, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "Tasks. In Appendix L, we verify our findings on: (1) a Transformer trained on the WikiText-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "2 language modeling task; (2) fully-connected tanh networks with one hidden layer, trained on a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "score": 1.0, + "content": "synthetic one-dimensional toy regression task; and (3) deep linear networks trained on Gaussian", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 466, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 445, + 480 + ], + "score": 1.0, + "content": "data. In each case, the sharpness rises until hovering right at, or just above, the value", + "type": "text" + }, + { + "bbox": [ + 446, + 467, + 462, + 479 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 466, + 466, + 480 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 434, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Standard networks on CIFAR-10. In Appendix M, we verify our findings on three standard ar-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "chitectures trained on the full CIFAR-10 dataset: a ResNet with BN, a VGG with BN, and a VGG", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "without BN. For all three architectures, we find that progressive sharpening occurs to a dramatic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "degree, and, relatedly, that stable step sizes are dramatically suboptimal. For example, when we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 194, + 542 + ], + "score": 1.0, + "content": "train the VGG-BN to", + "type": "text" + }, + { + "bbox": [ + 195, + 528, + 215, + 539 + ], + "score": 0.86, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "accuracy using gradient flow / Runge-Kutta, we find that the sharpness", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "rises from 6.3 at initialization to a peak sharpness of 2227.6. Since this is an architecture for which", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "gradient descent closely hews to the gradient flow trajectory, we can conclude that any stable step", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 312, + 574 + ], + "score": 1.0, + "content": "size for gradient descent would need to be less than", + "type": "text" + }, + { + "bbox": [ + 313, + 561, + 403, + 573 + ], + "score": 0.87, + "content": "2 / 2 2 2 7 . 6 = 0 . 0 0 0 8 9 7 .", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 561, + 506, + 574 + ], + "score": 1.0, + "content": ". Training finishes at time", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 393, + 585 + ], + "score": 1.0, + "content": "14.91, so gradient descent at any stable step size would require at least", + "type": "text" + }, + { + "bbox": [ + 394, + 572, + 502, + 584 + ], + "score": 0.7, + "content": "1 4 . 9 1 / 0 . 0 \\bar { 0 } 0 8 9 7 = 1 6 , 6 2 2", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 571, + 505, + 585 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "iterations. Yet empirically, this network can be trained to completion at the larger, “Edge of Stabil-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 170, + 606 + ], + "score": 1.0, + "content": "ity” step size of", + "type": "text" + }, + { + "bbox": [ + 170, + 595, + 208, + 605 + ], + "score": 0.89, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "in just 329 iterations. Therefore, training at a stable step size is suboptimal", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 197, + 618 + ], + "score": 1.0, + "content": "by a factor of at least", + "type": "text" + }, + { + "bbox": [ + 197, + 605, + 277, + 617 + ], + "score": 0.87, + "content": "1 6 6 2 2 / 3 2 9 = 5 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 604, + 505, + 618 + ], + "score": 1.0, + "content": ". The situation is similar for the other two architectures", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "we consider. In short, for standard architectures on the standard dataset CIFAR-10, stable step sizes", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "are not just suboptimally small, they are so suboptimal as to be completely unreasonable. For these", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "networks, gradient descent at any reasonable step size eventually enters the Edge of Stability regime.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 484, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "Tracking gradient flow. Recall that for some (but not all) architectures, gradient descent closely", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 407, + 678 + ], + "score": 1.0, + "content": "hews to the gradient flow trajectory so long as the sharpness is less than", + "type": "text" + }, + { + "bbox": [ + 407, + 666, + 424, + 678 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 666, + 505, + 678 + ], + "score": 1.0, + "content": ". Among the archi-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "tectures considered in Appendix J, we found this to be true for the architectures with continuously", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "differentiable components, as well as some, but not all, with ReLU, hardtanh, and max-pooling.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Among the architectures in Appendix L, we found this to be true for the tanh network, but not for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "the deep linear network or the Transformer. Finally, we did find this to be true for the three standard", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 399, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 399, + 732 + ], + "score": 1.0, + "content": "architectures in Appendix M, even though those architectures use ReLU.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 654, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 190, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 106, + 503, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 120 + ], + "score": 1.0, + "content": "We now explain why the behavior of gradient descent at the Edge of Stability contradicts several", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 118, + 297, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 297, + 130 + ], + "score": 1.0, + "content": "pieces of conventional wisdom in optimization.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 142, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 443, + 154 + ], + "score": 1.0, + "content": "At reasonable step sizes, gradient descent cannot be analyzed using (even local)", + "type": "text" + }, + { + "bbox": [ + 444, + 142, + 451, + 152 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "-smoothness", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "Many convergence analyses of gradient descent assume a bound on the sharpness — either globally", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 504, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 420, + 176 + ], + "score": 1.0, + "content": "or, at the very least, along the optimization trajectory. This condition, called", + "type": "text" + }, + { + "bbox": [ + 421, + 164, + 429, + 174 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 163, + 504, + 176 + ], + "score": 1.0, + "content": "-smoothness (Nes-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 173, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 189 + ], + "score": 1.0, + "content": "terov, 1998), is intended to guarantee that each gradient step will decrease the training objective by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 423, + 199 + ], + "score": 1.0, + "content": "a certain amount; the weakest guarantee is that if the local sharpness is less than", + "type": "text" + }, + { + "bbox": [ + 424, + 185, + 441, + 198 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 186, + 506, + 199 + ], + "score": 1.0, + "content": ", then a gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 165, + 209 + ], + "score": 1.0, + "content": "step with size", + "type": "text" + }, + { + "bbox": [ + 165, + 198, + 173, + 208 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "is guaranteed to decrease (rather than increase) the training objective. At a bare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 379, + 220 + ], + "score": 1.0, + "content": "minimum, any convergence analysis of gradient descent based on", + "type": "text" + }, + { + "bbox": [ + 379, + 208, + 387, + 218 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 208, + 506, + 220 + ], + "score": 1.0, + "content": "-smoothness will require the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 343, + 231 + ], + "score": 1.0, + "content": "sharpness along the optimization trajectory to be less than", + "type": "text" + }, + { + "bbox": [ + 344, + 218, + 360, + 230 + ], + "score": 0.84, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 218, + 506, + 231 + ], + "score": 1.0, + "content": ". Yet to the contrary, at the Edge of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 270, + 243 + ], + "score": 1.0, + "content": "Stability, the sharpness hovers just above", + "type": "text" + }, + { + "bbox": [ + 271, + 230, + 287, + 241 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 230, + 505, + 243 + ], + "score": 1.0, + "content": ". Therefore, at any step size for which gradient descent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "enters the Edge of Stability (which, on realistic architectures, includes any reasonable step size),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 280, + 264 + ], + "score": 1.0, + "content": "gradient descent cannot be analyzed using", + "type": "text" + }, + { + "bbox": [ + 280, + 252, + 288, + 261 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "-smoothness. Li et al. (2020b) previously argued that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 233, + 275 + ], + "score": 1.0, + "content": "convergence analyses based on", + "type": "text" + }, + { + "bbox": [ + 233, + 263, + 241, + 272 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "-smoothness do not apply to networks with both batch normaliza-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 474, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 474, + 287 + ], + "score": 1.0, + "content": "tion and weight decay; our paper empirically extends this to neural networks without either.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 114, + 307 + ], + "score": 0.61, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "-smoothness may be inappropriate when analyzing other optimization algorithms too It is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "common for optimization papers seemingly motivated by deep learning to analyze algorithms un-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 205, + 331 + ], + "score": 1.0, + "content": "der the “non-convex but", + "type": "text" + }, + { + "bbox": [ + 205, + 320, + 213, + 329 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "-smooth”’ setting (Reddi et al., 2016; Agarwal et al., 2017; Zaheer et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "2018; Zhou et al., 2018; Chen et al., 2019; Li & Orabona, 2019; Ward et al., 2019; You et al., 2020;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Vaswani et al., 2019; Sankararaman et al., 2019; Reddi et al., 2021; Défossez et al., 2020; Xie et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "2020; Liu et al., 2020; Defazio, 2020). Since our experiments focus on gradient descent, it does not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 201, + 376 + ], + "score": 1.0, + "content": "necessarily follow that", + "type": "text" + }, + { + "bbox": [ + 201, + 363, + 209, + 373 + ], + "score": 0.73, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "-smoothness assumptions are unjustified when analyzing other optimiza-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "tion algorithms. However, gradient descent is arguably the simplest optimization algorithm, so we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 257, + 397 + ], + "score": 1.0, + "content": "believe that the fact that (even local)", + "type": "text" + }, + { + "bbox": [ + 258, + 385, + 266, + 395 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "-smoothness fails even there should raise serious questions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 211, + 409 + ], + "score": 1.0, + "content": "about the suitability of the", + "type": "text" + }, + { + "bbox": [ + 211, + 397, + 219, + 406 + ], + "score": 0.74, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "-smoothness assumption in neural network optimization more generally.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 478, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 478, + 420 + ], + "score": 1.0, + "content": "In particular, the burden of proof should be on authors to empirically justify this assumption.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "score": 1.0, + "content": "At reasonable step sizes, gradient descent does not monotonically decrease the training loss", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "In neural network training, SGD does not monotonically decrease the training objective, in part", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "due to minibatch randomness. However, it is often assumed that full-batch gradient descent would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "monotonically decrease the training objective, were it used to train neural networks. For example,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 263, + 487 + ], + "score": 1.0, + "content": "Zhang et al. (2020) proposed a “relaxed", + "type": "text" + }, + { + "bbox": [ + 263, + 475, + 272, + 485 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "-smoothness” condition that is less restrictive than standard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 114, + 496 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "-smoothness, and proved a convergence guarantee for gradient descent under this condition which", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "asserted that the training objective will decrease monotonically. Likewise, some neural network", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "analyses such as Allen-Zhu et al. (2019) also assert that the training objective will monotonically", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "decrease. Yet, at the Edge of Stability, the training loss behaves non-monotonically over short", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "timescales even as it consistently decreases over long timescales. Therefore, convergence analyses", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 540, + 498, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 498, + 554 + ], + "score": 1.0, + "content": "which assert monotone descent cannot possibly apply to gradient descent at reasonable step sizes.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "The Edge of Stability is inherently non-quadratic It is tempting to try to reason about the behav-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "ior of gradient descent on neural network training objectives by analyzing, as a proxy, the behavior", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "of gradient descent on the local quadratic Taylor approximation (LeCun et al., 1998). However, at", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "the Edge of Stability, the behavior of gradient descent on the real neural training objective is irrec-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "oncilably different from the behavior of gradient descent on the quadratic Taylor approximation: the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "former makes consistent (if choppy) progress, whereas the latter would diverge (and this divergence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "would happen quickly, as we demonstrate in Appendix E). Thus, the behavior of gradient descent at", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 642, + 303, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 303, + 654 + ], + "score": 1.0, + "content": "the Edge of Stability is inherently non-quadratic.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Dogma for step size selection may be unjustified An influential piece of conventional wisdom", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "concerning step size selection has its roots in the quadratic Taylor approximation model of gradient", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "descent. This conventional wisdom (LeCun et al., 1993; 1998; Schaul et al., 2013) holds that if", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 196, + 712 + ], + "score": 1.0, + "content": "the sharpness at step", + "type": "text" + }, + { + "bbox": [ + 196, + 700, + 201, + 709 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 698, + 214, + 712 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 214, + 699, + 224, + 710 + ], + "score": 0.87, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 698, + 338, + 712 + ], + "score": 1.0, + "content": ", then the current step size", + "type": "text" + }, + { + "bbox": [ + 338, + 700, + 348, + 710 + ], + "score": 0.81, + "content": "\\eta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 698, + 468, + 712 + ], + "score": 1.0, + "content": "must be set no greater than", + "type": "text" + }, + { + "bbox": [ + 468, + 699, + 489, + 711 + ], + "score": 0.9, + "content": "2 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "(in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "order to prevent divergence); and furthermore, barring additional information about the objective", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 161, + 733 + ], + "score": 1.0, + "content": "function, that", + "type": "text" + }, + { + "bbox": [ + 161, + 722, + 171, + 732 + ], + "score": 0.84, + "content": "\\eta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 721, + 275, + 733 + ], + "score": 1.0, + "content": "should optimally be set to", + "type": "text" + }, + { + "bbox": [ + 275, + 721, + 296, + 732 + ], + "score": 0.89, + "content": "1 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 721, + 505, + 733 + ], + "score": 1.0, + "content": ". Our findings complicate this conventional wisdom.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 190, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 192, + 96 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 106, + 503, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 120 + ], + "score": 1.0, + "content": "We now explain why the behavior of gradient descent at the Edge of Stability contradicts several", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 118, + 297, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 297, + 130 + ], + "score": 1.0, + "content": "pieces of conventional wisdom in optimization.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 105, + 505, + 130 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 142, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 443, + 154 + ], + "score": 1.0, + "content": "At reasonable step sizes, gradient descent cannot be analyzed using (even local)", + "type": "text" + }, + { + "bbox": [ + 444, + 142, + 451, + 152 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "-smoothness", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "Many convergence analyses of gradient descent assume a bound on the sharpness — either globally", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 504, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 420, + 176 + ], + "score": 1.0, + "content": "or, at the very least, along the optimization trajectory. This condition, called", + "type": "text" + }, + { + "bbox": [ + 421, + 164, + 429, + 174 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 163, + 504, + 176 + ], + "score": 1.0, + "content": "-smoothness (Nes-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 173, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 189 + ], + "score": 1.0, + "content": "terov, 1998), is intended to guarantee that each gradient step will decrease the training objective by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 423, + 199 + ], + "score": 1.0, + "content": "a certain amount; the weakest guarantee is that if the local sharpness is less than", + "type": "text" + }, + { + "bbox": [ + 424, + 185, + 441, + 198 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 186, + 506, + 199 + ], + "score": 1.0, + "content": ", then a gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 165, + 209 + ], + "score": 1.0, + "content": "step with size", + "type": "text" + }, + { + "bbox": [ + 165, + 198, + 173, + 208 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "is guaranteed to decrease (rather than increase) the training objective. At a bare", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 379, + 220 + ], + "score": 1.0, + "content": "minimum, any convergence analysis of gradient descent based on", + "type": "text" + }, + { + "bbox": [ + 379, + 208, + 387, + 218 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 208, + 506, + 220 + ], + "score": 1.0, + "content": "-smoothness will require the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 343, + 231 + ], + "score": 1.0, + "content": "sharpness along the optimization trajectory to be less than", + "type": "text" + }, + { + "bbox": [ + 344, + 218, + 360, + 230 + ], + "score": 0.84, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 218, + 506, + 231 + ], + "score": 1.0, + "content": ". Yet to the contrary, at the Edge of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 270, + 243 + ], + "score": 1.0, + "content": "Stability, the sharpness hovers just above", + "type": "text" + }, + { + "bbox": [ + 271, + 230, + 287, + 241 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 230, + 505, + 243 + ], + "score": 1.0, + "content": ". Therefore, at any step size for which gradient descent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "enters the Edge of Stability (which, on realistic architectures, includes any reasonable step size),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 280, + 264 + ], + "score": 1.0, + "content": "gradient descent cannot be analyzed using", + "type": "text" + }, + { + "bbox": [ + 280, + 252, + 288, + 261 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 251, + 506, + 264 + ], + "score": 1.0, + "content": "-smoothness. Li et al. (2020b) previously argued that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 233, + 275 + ], + "score": 1.0, + "content": "convergence analyses based on", + "type": "text" + }, + { + "bbox": [ + 233, + 263, + 241, + 272 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "-smoothness do not apply to networks with both batch normaliza-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 474, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 474, + 287 + ], + "score": 1.0, + "content": "tion and weight decay; our paper empirically extends this to neural networks without either.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 142, + 506, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 114, + 307 + ], + "score": 0.61, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "-smoothness may be inappropriate when analyzing other optimization algorithms too It is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "common for optimization papers seemingly motivated by deep learning to analyze algorithms un-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 205, + 331 + ], + "score": 1.0, + "content": "der the “non-convex but", + "type": "text" + }, + { + "bbox": [ + 205, + 320, + 213, + 329 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "-smooth”’ setting (Reddi et al., 2016; Agarwal et al., 2017; Zaheer et al.,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "2018; Zhou et al., 2018; Chen et al., 2019; Li & Orabona, 2019; Ward et al., 2019; You et al., 2020;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Vaswani et al., 2019; Sankararaman et al., 2019; Reddi et al., 2021; Défossez et al., 2020; Xie et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "2020; Liu et al., 2020; Defazio, 2020). Since our experiments focus on gradient descent, it does not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 201, + 376 + ], + "score": 1.0, + "content": "necessarily follow that", + "type": "text" + }, + { + "bbox": [ + 201, + 363, + 209, + 373 + ], + "score": 0.73, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "-smoothness assumptions are unjustified when analyzing other optimiza-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "tion algorithms. However, gradient descent is arguably the simplest optimization algorithm, so we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 257, + 397 + ], + "score": 1.0, + "content": "believe that the fact that (even local)", + "type": "text" + }, + { + "bbox": [ + 258, + 385, + 266, + 395 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "-smoothness fails even there should raise serious questions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 211, + 409 + ], + "score": 1.0, + "content": "about the suitability of the", + "type": "text" + }, + { + "bbox": [ + 211, + 397, + 219, + 406 + ], + "score": 0.74, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "-smoothness assumption in neural network optimization more generally.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 407, + 478, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 478, + 420 + ], + "score": 1.0, + "content": "In particular, the burden of proof should be on authors to empirically justify this assumption.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 297, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 443 + ], + "score": 1.0, + "content": "At reasonable step sizes, gradient descent does not monotonically decrease the training loss", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "In neural network training, SGD does not monotonically decrease the training objective, in part", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "due to minibatch randomness. However, it is often assumed that full-batch gradient descent would", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "monotonically decrease the training objective, were it used to train neural networks. For example,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 263, + 487 + ], + "score": 1.0, + "content": "Zhang et al. (2020) proposed a “relaxed", + "type": "text" + }, + { + "bbox": [ + 263, + 475, + 272, + 485 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "-smoothness” condition that is less restrictive than standard", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 485, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 114, + 496 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 485, + 506, + 499 + ], + "score": 1.0, + "content": "-smoothness, and proved a convergence guarantee for gradient descent under this condition which", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "asserted that the training objective will decrease monotonically. Likewise, some neural network", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "analyses such as Allen-Zhu et al. (2019) also assert that the training objective will monotonically", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 531 + ], + "score": 1.0, + "content": "decrease. Yet, at the Edge of Stability, the training loss behaves non-monotonically over short", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "timescales even as it consistently decreases over long timescales. Therefore, convergence analyses", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 540, + 498, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 498, + 554 + ], + "score": 1.0, + "content": "which assert monotone descent cannot possibly apply to gradient descent at reasonable step sizes.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 432, + 506, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "The Edge of Stability is inherently non-quadratic It is tempting to try to reason about the behav-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "ior of gradient descent on neural network training objectives by analyzing, as a proxy, the behavior", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "of gradient descent on the local quadratic Taylor approximation (LeCun et al., 1998). However, at", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "the Edge of Stability, the behavior of gradient descent on the real neural training objective is irrec-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "oncilably different from the behavior of gradient descent on the quadratic Taylor approximation: the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "former makes consistent (if choppy) progress, whereas the latter would diverge (and this divergence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "would happen quickly, as we demonstrate in Appendix E). Thus, the behavior of gradient descent at", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 642, + 303, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 303, + 654 + ], + "score": 1.0, + "content": "the Edge of Stability is inherently non-quadratic.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 564, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Dogma for step size selection may be unjustified An influential piece of conventional wisdom", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "concerning step size selection has its roots in the quadratic Taylor approximation model of gradient", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "descent. This conventional wisdom (LeCun et al., 1993; 1998; Schaul et al., 2013) holds that if", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 196, + 712 + ], + "score": 1.0, + "content": "the sharpness at step", + "type": "text" + }, + { + "bbox": [ + 196, + 700, + 201, + 709 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 698, + 214, + 712 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 214, + 699, + 224, + 710 + ], + "score": 0.87, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 698, + 338, + 712 + ], + "score": 1.0, + "content": ", then the current step size", + "type": "text" + }, + { + "bbox": [ + 338, + 700, + 348, + 710 + ], + "score": 0.81, + "content": "\\eta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 698, + 468, + 712 + ], + "score": 1.0, + "content": "must be set no greater than", + "type": "text" + }, + { + "bbox": [ + 468, + 699, + 489, + 711 + ], + "score": 0.9, + "content": "2 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "(in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "order to prevent divergence); and furthermore, barring additional information about the objective", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 161, + 733 + ], + "score": 1.0, + "content": "function, that", + "type": "text" + }, + { + "bbox": [ + 161, + 722, + 171, + 732 + ], + "score": 0.84, + "content": "\\eta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 721, + 275, + 733 + ], + "score": 1.0, + "content": "should optimally be set to", + "type": "text" + }, + { + "bbox": [ + 275, + 721, + 296, + 732 + ], + "score": 0.89, + "content": "1 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 721, + 505, + 733 + ], + "score": 1.0, + "content": ". Our findings complicate this conventional wisdom.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 665, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "To start, it is nearly impossible to satisfy these prescriptions with a fixed step size: for any fixed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 214, + 107 + ], + "score": 1.0, + "content": "(and reasonable) step size", + "type": "text" + }, + { + "bbox": [ + 214, + 95, + 246, + 105 + ], + "score": 0.89, + "content": "\\eta _ { t } = \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ", progressive sharpening eventually drives gradient descent into", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 326, + 118 + ], + "score": 1.0, + "content": "regions where the sharpness is just a bit greater than", + "type": "text" + }, + { + "bbox": [ + 326, + 105, + 343, + 117 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 104, + 487, + 118 + ], + "score": 1.0, + "content": "— which means that the step size", + "type": "text" + }, + { + "bbox": [ + 487, + 106, + 494, + 116 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "score": 1.0, + "content": "purportedly impermissible. Furthermore, in Appendix F, we try running gradient descent with the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 190, + 139 + ], + "score": 1.0, + "content": "purportedly optimal", + "type": "text" + }, + { + "bbox": [ + 190, + 126, + 236, + 138 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "rule, and find that this algorithm is soundly outperformed by the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 137, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 379, + 151 + ], + "score": 1.0, + "content": "purportedly impermissible baseline of gradient descent with a fixed", + "type": "text" + }, + { + "bbox": [ + 380, + 137, + 424, + 149 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 137, + 492, + 151 + ], + "score": 1.0, + "content": "step size, where", + "type": "text" + }, + { + "bbox": [ + 492, + 137, + 504, + 149 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 261, + 161 + ], + "score": 1.0, + "content": "is the sharpness at initialization. The", + "type": "text" + }, + { + "bbox": [ + 262, + 148, + 307, + 160 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "rule continually anneals the step size, and in so", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 504, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 460, + 173 + ], + "score": 1.0, + "content": "doing ensures that the training objective will decrease at each iteration, whereas the fixed", + "type": "text" + }, + { + "bbox": [ + 461, + 159, + 504, + 171 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { 0 }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "step size often increases the training objective. However, this non-monotonicity turns out to be a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 376, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 376, + 194 + ], + "score": 1.0, + "content": "worthwhile price to pay in return for the ability to take larger steps.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 209, + 299, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 301, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 301, + 223 + ], + "score": 1.0, + "content": "6 STOCHASTIC GRADIENT DESCENT", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 233, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "Our precise characterization of the behavior of the sharpness only applies to full-batch gradient", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "score": 1.0, + "content": "descent. In contrast, during SGD, the sharpness does not always settle at any fixed value (Appendix", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "G), let alone one that can be numerically predicted from the hyperparameters. Nevertheless, prior", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "works (Jastrz˛ebski et al., 2017; 2019; 2020) have demonstrated that large step sizes do steer SGD", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 334, + 290 + ], + "score": 1.0, + "content": "into regions of the landscape with lower sharpness; the", + "type": "text" + }, + { + "bbox": [ + 334, + 277, + 351, + 289 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "rule we have identified for full-batch", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "gradient descent is a special case of this observation. Furthermore, small batch sizes also steer SGD", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "into regions with lower sharpness (Keskar et al., 2016; Jastrz˛ebski et al., 2017), as we illustrate in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "Appendix G. Jastrz˛ebski et al. (2020) attributed this phenomenon to the stability properties of SGD.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 504, + 339 + ], + "score": 1.0, + "content": "Even though our findings only strictly hold for gradient descent, they may have relevance to SGD", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "as well. First, since gradient descent is a special case of SGD, any general characterization of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "dynamics of SGD must reduce to the Edge of Stability in the full-batch special case. Second, there", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "are indications that the Edge of Stability may have some analogue for SGD. One way to interpret our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "main findings is that gradient descent “acclimates” to the step size in such a way that each update", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "sometimes increases and sometimes decrease the train loss, yet an update with a smaller step size", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "would consistently decrease the training loss. Along similar lines, in Appendix H we demonstrate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "that SGD “acclimates” to the step size and batch size in such a way that each SGD update sometimes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "increases and sometimes decreases the training loss in expectation, yet an SGD update with a smaller", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 462, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 462, + 439 + ], + "score": 1.0, + "content": "step size or larger batch size would consistently decrease the training loss in expectation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "In extending these findings to SGD, the question arises of how to model “stability” of SGD. This is a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "highly active area of research. Wu et al. (2018) proposed modeling stability in expectation, and gave", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "a sufficient (but not necessary) criterion for the stability of SGD in expectation. Building on this", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "framework, Jastrz˛ebski et al. (2020) argued that if the Hessian is aligned with the second moment", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "score": 1.0, + "content": "matrix of per-example gradients, then SGD is stable so long as a certain expression (involving the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "sharpness) is below a certain threshold. In the special full-batch case, their criterion reduces to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 507, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 207, + 523 + ], + "score": 1.0, + "content": "sharpness being beneath", + "type": "text" + }, + { + "bbox": [ + 208, + 509, + 225, + 520 + ], + "score": 0.85, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 507, + 505, + 523 + ], + "score": 1.0, + "content": "— a constraint which we have shown is “tight” throughout training.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "However, in the general SGD case, there is no evidence that their stability constraint is tight through-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "out training. Giladi et al. (2020) showed that the generalization gap in asynchronous SGD can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "mostly ameliorated by setting the step size so as to ensure that stability properties in expectation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "remain identical to those of a well-tuned implementation of synchronous SGD. Finally, a number", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 578 + ], + "score": 1.0, + "content": "of papers have attempted to mathematically model the propensity of SGD to “escape from sharp", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 573, + 349, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 349, + 587 + ], + "score": 1.0, + "content": "minima” (Hu et al., 2017; Zhu et al., 2019; Xie et al., 2021).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 601, + 195, + 614 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 197, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 197, + 618 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "We have empirically demonstrated that the behavior of gradient decent on neural training objectives", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "is both surprisingly consistent across architectures and tasks, and surprisingly different from that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "envisioned in the conventional wisdom. Our findings raise a number of questions. Why does pro-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "gressive sharpening occur? At the Edge of Stability, by what mechanism does gradient descent avoid", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "diverging entirely? Since the conventional wisdom for step size selection is wrong, how should the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "gradient descent step size be set during deep learning? Does the “Edge of Stability” effect generalize", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 104, + 693, + 505, + 706 + ], + "spans": [ + { + "bbox": [ + 104, + 693, + 505, + 706 + ], + "score": 1.0, + "content": "in some way to optimization algorithms beyond gradient descent, such as SGD? We hope to inspire", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 704, + 308, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 308, + 716 + ], + "score": 1.0, + "content": "future efforts aimed at addressing these questions.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "To start, it is nearly impossible to satisfy these prescriptions with a fixed step size: for any fixed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 214, + 107 + ], + "score": 1.0, + "content": "(and reasonable) step size", + "type": "text" + }, + { + "bbox": [ + 214, + 95, + 246, + 105 + ], + "score": 0.89, + "content": "\\eta _ { t } = \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ", progressive sharpening eventually drives gradient descent into", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 326, + 118 + ], + "score": 1.0, + "content": "regions where the sharpness is just a bit greater than", + "type": "text" + }, + { + "bbox": [ + 326, + 105, + 343, + 117 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 104, + 487, + 118 + ], + "score": 1.0, + "content": "— which means that the step size", + "type": "text" + }, + { + "bbox": [ + 487, + 106, + 494, + 116 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 507, + 129 + ], + "score": 1.0, + "content": "purportedly impermissible. Furthermore, in Appendix F, we try running gradient descent with the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 190, + 139 + ], + "score": 1.0, + "content": "purportedly optimal", + "type": "text" + }, + { + "bbox": [ + 190, + 126, + 236, + 138 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "rule, and find that this algorithm is soundly outperformed by the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 137, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 379, + 151 + ], + "score": 1.0, + "content": "purportedly impermissible baseline of gradient descent with a fixed", + "type": "text" + }, + { + "bbox": [ + 380, + 137, + 424, + 149 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 137, + 492, + 151 + ], + "score": 1.0, + "content": "step size, where", + "type": "text" + }, + { + "bbox": [ + 492, + 137, + 504, + 149 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 261, + 161 + ], + "score": 1.0, + "content": "is the sharpness at initialization. The", + "type": "text" + }, + { + "bbox": [ + 262, + 148, + 307, + 160 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "rule continually anneals the step size, and in so", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 504, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 460, + 173 + ], + "score": 1.0, + "content": "doing ensures that the training objective will decrease at each iteration, whereas the fixed", + "type": "text" + }, + { + "bbox": [ + 461, + 159, + 504, + 171 + ], + "score": 0.93, + "content": "\\eta _ { t } = 1 / \\lambda _ { 0 }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "step size often increases the training objective. However, this non-monotonicity turns out to be a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 376, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 376, + 194 + ], + "score": 1.0, + "content": "worthwhile price to pay in return for the ability to take larger steps.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 82, + 507, + 194 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 209, + 299, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 301, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 301, + 223 + ], + "score": 1.0, + "content": "6 STOCHASTIC GRADIENT DESCENT", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 233, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "Our precise characterization of the behavior of the sharpness only applies to full-batch gradient", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 258 + ], + "score": 1.0, + "content": "descent. In contrast, during SGD, the sharpness does not always settle at any fixed value (Appendix", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "G), let alone one that can be numerically predicted from the hyperparameters. Nevertheless, prior", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "works (Jastrz˛ebski et al., 2017; 2019; 2020) have demonstrated that large step sizes do steer SGD", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 334, + 290 + ], + "score": 1.0, + "content": "into regions of the landscape with lower sharpness; the", + "type": "text" + }, + { + "bbox": [ + 334, + 277, + 351, + 289 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "rule we have identified for full-batch", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "gradient descent is a special case of this observation. Furthermore, small batch sizes also steer SGD", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "into regions with lower sharpness (Keskar et al., 2016; Jastrz˛ebski et al., 2017), as we illustrate in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "Appendix G. Jastrz˛ebski et al. (2020) attributed this phenomenon to the stability properties of SGD.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 234, + 505, + 323 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 504, + 339 + ], + "score": 1.0, + "content": "Even though our findings only strictly hold for gradient descent, they may have relevance to SGD", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "as well. First, since gradient descent is a special case of SGD, any general characterization of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "dynamics of SGD must reduce to the Edge of Stability in the full-batch special case. Second, there", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "are indications that the Edge of Stability may have some analogue for SGD. One way to interpret our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "main findings is that gradient descent “acclimates” to the step size in such a way that each update", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "sometimes increases and sometimes decrease the train loss, yet an update with a smaller step size", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "would consistently decrease the training loss. Along similar lines, in Appendix H we demonstrate", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "that SGD “acclimates” to the step size and batch size in such a way that each SGD update sometimes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "increases and sometimes decreases the training loss in expectation, yet an SGD update with a smaller", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 462, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 462, + 439 + ], + "score": 1.0, + "content": "step size or larger batch size would consistently decrease the training loss in expectation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 328, + 506, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "In extending these findings to SGD, the question arises of how to model “stability” of SGD. This is a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "highly active area of research. Wu et al. (2018) proposed modeling stability in expectation, and gave", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "a sufficient (but not necessary) criterion for the stability of SGD in expectation. Building on this", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "framework, Jastrz˛ebski et al. (2020) argued that if the Hessian is aligned with the second moment", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 499 + ], + "score": 1.0, + "content": "matrix of per-example gradients, then SGD is stable so long as a certain expression (involving the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "sharpness) is below a certain threshold. In the special full-batch case, their criterion reduces to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 507, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 207, + 523 + ], + "score": 1.0, + "content": "sharpness being beneath", + "type": "text" + }, + { + "bbox": [ + 208, + 509, + 225, + 520 + ], + "score": 0.85, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 507, + 505, + 523 + ], + "score": 1.0, + "content": "— a constraint which we have shown is “tight” throughout training.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "However, in the general SGD case, there is no evidence that their stability constraint is tight through-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "out training. Giladi et al. (2020) showed that the generalization gap in asynchronous SGD can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "mostly ameliorated by setting the step size so as to ensure that stability properties in expectation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "remain identical to those of a well-tuned implementation of synchronous SGD. Finally, a number", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 562, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 578 + ], + "score": 1.0, + "content": "of papers have attempted to mathematically model the propensity of SGD to “escape from sharp", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 573, + 349, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 349, + 587 + ], + "score": 1.0, + "content": "minima” (Hu et al., 2017; Zhu et al., 2019; Xie et al., 2021).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 442, + 506, + 587 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 601, + 195, + 614 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 197, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 197, + 618 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "We have empirically demonstrated that the behavior of gradient decent on neural training objectives", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "is both surprisingly consistent across architectures and tasks, and surprisingly different from that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "envisioned in the conventional wisdom. 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We hope to inspire", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 704, + 308, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 308, + 716 + ], + "score": 1.0, + "content": "future efforts aimed at addressing these questions.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 104, + 627, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 243, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "score": 1.0, + "content": "8 ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 505, + 118 + ], + "score": 1.0, + "content": "This work was supported in part by DARPA FA875017C0141, the National Science Foundation", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 104, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "grants IIS1705121 and IIS1838017, an Amazon Web Services Award, a JP Morgan A.I. Research", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "score": 1.0, + "content": "Faculty Award, a Carnegie Bosch Institute Research Award, a Facebook Faculty Research Award,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "and a Block Center Grant. Any opinions, findings and conclusions or recommendations expressed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 162 + ], + "score": 1.0, + "content": "in this material are those of the author(s) and do not necessarily reflect the views of DARPA, the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 343, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 343, + 175 + ], + "score": 1.0, + "content": "National Science Foundation, or any other funding agency.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 190, + 175, + 202 + ], + "lines": [ + { + "bbox": [ + 106, + 190, + 176, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 176, + 203 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "Naman Agarwal, Zeyuan Allen-Zhu, Brian Bullins, Elad Hazan, and Tengyu Ma. 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The anisotropic noise in stochas-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 116, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 116, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "tic gradient descent: Its behavior of escaping from sharp minima and regularization effects. In", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 116, + 542, + 465, + 556 + ], + "spans": [ + { + "bbox": [ + 116, + 542, + 465, + 556 + ], + "score": 1.0, + "content": "Proceedings of the 36th International Conference on Machine Learning. PMLR, 2019.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 521, + 505, + 556 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 178, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 178, + 96 + ], + "score": 1.0, + "content": "A CAVEATS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we list several caveats to our generic characterization of the dynamics of full-batch", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 327, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 327, + 129 + ], + "score": 1.0, + "content": "gradient descent on neural network training objectives.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 505, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 153 + ], + "score": 1.0, + "content": "1. With cross-entropy loss, the sharpness often drops at the end of training As mentioned in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "the main text, when neural networks are trained on classification tasks using the cross-entropy loss,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 176 + ], + "score": 1.0, + "content": "the sharpness frequently drops near the end of training, once the classification accuracy begins to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 309, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 309, + 186 + ], + "score": 1.0, + "content": "approach 1. We explain this effect in Appendix C.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 197, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "2. For shallow or wide networks, or on simple datasets, sharpness doesn’t rise that much", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 207, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 504, + 221 + ], + "score": 1.0, + "content": "When the network is shallow (Figure 16 - 17) or wide (Figure 12 - 14), or when the dataset is “easy”", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "or small (Figure 18), the sharpness may rise only a small amount over the gradient flow trajectory.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "score": 1.0, + "content": "For these optimization problems, “stable step sizes” (those for which gradient descent never enters", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "the Edge of Stability) may be quite reasonable, and the range of “Edge of Stability” step sizes may", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 168, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 168, + 265 + ], + "score": 1.0, + "content": "be quite small.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "3. Sharpness sometimes drops at the beginning of training We sometimes observe that the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "sharpness drops at the very beginning of training, as the network leaves its initialization. This was", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "more common when training with MSE loss than with cross-entropy loss. For most networks, this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "drop was very slight. However, the combination of both batch normalization and MSE loss some-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "score": 1.0, + "content": "times caused situations where the sharpness was considerably large at initialization, and dropped", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 331, + 419, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 419, + 343 + ], + "score": 1.0, + "content": "precipitously as soon as training began. Figure 8 illustrates one such network.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "image", + "bbox": [ + 133, + 355, + 478, + 423 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 133, + 355, + 478, + 423 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 355, + 478, + 423 + ], + "spans": [ + { + "bbox": [ + 133, + 355, + 478, + 423 + ], + "score": 0.966, + "type": "image", + "image_path": "4f65d59ff48c406a01dc0bef3cbe8d711ac190b425e7e02746021efd9dd37558.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 133, + 355, + 478, + 377.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 133, + 377.6666666666667, + 478, + 400.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 133, + 400.33333333333337, + 478, + 423.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 435, + 506, + 479 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "Figure 8: With batch norm and MSE loss, the sharpness sometimes drops precipitously: We use", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "gradient flow to train a CNN with ReLU activations and batch norm using the MSE loss. Observe", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "that the sharpness drops substantially at the beginning of training, and never recovers to its initial", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 482, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 482, + 480 + ], + "score": 1.0, + "content": "value. This seems to occur due to the combination of both batch normalization and MSE loss.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + } + ], + "index": 21.75 + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 517 + ], + "score": 1.0, + "content": "4. With batch normalization, need to look at sharpness between iterates As detailed in Ap-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 435, + 528 + ], + "score": 1.0, + "content": "pendix K, when we trained batch-normalized networks using very small step sizes", + "type": "text" + }, + { + "bbox": [ + 435, + 517, + 442, + 527 + ], + "score": 0.75, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 515, + 506, + 528 + ], + "score": 1.0, + "content": ", we sometimes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 463, + 538 + ], + "score": 1.0, + "content": "observed that the sharpness at the gradient descent iterates themselves plateaued below", + "type": "text" + }, + { + "bbox": [ + 463, + 526, + 480, + 538 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 525, + 505, + 538 + ], + "score": 1.0, + "content": ", even", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 537, + 439, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 368, + 549 + ], + "score": 1.0, + "content": "though the sharpness in between the iterates plateaued just above", + "type": "text" + }, + { + "bbox": [ + 368, + 537, + 385, + 549 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 537, + 439, + 549 + ], + "score": 1.0, + "content": ", as expected.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "5. With non-differentiable components, instability sometimes begins when the sharpness", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 183, + 584 + ], + "score": 1.0, + "content": "is a bit less than", + "type": "text" + }, + { + "bbox": [ + 183, + 571, + 201, + 583 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "When training networks with ReLU or hardtanh activations, we some-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "times observed that the sharpness started to plateau (and the training loss started to behave non-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 593, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 378, + 606 + ], + "score": 1.0, + "content": "monotonically) a bit before the instant when the sharpness crossed", + "type": "text" + }, + { + "bbox": [ + 379, + 593, + 395, + 605 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 594, + 504, + 606 + ], + "score": 1.0, + "content": ". For example, see Figures", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "41, 43 or Figures 45, 47. One potential explanation is that for such networks, the training objective", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "is not continuously differentiable, so the second-order Taylor approximation around a given iterate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "may be a poor local model for the training objective at even an tiny distance away in weight space, if", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 502, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 397, + 650 + ], + "score": 1.0, + "content": "that weight change causes some activations to switch “ReLU case” from", + "type": "text" + }, + { + "bbox": [ + 397, + 638, + 414, + 648 + ], + "score": 0.87, + "content": "\\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 637, + 425, + 650 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 425, + 638, + 442, + 648 + ], + "score": 0.84, + "content": "> 0", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 637, + 502, + 650 + ], + "score": 1.0, + "content": ", or visa versa.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 178, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 178, + 96 + ], + "score": 1.0, + "content": "A CAVEATS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we list several caveats to our generic characterization of the dynamics of full-batch", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 327, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 327, + 129 + ], + "score": 1.0, + "content": "gradient descent on neural network training objectives.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 106, + 505, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 505, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 153 + ], + "score": 1.0, + "content": "1. With cross-entropy loss, the sharpness often drops at the end of training As mentioned in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "the main text, when neural networks are trained on classification tasks using the cross-entropy loss,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 176 + ], + "score": 1.0, + "content": "the sharpness frequently drops near the end of training, once the classification accuracy begins to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 309, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 309, + 186 + ], + "score": 1.0, + "content": "approach 1. We explain this effect in Appendix C.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 141, + 506, + 186 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 197, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "2. For shallow or wide networks, or on simple datasets, sharpness doesn’t rise that much", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 207, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 504, + 221 + ], + "score": 1.0, + "content": "When the network is shallow (Figure 16 - 17) or wide (Figure 12 - 14), or when the dataset is “easy”", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "or small (Figure 18), the sharpness may rise only a small amount over the gradient flow trajectory.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 243 + ], + "score": 1.0, + "content": "For these optimization problems, “stable step sizes” (those for which gradient descent never enters", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "the Edge of Stability) may be quite reasonable, and the range of “Edge of Stability” step sizes may", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 168, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 168, + 265 + ], + "score": 1.0, + "content": "be quite small.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 197, + 505, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "3. Sharpness sometimes drops at the beginning of training We sometimes observe that the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "sharpness drops at the very beginning of training, as the network leaves its initialization. This was", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "more common when training with MSE loss than with cross-entropy loss. For most networks, this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "drop was very slight. However, the combination of both batch normalization and MSE loss some-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "score": 1.0, + "content": "times caused situations where the sharpness was considerably large at initialization, and dropped", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 331, + 419, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 419, + 343 + ], + "score": 1.0, + "content": "precipitously as soon as training began. 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Observe", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "that the sharpness drops substantially at the beginning of training, and never recovers to its initial", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 482, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 482, + 480 + ], + "score": 1.0, + "content": "value. This seems to occur due to the combination of both batch normalization and MSE loss.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + } + ], + "index": 21.75 + }, + { + "type": "text", + "bbox": [ + 107, + 504, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 517 + ], + "score": 1.0, + "content": "4. With batch normalization, need to look at sharpness between iterates As detailed in Ap-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 435, + 528 + ], + "score": 1.0, + "content": "pendix K, when we trained batch-normalized networks using very small step sizes", + "type": "text" + }, + { + "bbox": [ + 435, + 517, + 442, + 527 + ], + "score": 0.75, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 515, + 506, + 528 + ], + "score": 1.0, + "content": ", we sometimes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 463, + 538 + ], + "score": 1.0, + "content": "observed that the sharpness at the gradient descent iterates themselves plateaued below", + "type": "text" + }, + { + "bbox": [ + 463, + 526, + 480, + 538 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 525, + 505, + 538 + ], + "score": 1.0, + "content": ", even", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 537, + 439, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 368, + 549 + ], + "score": 1.0, + "content": "though the sharpness in between the iterates plateaued just above", + "type": "text" + }, + { + "bbox": [ + 368, + 537, + 385, + 549 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 537, + 439, + 549 + ], + "score": 1.0, + "content": ", as expected.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 502, + 506, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "5. 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For example, see Figures", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "41, 43 or Figures 45, 47. One potential explanation is that for such networks, the training objective", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "is not continuously differentiable, so the second-order Taylor approximation around a given iterate", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "may be a poor local model for the training objective at even an tiny distance away in weight space, if", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 637, + 502, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 397, + 650 + ], + "score": 1.0, + "content": "that weight change causes some activations to switch “ReLU case” from", + "type": "text" + }, + { + "bbox": [ + 397, + 638, + 414, + 648 + ], + "score": 0.87, + "content": "\\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 637, + 425, + 650 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 425, + 638, + 442, + 648 + ], + "score": 0.84, + "content": "> 0", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 637, + 502, + 650 + ], + "score": 1.0, + "content": ", or visa versa.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 560, + 505, + 650 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 451, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 451, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 451, + 96 + ], + "score": 1.0, + "content": "B STABILITY OF GRADIENT DESCENT ON QUADRATIC FUNCTIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "This appendix describes the stability properties of gradient descent (and its momentum variants)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 305, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 305, + 130 + ], + "score": 1.0, + "content": "when optimizing the quadratic objective function", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 133, + 365, + 158 + ], + "lines": [ + { + "bbox": [ + 246, + 133, + 365, + 158 + ], + "spans": [ + { + "bbox": [ + 246, + 133, + 365, + 158 + ], + "score": 0.94, + "content": "f ( \\mathbf { x } ) = { \\frac { 1 } { 2 } } \\mathbf { x } ^ { T } \\mathbf { A } \\mathbf { x } + \\mathbf { b } ^ { T } \\mathbf { x } + c", + "type": "interline_equation", + "image_path": "b08f3fb6d94c4e99f30a8748c8f3362b60436c7cae05531e3748484082a4787b.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 246, + 133, + 365, + 158 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 162, + 243, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 244, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 228, + 176 + ], + "score": 1.0, + "content": "starting from the initialization", + "type": "text" + }, + { + "bbox": [ + 228, + 164, + 240, + 173 + ], + "score": 0.83, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 160, + 244, + 176 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 353, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 353, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 353, + 192 + ], + "score": 1.0, + "content": "To review, vanilla gradient descent is defined by the iteration:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 196, + 354, + 210 + ], + "lines": [ + { + "bbox": [ + 257, + 196, + 354, + 210 + ], + "spans": [ + { + "bbox": [ + 257, + 196, + 354, + 210 + ], + "score": 0.93, + "content": "\\mathbf x _ { t + 1 } = \\mathbf x _ { t } - \\eta \\nabla f ( \\mathbf x _ { t } ) .", + "type": "interline_equation", + "image_path": "be3e753aacdaf65d6bd046fde0850c2c27d5a97c3867beb45c543199b2f64d86.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 257, + 196, + 354, + 210 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "Meanwhile, gradient descent with Polyak (also called “heavy ball”) momentum (Polyak, 1964;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 225, + 404, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 404, + 239 + ], + "score": 1.0, + "content": "Sutskever et al., 2013; Goodfellow et al., 2016) is defined by the iteration:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 242, + 356, + 272 + ], + "lines": [ + { + "bbox": [ + 254, + 242, + 356, + 272 + ], + "spans": [ + { + "bbox": [ + 254, + 242, + 356, + 272 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathbf { v } _ { t + 1 } = \\beta \\mathbf { v } _ { t } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } } \\end{array}", + "type": "interline_equation", + "image_path": "4b2dbb2dcfe8b7269da72de452d53cbcd37299bf3eac43901b6686fb742e4530.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 254, + 242, + 356, + 257.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 254, + 257.0, + 356, + 272.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 504, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 132, + 288 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 277, + 144, + 286 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 275, + 245, + 288 + ], + "score": 1.0, + "content": "is a “velocity” vector and", + "type": "text" + }, + { + "bbox": [ + 246, + 276, + 289, + 287 + ], + "score": 0.93, + "content": "0 \\leq \\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 275, + 423, + 288 + ], + "score": 1.0, + "content": "is the momentum coefficient. For", + "type": "text" + }, + { + "bbox": [ + 423, + 276, + 449, + 287 + ], + "score": 0.89, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "the algorithm", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 287, + 198, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 198, + 298 + ], + "score": 1.0, + "content": "reduces to vanilla GD.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 503, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Finally, Nesterov momentum Sutskever et al. (2013); Goodfellow et al. (2016) is an adaptation of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 314, + 471, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 471, + 327 + ], + "score": 1.0, + "content": "Nesterov’s accelerated gradient (Nesterov, 1983) for deep learning defined by the iteration:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 331, + 370, + 360 + ], + "lines": [ + { + "bbox": [ + 240, + 331, + 370, + 360 + ], + "spans": [ + { + "bbox": [ + 240, + 331, + 370, + 360 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathbf { v } _ { t + 1 } = \\beta \\mathbf { v } _ { t } - \\eta \\nabla f ( \\mathbf { x } _ { t } + \\beta \\mathbf { v } _ { t } ) } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } } \\end{array}", + "type": "interline_equation", + "image_path": "2fc7900f4d1874e51fa52179567b0feb115f1698d8ef9f5e3ac6fa657621cf57.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 331, + 370, + 345.5 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 240, + 345.5, + 370, + 360.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 502, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 364, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 132, + 376 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 366, + 144, + 375 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 364, + 245, + 376 + ], + "score": 1.0, + "content": "is a “velocity” vector and", + "type": "text" + }, + { + "bbox": [ + 245, + 365, + 289, + 376 + ], + "score": 0.94, + "content": "0 \\leq \\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 364, + 423, + 376 + ], + "score": 1.0, + "content": "is the momentum coefficient. For", + "type": "text" + }, + { + "bbox": [ + 423, + 365, + 449, + 376 + ], + "score": 0.85, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 364, + 504, + 376 + ], + "score": 1.0, + "content": "the algorithm", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 198, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 198, + 386 + ], + "score": 1.0, + "content": "reduces to vanilla GD.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "All three of these algorithms share a special property: on quadratic functions, they act independently", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "along each Hessian eigenvector. That is, if we express the iterates in the Hessian eigenvector basis,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 485, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 485, + 427 + ], + "score": 1.0, + "content": "then in this basis the coordinates evolve independent from one another under gradient descent.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 442 + ], + "score": 1.0, + "content": "Proposition 1. Consider running vanilla gradient descent on the quadratic objective (2) starting", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 127, + 453 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 127, + 441, + 139, + 451 + ], + "score": 0.8, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 440, + 160, + 453 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 160, + 440, + 184, + 452 + ], + "score": 0.89, + "content": "( \\mathbf { q } , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 440, + 357, + 453 + ], + "score": 1.0, + "content": "be an eigenvector/eigenvalue pair of A. If", + "type": "text" + }, + { + "bbox": [ + 358, + 440, + 393, + 452 + ], + "score": 0.91, + "content": "a > 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 440, + 471, + 453 + ], + "score": 1.0, + "content": ", then the sequence", + "type": "text" + }, + { + "bbox": [ + 471, + 439, + 504, + 452 + ], + "score": 0.92, + "content": "\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 449, + 159, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 159, + 465 + ], + "score": 1.0, + "content": "will diverge.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 399, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 399, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 399, + 488 + ], + "score": 1.0, + "content": "Proof. 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\\eta \\nabla f ( \\mathbf { x } _ { t } ) } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } } \\end{array}", + "type": "interline_equation", + "image_path": "4b2dbb2dcfe8b7269da72de452d53cbcd37299bf3eac43901b6686fb742e4530.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 254, + 242, + 356, + 257.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 254, + 257.0, + 356, + 272.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 504, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 132, + 288 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 277, + 144, + 286 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 275, + 245, + 288 + ], + "score": 1.0, + "content": "is a “velocity” vector and", + "type": "text" + }, + { + "bbox": [ + 246, + 276, + 289, + 287 + ], + "score": 0.93, + "content": "0 \\leq \\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 275, + 423, + 288 + ], + "score": 1.0, + "content": "is the momentum coefficient. For", + "type": "text" + }, + { + "bbox": [ + 423, + 276, + 449, + 287 + ], + "score": 0.89, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "the algorithm", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 287, + 198, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 198, + 298 + ], + "score": 1.0, + "content": "reduces to vanilla GD.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 275, + 505, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 503, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Finally, Nesterov momentum Sutskever et al. (2013); Goodfellow et al. 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For", + "type": "text" + }, + { + "bbox": [ + 423, + 365, + 449, + 376 + ], + "score": 0.85, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 364, + 504, + 376 + ], + "score": 1.0, + "content": "the algorithm", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 198, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 198, + 386 + ], + "score": 1.0, + "content": "reduces to vanilla GD.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 364, + 504, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "All three of these algorithms share a special property: on quadratic functions, they act independently", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "along each Hessian eigenvector. That is, if we express the iterates in the Hessian eigenvector basis,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 485, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 485, + 427 + ], + "score": 1.0, + "content": "then in this basis the coordinates evolve independent from one another under gradient descent.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 392, + 505, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 442 + ], + "score": 1.0, + "content": "Proposition 1. Consider running vanilla gradient descent on the quadratic objective (2) starting", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 504, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 127, + 453 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 127, + 441, + 139, + 451 + ], + "score": 0.8, + "content": "\\mathbf { x } _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 440, + 160, + 453 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 160, + 440, + 184, + 452 + ], + "score": 0.89, + "content": "( \\mathbf { q } , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 440, + 357, + 453 + ], + "score": 1.0, + "content": "be an eigenvector/eigenvalue pair of A. If", + "type": "text" + }, + { + "bbox": [ + 358, + 440, + 393, + 452 + ], + "score": 0.91, + "content": "a > 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 440, + 471, + 453 + ], + "score": 1.0, + "content": ", then the sequence", + "type": "text" + }, + { + "bbox": [ + 471, + 439, + 504, + 452 + ], + "score": 0.92, + "content": "\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 449, + 159, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 159, + 465 + ], + "score": 1.0, + "content": "will diverge.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 427, + 505, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 399, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 399, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 399, + 488 + ], + "score": 1.0, + "content": "Proof. The update rule for gradient descent on this quadratic function is:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 474, + 399, + 488 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 491, + 361, + 521 + ], + "lines": [ + { + "bbox": [ + 250, + 491, + 361, + 521 + ], + "spans": [ + { + "bbox": [ + 250, + 491, + 361, + 521 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } - \\eta ( \\mathbf { A } \\mathbf { x } _ { t } + \\mathbf { b } ) } \\\\ { = ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\eta \\mathbf { b } . } \\end{array}", + "type": "interline_equation", + "image_path": "cc33d683ab17cdba521de10972b9913d629706f5672fa14ed396a67497bcf2d9.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 491, + 361, + 506.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 250, + 506.0, + 361, + 521.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 525, + 363, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 524, + 364, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 200, + 540 + ], + "score": 1.0, + "content": "Therefore, the quantity", + "type": "text" + }, + { + "bbox": [ + 200, + 525, + 223, + 538 + ], + "score": 0.92, + "content": "\\mathbf { q } ^ { T } \\mathbf { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 524, + 364, + 540 + ], + "score": 1.0, + "content": "evolves under gradient descent as:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 524, + 364, + 540 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 542, + 379, + 576 + ], + "lines": [ + { + "bbox": [ + 232, + 542, + 379, + 576 + ], + "spans": [ + { + "bbox": [ + 232, + 542, + 379, + 576 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbf { q } ^ { T } \\mathbf { x } _ { t + 1 } = \\mathbf { q } ^ { T } ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\eta \\mathbf { b } } \\\\ & { \\qquad = ( 1 - \\eta a ) \\mathbf { q } ^ { T } \\mathbf { x } _ { t } - \\eta \\mathbf { q } ^ { T } \\mathbf { b } . } \\end{array}", + "type": "interline_equation", + "image_path": "9cab90bcb3a949bb22abc1fff25329327a36d8ce439c3b0a566b603eaa7c3cba.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 542, + 379, + 559.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 232, + 559.0, + 379, + 576.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 449, + 559, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 449, + 559, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 449, + 559, + 505, + 574 + ], + "score": 0.85, + "content": "( \\mathbf { q } ^ { T } \\mathbf { A } = a \\mathbf { q } )", + "type": "interline_equation", + "image_path": "4df55f00b4539a762d0888911b989e2ae1553327b3052871f8292528df217304.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 449, + 559, + 505, + 574 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 586, + 460, + 600 + ], + "lines": [ + { + "bbox": [ + 104, + 584, + 462, + 602 + ], + "spans": [ + { + "bbox": [ + 104, + 584, + 135, + 602 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 136, + 586, + 219, + 600 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { x } _ { t } = \\mathbf q ^ { T } \\mathbf x _ { t } + \\frac { 1 } { a } \\mathbf q ^ { T } \\mathbf b } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 584, + 276, + 602 + ], + "score": 1.0, + "content": ", and note that", + "type": "text" + }, + { + "bbox": [ + 277, + 586, + 309, + 600 + ], + "score": 0.93, + "content": "\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 584, + 401, + 602 + ], + "score": 1.0, + "content": "diverges if and only if", + "type": "text" + }, + { + "bbox": [ + 401, + 587, + 421, + 599 + ], + "score": 0.92, + "content": "\\{ \\tilde { x } _ { t } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 584, + 462, + 602 + ], + "score": 1.0, + "content": "diverges.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 584, + 462, + 602 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 604, + 414, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 413, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 159, + 618 + ], + "score": 1.0, + "content": "The quantity", + "type": "text" + }, + { + "bbox": [ + 160, + 605, + 170, + 615 + ], + "score": 0.89, + "content": "\\tilde { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 603, + 413, + 618 + ], + "score": 1.0, + "content": "evolves under gradient descent according to the simple rule:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 603, + 413, + 618 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 621, + 345, + 635 + ], + "lines": [ + { + "bbox": [ + 265, + 621, + 345, + 635 + ], + "spans": [ + { + "bbox": [ + 265, + 621, + 345, + 635 + ], + "score": 0.91, + "content": "\\tilde { x } _ { t + 1 } = ( 1 - \\eta a ) \\tilde { x } _ { t }", + "type": "interline_equation", + "image_path": "b927b8e51febf2c128a19445c33dec5e8a7dd8448c5225dab175fc47f6e156d2.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 265, + 621, + 345, + 635 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 645, + 429, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 430, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 131, + 660 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 646, + 156, + 658 + ], + "score": 0.9, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 644, + 168, + 660 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 168, + 646, + 204, + 658 + ], + "score": 0.92, + "content": "a > 2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 644, + 225, + 660 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 225, + 646, + 287, + 658 + ], + "score": 0.91, + "content": "( 1 - \\eta a ) < - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 644, + 356, + 660 + ], + "score": 1.0, + "content": ", so the sequence", + "type": "text" + }, + { + "bbox": [ + 356, + 646, + 376, + 658 + ], + "score": 0.92, + "content": "\\{ \\tilde { x } _ { t } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 644, + 430, + 660 + ], + "score": 1.0, + "content": "will diverge.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 644, + 430, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 387, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 387, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 387, + 684 + ], + "score": 1.0, + "content": "We now prove analogous results for Nesterov and Polyak momentum.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 668, + 387, + 684 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 125 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Theorem 1. Consider running Nesterov momentum on the quadratic objective (2) starting from", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 104, + 95, + 199, + 111 + ], + "score": 1.0, + "content": "any initialization. Let", + "type": "text" + }, + { + "bbox": [ + 199, + 97, + 223, + 109 + ], + "score": 0.91, + "content": "( \\mathbf { q } , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 95, + 402, + 111 + ], + "score": 1.0, + "content": "be an eigenvector/eigenvalue pair of A. If", + "type": "text" + }, + { + "bbox": [ + 402, + 93, + 465, + 113 + ], + "score": 0.93, + "content": "\\begin{array} { r } { a > \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 93, + 505, + 113 + ], + "score": 1.0, + "content": ", then the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 231, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 146, + 127 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 111, + 178, + 125 + ], + "score": 0.92, + "content": "\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 111, + 231, + 127 + ], + "score": 1.0, + "content": "will diverge.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 105, + 136, + 426, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 428, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 428, + 150 + ], + "score": 1.0, + "content": "Proof. The update rules for Nesterov momentum on this quadratic function are:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 153, + 380, + 182 + ], + "lines": [ + { + "bbox": [ + 230, + 153, + 380, + 182 + ], + "spans": [ + { + "bbox": [ + 230, + 153, + 380, + 182 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbf { v } _ { t + 1 } = \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { v } _ { t } - \\eta \\mathbf { b } - \\eta \\mathbf { A } \\mathbf { x } _ { t } } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "f616dcc37ade3672fcc0d5f0ddd3fffc41aad18dd30b5e6fe7a6438f4b489daa.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 230, + 153, + 380, + 182 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 435, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 437, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 182, + 206 + ], + "score": 1.0, + "content": "Using the fact that", + "type": "text" + }, + { + "bbox": [ + 182, + 194, + 248, + 204 + ], + "score": 0.9, + "content": "\\mathbf { v } _ { t } = \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 192, + 396, + 206 + ], + "score": 1.0, + "content": ", we can rewrite this as a recursion in", + "type": "text" + }, + { + "bbox": [ + 397, + 195, + 407, + 204 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 192, + 437, + 206 + ], + "score": 1.0, + "content": "alone:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 209, + 413, + 239 + ], + "lines": [ + { + "bbox": [ + 197, + 209, + 413, + 239 + ], + "spans": [ + { + "bbox": [ + 197, + 209, + 413, + 239 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) ( \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 } ) - \\eta \\mathbf { b } - \\eta \\mathbf { A } \\mathbf { x } _ { t } } \\\\ { = ( 1 + \\beta ) ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { b } } \\end{array}", + "type": "interline_equation", + "image_path": "023dd9f49c717cdb578bdfa3754ebf0214d1b0c545df2e63a91a65620addb629.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 197, + 209, + 413, + 239 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 249, + 504, + 273 + ], + "lines": [ + { + "bbox": [ + 104, + 248, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 136, + 264 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 137, + 249, + 224, + 263 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { x } _ { t } = { \\bf q } ^ { T } { \\bf x } _ { t } + \\frac { 1 } { a } { \\bf q } ^ { T } { \\bf b } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 248, + 285, + 264 + ], + "score": 1.0, + "content": ", and note that", + "type": "text" + }, + { + "bbox": [ + 286, + 249, + 309, + 262 + ], + "score": 0.92, + "content": "\\mathbf { q } ^ { T } \\mathbf { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 248, + 360, + 264 + ], + "score": 1.0, + "content": "diverges iff", + "type": "text" + }, + { + "bbox": [ + 360, + 251, + 371, + 261 + ], + "score": 0.88, + "content": "\\tilde { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 248, + 493, + 264 + ], + "score": 1.0, + "content": "diverges. It can be seen that", + "type": "text" + }, + { + "bbox": [ + 493, + 251, + 504, + 261 + ], + "score": 0.85, + "content": "\\tilde { x } _ { t }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 259, + 153, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 153, + 274 + ], + "score": 1.0, + "content": "evolves as:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 278, + 398, + 292 + ], + "lines": [ + { + "bbox": [ + 213, + 278, + 398, + 292 + ], + "spans": [ + { + "bbox": [ + 213, + 278, + 398, + 292 + ], + "score": 0.9, + "content": "\\tilde { x } _ { t + 1 } = ( 1 + \\beta ) ( 1 - \\eta a ) \\tilde { x } _ { t } - \\beta ( 1 - \\eta a ) \\tilde { x } _ { t - 1 }", + "type": "interline_equation", + "image_path": "86072fb8138371a0270d0e5423ca362d419de9fbe0334bd4af1a3e076fee7bcb.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 213, + 278, + 398, + 292 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 504, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "This is a linear homogenous second-order difference equation. By Theorem 2.37 in Elaydi (2005),", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 101, + 309, + 394, + 338 + ], + "spans": [ + { + "bbox": [ + 101, + 309, + 129, + 338 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 318, + 154, + 329 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 309, + 172, + 338 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 317, + 198, + 329 + ], + "score": 0.9, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 309, + 210, + 338 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 211, + 313, + 271, + 334 + ], + "score": 0.95, + "content": "\\begin{array} { r } { a > \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 309, + 394, + 338 + ], + "score": 1.0, + "content": "then this recurrence diverges.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 334, + 373 + ], + "lines": [ + { + "bbox": [ + 107, + 361, + 333, + 374 + ], + "spans": [ + { + "bbox": [ + 107, + 361, + 333, + 374 + ], + "score": 1.0, + "content": "The following result previously appeared in Goh (2017).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 506, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 390 + ], + "score": 1.0, + "content": "Theorem 2. Consider running Polyak momentum on the quadratic objective (2) starting from any", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 386, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 184, + 403 + ], + "score": 1.0, + "content": "initialization. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 388, + 208, + 399 + ], + "score": 0.91, + "content": "( \\mathbf { q } , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 386, + 394, + 403 + ], + "score": 1.0, + "content": "be an eigenvector/eigenvalue pair of A. If", + "type": "text" + }, + { + "bbox": [ + 394, + 387, + 463, + 402 + ], + "score": 0.94, + "content": "a ~ > ~ \\frac { 1 } { \\eta } \\left( 2 + 2 \\beta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 386, + 506, + 403 + ], + "score": 1.0, + "content": ", then the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 400, + 232, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 145, + 416 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 401, + 178, + 414 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { q } ^ { T } \\mathbf { x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 400, + 232, + 416 + ], + "score": 1.0, + "content": "will diverge.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 504, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 214, + 439 + ], + "score": 1.0, + "content": "Proof. Using the fact that", + "type": "text" + }, + { + "bbox": [ + 214, + 427, + 282, + 437 + ], + "score": 0.9, + "content": "\\mathbf v _ { t } = \\mathbf x _ { t } - \\mathbf x _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 424, + 506, + 439 + ], + "score": 1.0, + "content": ", we can re-write the Polyak momentum recursion as a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 436, + 192, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 156, + 449 + ], + "score": 1.0, + "content": "recursion in", + "type": "text" + }, + { + "bbox": [ + 156, + 438, + 164, + 446 + ], + "score": 0.69, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 436, + 192, + 449 + ], + "score": 1.0, + "content": "alone:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 452, + 390, + 497 + ], + "lines": [ + { + "bbox": [ + 220, + 452, + 390, + 497 + ], + "spans": [ + { + "bbox": [ + 220, + 452, + 390, + 497 + ], + "score": 0.93, + "content": "\\begin{array} { r l } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\beta \\mathbf { v } _ { t } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) } & { } \\\\ { = \\mathbf { x } _ { t } + \\left( \\beta ( \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 } ) - \\eta \\nabla f ( \\mathbf { x } _ { t } ) \\right) } & { } \\\\ { = ( 1 + \\beta ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "b2e7fb1644cf9b35f467a98b39d645fdde2da527ee28ef5991b877863faaa65d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 220, + 452, + 390, + 467.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 220, + 467.0, + 390, + 482.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 220, + 482.0, + 390, + 497.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 345, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 345, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 345, + 514 + ], + "score": 1.0, + "content": "For the quadratic objective (2), this update rule amounts to:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 516, + 394, + 546 + ], + "lines": [ + { + "bbox": [ + 217, + 516, + 394, + 546 + ], + "spans": [ + { + "bbox": [ + 217, + 516, + 394, + 546 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathbf { x } _ { t + 1 } = ( 1 + \\beta ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta ( \\mathbf { A } \\mathbf { x } _ { t } + \\mathbf { b } ) } \\\\ & { \\qquad = ( 1 + \\beta - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { b } . } \\end{array}", + "type": "interline_equation", + "image_path": "2baf7033190aa82ab06c7ec5f6b2926fcd86e37253a84a1305f697795aba186a.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 217, + 516, + 394, + 546 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 231, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 230, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 169, + 565 + ], + "score": 1.0, + "content": "Multiplying by", + "type": "text" + }, + { + "bbox": [ + 169, + 551, + 182, + 564 + ], + "score": 0.88, + "content": "\\mathbf { q } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 550, + 230, + 565 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 568, + 412, + 583 + ], + "lines": [ + { + "bbox": [ + 198, + 568, + 412, + 583 + ], + "spans": [ + { + "bbox": [ + 198, + 568, + 412, + 583 + ], + "score": 0.89, + "content": "\\mathbf { q } ^ { T } \\mathbf { x } _ { t + 1 } = ( 1 + \\beta - \\eta a ) \\mathbf { q } ^ { T } \\mathbf { x } _ { t } - \\beta \\mathbf { q } ^ { T } \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { q } ^ { T } \\mathbf { b } .", + "type": "interline_equation", + "image_path": "a2d6477c3eb33e96773ff2908186d1aa508a8f03a3c5b6c188acaf52cf3a8f24.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 198, + 568, + 412, + 583 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 504, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 158, + 604 + ], + "score": 1.0, + "content": "Now, define", + "type": "text" + }, + { + "bbox": [ + 158, + 589, + 244, + 604 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { x } _ { t } = \\mathbf q ^ { T } \\mathbf x _ { t } + \\frac { 1 } { a } \\mathbf q ^ { T } \\mathbf b } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 588, + 290, + 604 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 290, + 590, + 313, + 602 + ], + "score": 0.92, + "content": "\\mathbf { q } ^ { T } \\mathbf { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 588, + 363, + 604 + ], + "score": 1.0, + "content": "diverges iff", + "type": "text" + }, + { + "bbox": [ + 363, + 591, + 374, + 601 + ], + "score": 0.89, + "content": "\\tilde { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 588, + 493, + 604 + ], + "score": 1.0, + "content": "diverges. It can be seen that", + "type": "text" + }, + { + "bbox": [ + 493, + 591, + 504, + 601 + ], + "score": 0.86, + "content": "\\tilde { x } _ { t }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 601, + 153, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 153, + 614 + ], + "score": 1.0, + "content": "evolves as:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 618, + 375, + 632 + ], + "lines": [ + { + "bbox": [ + 236, + 618, + 375, + 632 + ], + "spans": [ + { + "bbox": [ + 236, + 618, + 375, + 632 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\tilde { x } _ { t + 1 } = ( 1 + \\beta - \\eta a ) \\tilde { x } _ { t } - \\beta \\tilde { x } _ { t - 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "6207d746a126d94a118c85d3743bd9436b973920f5defcc09e6f3c46fd43b8fd.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 236, + 618, + 375, + 632 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 643, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "This is a linear homogenous second-order difference equation. By Theorem 2.37 in Elaydi (2005),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 129, + 667 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 655, + 154, + 666 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 653, + 172, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 654, + 198, + 666 + ], + "score": 0.91, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 653, + 210, + 667 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 211, + 653, + 274, + 669 + ], + "score": 0.94, + "content": "\\begin{array} { r } { a > \\frac { 1 } { \\eta } ( 2 + 2 \\beta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 653, + 395, + 667 + ], + "score": 1.0, + "content": "then this recurrence diverges.", + "type": "text" + }, + { + "bbox": [ + 494, + 654, + 506, + 666 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 338, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 495, + 339, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 495, + 339, + 505, + 349 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 125 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Theorem 1. Consider running Nesterov momentum on the quadratic objective (2) starting from", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 104, + 95, + 199, + 111 + ], + "score": 1.0, + "content": "any initialization. Let", + "type": "text" + }, + { + "bbox": [ + 199, + 97, + 223, + 109 + ], + "score": 0.91, + "content": "( \\mathbf { q } , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 95, + 402, + 111 + ], + "score": 1.0, + "content": "be an eigenvector/eigenvalue pair of A. If", + "type": "text" + }, + { + "bbox": [ + 402, + 93, + 465, + 113 + ], + "score": 0.93, + "content": "\\begin{array} { r } { a > \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 93, + 505, + 113 + ], + "score": 1.0, + "content": ", then the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 231, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 146, + 127 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 111, + 178, + 125 + ], + "score": 0.92, + "content": "\\left\\{ { \\bf q } ^ { T } { \\bf x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 111, + 231, + 127 + ], + "score": 1.0, + "content": "will diverge.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 104, + 81, + 506, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 136, + 426, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 135, + 428, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 428, + 150 + ], + "score": 1.0, + "content": "Proof. The update rules for Nesterov momentum on this quadratic function are:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 135, + 428, + 150 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 153, + 380, + 182 + ], + "lines": [ + { + "bbox": [ + 230, + 153, + 380, + 182 + ], + "spans": [ + { + "bbox": [ + 230, + 153, + 380, + 182 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbf { v } _ { t + 1 } = \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { v } _ { t } - \\eta \\mathbf { b } - \\eta \\mathbf { A } \\mathbf { x } _ { t } } \\\\ & { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\mathbf { v } _ { t + 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "f616dcc37ade3672fcc0d5f0ddd3fffc41aad18dd30b5e6fe7a6438f4b489daa.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 230, + 153, + 380, + 182 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 435, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 437, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 182, + 206 + ], + "score": 1.0, + "content": "Using the fact that", + "type": "text" + }, + { + "bbox": [ + 182, + 194, + 248, + 204 + ], + "score": 0.9, + "content": "\\mathbf { v } _ { t } = \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 192, + 396, + 206 + ], + "score": 1.0, + "content": ", we can rewrite this as a recursion in", + "type": "text" + }, + { + "bbox": [ + 397, + 195, + 407, + 204 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 192, + 437, + 206 + ], + "score": 1.0, + "content": "alone:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 192, + 437, + 206 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 209, + 413, + 239 + ], + "lines": [ + { + "bbox": [ + 197, + 209, + 413, + 239 + ], + "spans": [ + { + "bbox": [ + 197, + 209, + 413, + 239 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) ( \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 } ) - \\eta \\mathbf { b } - \\eta \\mathbf { A } \\mathbf { x } _ { t } } \\\\ { = ( 1 + \\beta ) ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\beta ( \\mathbf { I } - \\eta \\mathbf { A } ) \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { b } } \\end{array}", + "type": "interline_equation", + "image_path": "023dd9f49c717cdb578bdfa3754ebf0214d1b0c545df2e63a91a65620addb629.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 197, + 209, + 413, + 239 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 249, + 504, + 273 + ], + "lines": [ + { + "bbox": [ + 104, + 248, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 136, + 264 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 137, + 249, + 224, + 263 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { x } _ { t } = { \\bf q } ^ { T } { \\bf x } _ { t } + \\frac { 1 } { a } { \\bf q } ^ { T } { \\bf b } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 248, + 285, + 264 + ], + "score": 1.0, + "content": ", and note that", + "type": "text" + }, + { + "bbox": [ + 286, + 249, + 309, + 262 + ], + "score": 0.92, + "content": "\\mathbf { q } ^ { T } \\mathbf { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 248, + 360, + 264 + ], + "score": 1.0, + "content": "diverges iff", + "type": "text" + }, + { + "bbox": [ + 360, + 251, + 371, + 261 + ], + "score": 0.88, + "content": "\\tilde { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 248, + 493, + 264 + ], + "score": 1.0, + "content": "diverges. It can be seen that", + "type": "text" + }, + { + "bbox": [ + 493, + 251, + 504, + 261 + ], + "score": 0.85, + "content": "\\tilde { x } _ { t }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 259, + 153, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 153, + 274 + ], + "score": 1.0, + "content": "evolves as:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 248, + 504, + 274 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 278, + 398, + 292 + ], + "lines": [ + { + "bbox": [ + 213, + 278, + 398, + 292 + ], + "spans": [ + { + "bbox": [ + 213, + 278, + 398, + 292 + ], + "score": 0.9, + "content": "\\tilde { x } _ { t + 1 } = ( 1 + \\beta ) ( 1 - \\eta a ) \\tilde { x } _ { t } - \\beta ( 1 - \\eta a ) \\tilde { x } _ { t - 1 }", + "type": "interline_equation", + "image_path": "86072fb8138371a0270d0e5423ca362d419de9fbe0334bd4af1a3e076fee7bcb.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 213, + 278, + 398, + 292 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 302, + 504, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "This is a linear homogenous second-order difference equation. By Theorem 2.37 in Elaydi (2005),", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 101, + 309, + 394, + 338 + ], + "spans": [ + { + "bbox": [ + 101, + 309, + 129, + 338 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 318, + 154, + 329 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 309, + 172, + 338 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 173, + 317, + 198, + 329 + ], + "score": 0.9, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 309, + 210, + 338 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 211, + 313, + 271, + 334 + ], + "score": 0.95, + "content": "\\begin{array} { r } { a > \\frac { 1 } { \\eta } \\left( \\frac { 2 + 2 \\beta } { 1 + 2 \\beta } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 309, + 394, + 338 + ], + "score": 1.0, + "content": "then this recurrence diverges.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 101, + 302, + 505, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 334, + 373 + ], + "lines": [ + { + "bbox": [ + 107, + 361, + 333, + 374 + ], + "spans": [ + { + "bbox": [ + 107, + 361, + 333, + 374 + ], + "score": 1.0, + "content": "The following result previously appeared in Goh (2017).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 107, + 361, + 333, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 506, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 390 + ], + "score": 1.0, + "content": "Theorem 2. Consider running Polyak momentum on the quadratic objective (2) starting from any", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 386, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 184, + 403 + ], + "score": 1.0, + "content": "initialization. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 388, + 208, + 399 + ], + "score": 0.91, + "content": "( \\mathbf { q } , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 386, + 394, + 403 + ], + "score": 1.0, + "content": "be an eigenvector/eigenvalue pair of A. If", + "type": "text" + }, + { + "bbox": [ + 394, + 387, + 463, + 402 + ], + "score": 0.94, + "content": "a ~ > ~ \\frac { 1 } { \\eta } \\left( 2 + 2 \\beta \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 386, + 506, + 403 + ], + "score": 1.0, + "content": ", then the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 400, + 232, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 145, + 416 + ], + "score": 1.0, + "content": "sequence", + "type": "text" + }, + { + "bbox": [ + 146, + 401, + 178, + 414 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { q } ^ { T } \\mathbf { x } _ { t } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 400, + 232, + 416 + ], + "score": 1.0, + "content": "will diverge.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 374, + 506, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 504, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 214, + 439 + ], + "score": 1.0, + "content": "Proof. Using the fact that", + "type": "text" + }, + { + "bbox": [ + 214, + 427, + 282, + 437 + ], + "score": 0.9, + "content": "\\mathbf v _ { t } = \\mathbf x _ { t } - \\mathbf x _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 424, + 506, + 439 + ], + "score": 1.0, + "content": ", we can re-write the Polyak momentum recursion as a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 436, + 192, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 156, + 449 + ], + "score": 1.0, + "content": "recursion in", + "type": "text" + }, + { + "bbox": [ + 156, + 438, + 164, + 446 + ], + "score": 0.69, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 436, + 192, + 449 + ], + "score": 1.0, + "content": "alone:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 424, + 506, + 449 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 452, + 390, + 497 + ], + "lines": [ + { + "bbox": [ + 220, + 452, + 390, + 497 + ], + "spans": [ + { + "bbox": [ + 220, + 452, + 390, + 497 + ], + "score": 0.93, + "content": "\\begin{array} { r l } { \\mathbf { x } _ { t + 1 } = \\mathbf { x } _ { t } + \\beta \\mathbf { v } _ { t } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) } & { } \\\\ { = \\mathbf { x } _ { t } + \\left( \\beta ( \\mathbf { x } _ { t } - \\mathbf { x } _ { t - 1 } ) - \\eta \\nabla f ( \\mathbf { x } _ { t } ) \\right) } & { } \\\\ { = ( 1 + \\beta ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta \\nabla f ( \\mathbf { x } _ { t } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "b2e7fb1644cf9b35f467a98b39d645fdde2da527ee28ef5991b877863faaa65d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 220, + 452, + 390, + 467.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 220, + 467.0, + 390, + 482.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 220, + 482.0, + 390, + 497.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 345, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 345, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 345, + 514 + ], + "score": 1.0, + "content": "For the quadratic objective (2), this update rule amounts to:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 500, + 345, + 514 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 516, + 394, + 546 + ], + "lines": [ + { + "bbox": [ + 217, + 516, + 394, + 546 + ], + "spans": [ + { + "bbox": [ + 217, + 516, + 394, + 546 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathbf { x } _ { t + 1 } = ( 1 + \\beta ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta ( \\mathbf { A } \\mathbf { x } _ { t } + \\mathbf { b } ) } \\\\ & { \\qquad = ( 1 + \\beta - \\eta \\mathbf { A } ) \\mathbf { x } _ { t } - \\beta \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { b } . } \\end{array}", + "type": "interline_equation", + "image_path": "2baf7033190aa82ab06c7ec5f6b2926fcd86e37253a84a1305f697795aba186a.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 217, + 516, + 394, + 546 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 231, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 230, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 169, + 565 + ], + "score": 1.0, + "content": "Multiplying by", + "type": "text" + }, + { + "bbox": [ + 169, + 551, + 182, + 564 + ], + "score": 0.88, + "content": "\\mathbf { q } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 550, + 230, + 565 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 550, + 230, + 565 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 568, + 412, + 583 + ], + "lines": [ + { + "bbox": [ + 198, + 568, + 412, + 583 + ], + "spans": [ + { + "bbox": [ + 198, + 568, + 412, + 583 + ], + "score": 0.89, + "content": "\\mathbf { q } ^ { T } \\mathbf { x } _ { t + 1 } = ( 1 + \\beta - \\eta a ) \\mathbf { q } ^ { T } \\mathbf { x } _ { t } - \\beta \\mathbf { q } ^ { T } \\mathbf { x } _ { t - 1 } - \\eta \\mathbf { q } ^ { T } \\mathbf { b } .", + "type": "interline_equation", + "image_path": "a2d6477c3eb33e96773ff2908186d1aa508a8f03a3c5b6c188acaf52cf3a8f24.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 198, + 568, + 412, + 583 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 504, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 158, + 604 + ], + "score": 1.0, + "content": "Now, define", + "type": "text" + }, + { + "bbox": [ + 158, + 589, + 244, + 604 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\tilde { x } _ { t } = \\mathbf q ^ { T } \\mathbf x _ { t } + \\frac { 1 } { a } \\mathbf q ^ { T } \\mathbf b } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 588, + 290, + 604 + ], + "score": 1.0, + "content": ". 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y ) = p ( 1 - p ) .", + "type": "interline_equation", + "image_path": "2e572575ad892fffd793f234a5de49bd9989984531caac9824efba32903f4758.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 263, + 262, + 348, + 277 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 224, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 225, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 225, + 294 + ], + "score": 1.0, + "content": "The full training objective is:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 297, + 420, + 330 + ], + "lines": [ + { + "bbox": [ + 190, + 297, + 420, + 330 + ], + "spans": [ + { + "bbox": [ + 190, + 297, + 420, + 330 + ], + "score": 0.93, + "content": "f ( \\theta ) = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } f _ { i } ( \\theta ) \\qquad { \\mathrm { w h e r e } } \\quad f _ { i } ( \\theta ) = \\ell ( f ( \\mathbf { x } _ { i } ; 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Bottou et al., 2018) of the per-example Hessian", + "type": "text" + }, + { + "bbox": [ + 323, + 397, + 357, + 410 + ], + "score": 0.93, + "content": "\\nabla ^ { 2 } f _ { i } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 396, + 362, + 411 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 414, + 494, + 429 + ], + "lines": [ + { + "bbox": [ + 116, + 414, + 494, + 429 + ], + "spans": [ + { + "bbox": [ + 116, + 414, + 494, + 429 + ], + "score": 0.91, + "content": "\\nabla ^ { 2 } f _ { i } ( \\theta ) = \\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) ^ { T } + \\ell ^ { \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } ^ { 2 } h ( \\mathbf { x } _ { i } ; \\theta ) \\quad { \\mathrm { w h e r e } } \\quad z _ { i } = h ( \\mathbf { x } _ { i } ; \\theta )", + "type": "interline_equation", + "image_path": "06cf4cb23653fdf1956e73435121d01a337c981ed345194e99d2d153fc78b5b0.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 116, + 414, + 494, + 429 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 433, + 502, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 504, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 134, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 433, + 207, + 446 + ], + "score": 0.92, + "content": "\\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 432, + 495, + 447 + ], + "score": 1.0, + "content": "is the gradient of the network output with respect to the weights, and", + "type": "text" + }, + { + "bbox": [ + 496, + 434, + 504, + 444 + ], + "score": 0.83, + "content": "\\ell ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 444, + 390, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 209, + 458 + ], + "score": 1.0, + "content": "refers to the derivative of", + "type": "text" + }, + { + "bbox": [ + 209, + 446, + 214, + 455 + ], + "score": 0.76, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 444, + 390, + 458 + ], + "score": 1.0, + "content": "with respect to its first argument, the score.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 502, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Empirically, the first term in the decomposition (usually called the “Gauss-Newton matrix”) tends to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 472, + 500, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 500, + 485 + ], + "score": 1.0, + "content": "dominate the second, which implies the following “Gauss-Newton approximation” to the Hessian:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 489, + 411, + 522 + ], + "lines": [ + { + "bbox": [ + 199, + 489, + 411, + 522 + ], + "spans": [ + { + "bbox": [ + 199, + 489, + 411, + 522 + ], + "score": 0.94, + "content": "\\nabla ^ { 2 } f ( \\theta ) \\approx \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) ^ { T }", + "type": "interline_equation", + "image_path": "4072d81c2b0cfde599bfad7c8be7c96d9b8608c05fbb300466a53f97e8b781f6.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 489, + 411, + 505.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 199, + 505.5, + 411, + 522.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 343, + 546 + ], + "score": 1.0, + "content": "In our experience, progressive sharpening affects", + "type": "text" + }, + { + "bbox": [ + 343, + 532, + 443, + 545 + ], + "score": 0.92, + "content": "\\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) \\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; 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Because the loss", + "type": "text" + }, + { + "bbox": [ + 416, + 605, + 422, + 615 + ], + "score": 0.74, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 602, + 489, + 619 + ], + "score": 1.0, + "content": "is decreasing in", + "type": "text" + }, + { + "bbox": [ + 489, + 607, + 501, + 616 + ], + "score": 0.78, + "content": "y z", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 602, + 506, + 619 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 615, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 176, + 628 + ], + "score": 1.0, + "content": "once an example", + "type": "text" + }, + { + "bbox": [ + 176, + 617, + 181, + 626 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 615, + 287, + 628 + ], + "score": 1.0, + "content": "is classified correctly (i.e.", + "type": "text" + }, + { + "bbox": [ + 287, + 617, + 324, + 627 + ], + "score": 0.88, + "content": "y _ { i } z _ { i } > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 615, + 504, + 628 + ], + "score": 1.0, + "content": "), the training objective can be optimized fur-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 225, + 640 + ], + "score": 1.0, + "content": "ther by increasing the margin", + "type": "text" + }, + { + "bbox": [ + 225, + 628, + 242, + 639 + ], + "score": 0.87, + "content": "y _ { i } z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 626, + 282, + 640 + ], + "score": 1.0, + "content": ". 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Near the end of training, once most examples are classified correctly,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "gradient descent can easily increase the margins of all these examples by simply scaling up the final", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 252, + 673 + ], + "score": 1.0, + "content": "layer weight matrix. This causes the", + "type": "text" + }, + { + "bbox": [ + 252, + 660, + 291, + 672 + ], + "score": 0.9, + "content": "\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "to drop. Therefore, even though progressive sharpen-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 182, + 684 + ], + "score": 1.0, + "content": "ing still applies to", + "type": "text" + }, + { + "bbox": [ + 182, + 671, + 281, + 682 + ], + "score": 0.91, + "content": "\\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) \\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 670, + 363, + 684 + ], + "score": 1.0, + "content": ", the decrease in the", + "type": "text" + }, + { + "bbox": [ + 364, + 671, + 402, + 683 + ], + "score": 0.93, + "content": "\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "’s pulls down the leading", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 681, + 324, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 324, + 694 + ], + "score": 1.0, + "content": "eigenvalue of the Gauss-Newton matrix in equation 3.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "This effect is illustrated in Figure 10. Here, we train a network on the binary classification task of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 708, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 486, + 723 + ], + "score": 1.0, + "content": "CIFAR-10 airplane vs. automobile, using the logistic loss. In Figure 10(e), we plot the margin", + "type": "text" + }, + { + "bbox": [ + 486, + 711, + 504, + 721 + ], + "score": 0.83, + "content": "y _ { i } z _ { i }", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "for 10 examples in the dataset. 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\\theta )", + "type": "interline_equation", + "image_path": "06cf4cb23653fdf1956e73435121d01a337c981ed345194e99d2d153fc78b5b0.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 116, + 414, + 494, + 429 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 433, + 502, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 504, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 134, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 433, + 207, + 446 + ], + "score": 0.92, + "content": "\\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\in \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 432, + 495, + 447 + ], + "score": 1.0, + "content": "is the gradient of the network output with respect to the weights, and", + "type": "text" + }, + { + "bbox": [ + 496, + 434, + 504, + 444 + ], + "score": 0.83, + "content": "\\ell ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 444, + 390, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 209, + 458 + ], + "score": 1.0, + "content": "refers to the derivative of", + "type": "text" + }, + { + "bbox": [ + 209, + 446, + 214, + 455 + ], + "score": 0.76, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 444, + 390, + 458 + ], + "score": 1.0, + "content": "with respect to its first argument, the score.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 432, + 504, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 502, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "Empirically, the first term in the decomposition (usually called the “Gauss-Newton matrix”) tends to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 472, + 500, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 500, + 485 + ], + "score": 1.0, + "content": "dominate the second, which implies the following “Gauss-Newton approximation” to the Hessian:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 461, + 505, + 485 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 489, + 411, + 522 + ], + "lines": [ + { + "bbox": [ + 199, + 489, + 411, + 522 + ], + "spans": [ + { + "bbox": [ + 199, + 489, + 411, + 522 + ], + "score": 0.94, + "content": "\\nabla ^ { 2 } f ( \\theta ) \\approx \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) ^ { T }", + "type": "interline_equation", + "image_path": "4072d81c2b0cfde599bfad7c8be7c96d9b8608c05fbb300466a53f97e8b781f6.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 489, + 411, + 505.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 199, + 505.5, + 411, + 522.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 104, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 343, + 546 + ], + "score": 1.0, + "content": "In our experience, progressive sharpening affects", + "type": "text" + }, + { + "bbox": [ + 343, + 532, + 443, + 545 + ], + "score": 0.92, + "content": "\\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; \\boldsymbol { \\theta } ) \\nabla _ { \\boldsymbol { \\theta } } h ( \\mathbf { x } _ { i } ; 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In contrast, for the logistic loss, many of the", + "type": "text" + }, + { + "bbox": [ + 355, + 566, + 393, + 578 + ], + "score": 0.93, + "content": "\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "decrease at the very end of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 363, + 589 + ], + "score": 1.0, + "content": "training. Why is this? In Figure 9, we plot both the logistic loss", + "type": "text" + }, + { + "bbox": [ + 363, + 578, + 368, + 587 + ], + "score": 0.73, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 576, + 473, + 589 + ], + "score": 1.0, + "content": ", and its second derivative", + "type": "text" + }, + { + "bbox": [ + 473, + 577, + 483, + 587 + ], + "score": 0.85, + "content": "\\ell ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 576, + 506, + 589 + ], + "score": 1.0, + "content": ", as a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 588, + 358, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 203, + 601 + ], + "score": 1.0, + "content": "function of the quantity", + "type": "text" + }, + { + "bbox": [ + 203, + 590, + 214, + 600 + ], + "score": 0.83, + "content": "y z", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 588, + 358, + 601 + ], + "score": 1.0, + "content": ", which is often called the “margin.”", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 532, + 506, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 221, + 619 + ], + "score": 1.0, + "content": "Crucially, observe that both", + "type": "text" + }, + { + "bbox": [ + 221, + 605, + 227, + 615 + ], + "score": 0.71, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 602, + 245, + 619 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 245, + 605, + 256, + 615 + ], + "score": 0.86, + "content": "\\ell ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 602, + 329, + 619 + ], + "score": 1.0, + "content": "are decreasing in", + "type": "text" + }, + { + "bbox": [ + 329, + 607, + 341, + 616 + ], + "score": 0.73, + "content": "y z", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 602, + 416, + 619 + ], + "score": 1.0, + "content": ". Because the loss", + "type": "text" + }, + { + "bbox": [ + 416, + 605, + 422, + 615 + ], + "score": 0.74, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 602, + 489, + 619 + ], + "score": 1.0, + "content": "is decreasing in", + "type": "text" + }, + { + "bbox": [ + 489, + 607, + 501, + 616 + ], + "score": 0.78, + "content": "y z", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 602, + 506, + 619 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 615, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 176, + 628 + ], + "score": 1.0, + "content": "once an example", + "type": "text" + }, + { + "bbox": [ + 176, + 617, + 181, + 626 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 615, + 287, + 628 + ], + "score": 1.0, + "content": "is classified correctly (i.e.", + "type": "text" + }, + { + "bbox": [ + 287, + 617, + 324, + 627 + ], + "score": 0.88, + "content": "y _ { i } z _ { i } > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 615, + 504, + 628 + ], + "score": 1.0, + "content": "), the training objective can be optimized fur-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 225, + 640 + ], + "score": 1.0, + "content": "ther by increasing the margin", + "type": "text" + }, + { + "bbox": [ + 225, + 628, + 242, + 639 + ], + "score": 0.87, + "content": "y _ { i } z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 626, + 282, + 640 + ], + "score": 1.0, + "content": ". 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Near the end of training, once most examples are classified correctly,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "gradient descent can easily increase the margins of all these examples by simply scaling up the final", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 252, + 673 + ], + "score": 1.0, + "content": "layer weight matrix. This causes the", + "type": "text" + }, + { + "bbox": [ + 252, + 660, + 291, + 672 + ], + "score": 0.9, + "content": "\\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "to drop. 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Here, we train a network on the binary classification task of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 708, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 486, + 723 + ], + "score": 1.0, + "content": "CIFAR-10 airplane vs. automobile, using the logistic loss. In Figure 10(e), we plot the margin", + "type": "text" + }, + { + "bbox": [ + 486, + 711, + 504, + 721 + ], + "score": 0.83, + "content": "y _ { i } z _ { i }", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "for 10 examples in the dataset. 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Now, all the while, thekeeps rising, as can be seen from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 239, + 248, + 369, + 261 + ], + "spans": [ + { + "bbox": [ + 239, + 248, + 369, + 261 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla _ { \\theta } \\hat { h } ( \\mathbf x _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf x _ { i } ; \\theta ) ^ { T } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 253, + 272 + ], + "score": 1.0, + "content": "Figure 10(d). However, because the", + "type": "text" + }, + { + "bbox": [ + 253, + 261, + 268, + 269 + ], + "score": 0.78, + "content": "{ \\ell ^ { \\prime \\prime } s }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "are dropping, the leading eigenvalue of the Gauss-Newton", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 267, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 135, + 283 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 136, + 270, + 307, + 284 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell ^ { \\prime \\prime } ( z _ { i } ; y _ { i } ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) \\nabla _ { \\theta } h ( \\mathbf { x } _ { i } ; \\theta ) ^ { \\hat { T } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 267, + 506, + 285 + ], + "score": 1.0, + "content": "starts to decrease at the end of training, as can be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "seen from the green line in Figure 10(c). 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Observe that for all of these examples, this margin", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "rises at the end of training. In Figure 11(f), for those same ten examples, we plot the quantity", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 379, + 508, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 182, + 392 + ], + "score": 0.89, + "content": "{ \\bf p } _ { i } [ y _ { i } ] ( 1 - { \\bf p } _ { i } [ y _ { i } ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 379, + 421, + 410 + ], + "score": 1.0, + "content": ".Observe that for all of these examples, this quantity con. Now, all the while, the leading eigenvalue of the matrix", + "type": "text" + }, + { + "bbox": [ + 478, + 379, + 508, + 410 + ], + "score": 1.0, + "content": "ases atkeeps", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 421, + 390, + 478, + 404 + ], + "spans": [ + { + "bbox": [ + 421, + 390, + 478, + 404 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n ^ { * } } \\mathbf { J } _ { i } ^ { T } \\mathbf { J } _ { i } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 100, + 406, + 511, + 438 + ], + "spans": [ + { + "bbox": [ + 100, + 406, + 511, + 433 + ], + "score": 1.0, + "content": "i i eigenvalue of the Gauss-Newton matrix 1n Pni=1 JTi ∇2`(zi; yi) Ji starts to decrease at the end of", + "type": "text" + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "training, as can be seen from the green line in Figure 11(c). Finally, since the leading eigenvalue", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "of the Gauss-Newton matrix is an excellent approximation to the leading eigenvalue of the Hessian", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "(i.e. the sharpness), the sharpness also drops at the end of training, as can be seen from the orange", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 189, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 189, + 471 + ], + "score": 1.0, + "content": "line in Figure 11(c).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 100, + 323, + 511, + 471 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 281, + 505, + 439 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 281, + 505, + 439 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 281, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 110, + 281, + 505, + 439 + ], + "score": 0.971, + "type": "image", + "image_path": "3afa4026e93af581c17185584cf5281249dbef1d4b1b3a1e05e3bd52c68806bf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 281, + 505, + 333.6666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 333.6666666666667, + 505, + 386.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 386.33333333333337, + 505, + 439.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 452, + 506, + 530 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "Figure 11: We train a network using the cross-entropy loss on CIFAR-10. (a) The train loss. (b) The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "train accuracy. (c) The leading eigenvalue of the Hessian and of the Gauss-Newton matrix; observe", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "that the latter is a great approximation to the former. (d) The leading eigenvalue of the matrix", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 481, + 509, + 500 + ], + "spans": [ + { + "bbox": [ + 107, + 484, + 163, + 498 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbf { J } _ { i } ^ { T } \\mathbf { J } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 481, + 360, + 500 + ], + "score": 1.0, + "content": ", which is the Gauss-Newton matrix except the", + "type": "text" + }, + { + "bbox": [ + 360, + 484, + 379, + 495 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 481, + 509, + 500 + ], + "score": 1.0, + "content": "terms; observe that this value", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 390, + 509 + ], + "score": 1.0, + "content": "constantly rises — it does not dip at the end of training. (e) The margin", + "type": "text" + }, + { + "bbox": [ + 390, + 496, + 481, + 508 + ], + "score": 0.9, + "content": "\\mathbf { z } _ { i } [ y _ { i } ] - \\operatorname* { m a x } _ { j \\neq y _ { i } } \\mathbf { z } _ { i } [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "of 10", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 506, + 504, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 430, + 520 + ], + "score": 1.0, + "content": "examples; observe that all these margins rise at the end of training. 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(a) The train loss. (b) The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "train accuracy. (c) The leading eigenvalue of the Hessian and of the Gauss-Newton matrix; observe", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "that the latter is a great approximation to the former. (d) The leading eigenvalue of the matrix", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 481, + 509, + 500 + ], + "spans": [ + { + "bbox": [ + 107, + 484, + 163, + 498 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbf { J } _ { i } ^ { T } \\mathbf { J } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 481, + 360, + 500 + ], + "score": 1.0, + "content": ", which is the Gauss-Newton matrix except the", + "type": "text" + }, + { + "bbox": [ + 360, + 484, + 379, + 495 + ], + "score": 0.92, + "content": "\\nabla ^ { 2 } \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 481, + 509, + 500 + ], + "score": 1.0, + "content": "terms; observe that this value", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 495, + 390, + 509 + ], + "score": 1.0, + "content": "constantly rises — it does not dip at the end of training. (e) The margin", + "type": "text" + }, + { + "bbox": [ + 390, + 496, + 481, + 508 + ], + "score": 0.9, + "content": "\\mathbf { z } _ { i } [ y _ { i } ] - \\operatorname* { m a x } _ { j \\neq y _ { i } } \\mathbf { z } _ { i } [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "of 10", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 506, + 504, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 430, + 520 + ], + "score": 1.0, + "content": "examples; observe that all these margins rise at the end of training. (f) the value", + "type": "text" + }, + { + "bbox": [ + 430, + 507, + 504, + 519 + ], + "score": 0.9, + "content": "\\mathbf { p } _ { i } [ y _ { i } ] ( 1 - \\mathbf { p } _ { i } [ y _ { i } ] )", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 516, + 421, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 421, + 532 + ], + "score": 1.0, + "content": "for 10 examples; observe that all of these curves decline at the end of training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 384, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 385, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 385, + 96 + ], + "score": 1.0, + "content": "D EMPIRICAL STUDY OF PROGRESSIVE SHARPENING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we empirically study how problem parameters such as network width, network", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "depth, and dataset size affect the degree to which progressive sharpening occurs. To study progres-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "sive sharpening on its own, without the confounding factor of instability, we train neural networks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "using gradient flow (Runge-Kutta) rather than gradient descent. Informally speaking, gradient flow", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 493, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 493, + 163 + ], + "score": 1.0, + "content": "does “what gradient descent would do if gradient descent didn’t have to worry about instability.”", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 504, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "We observe that progressive sharpening occurs to a greater degree: (1) for narrower networks than", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 189 + ], + "score": 1.0, + "content": "for wider networks (which is consistent with infinite-width NTK theory), (2) for deeper networks", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 188, + 430, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 430, + 201 + ], + "score": 1.0, + "content": "than for shallower networks, and (3) for larger datasets than for smaller datasets.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 212, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "The effect of width When networks parameterized in a certain way (the “NTK parameterization”)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "are made infinitely wide and trained using gradient flow, the Hessian moves a vanishingly small", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "score": 1.0, + "content": "amount during training, implying that no progressive sharpening occurs (Jacot et al., 2018; Lee", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "et al., 2019; Jacot et al., 2020; Li & Liang, 2018). Therefore, a natural hypothesis is that progressive", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "sharpening might attenuate as network width increases. We now run an experiment which supports", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 171, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 171, + 280 + ], + "score": 1.0, + "content": "this hypothesis.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "We consider fully-connected architectures with two hidden layers and tanh activations, with widths", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 107, + 295, + 225, + 307 + ], + "score": 0.64, + "content": "\\{ 3 2 , 6 4 , 1 2 8 , 2 5 6 , 5 1 2 , 1 0 2 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 294, + 505, + 308 + ], + "score": 1.0, + "content": ". We train on a size-5,000 subset of CIFAR-10 using the cross-entropy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 293, + 318 + ], + "score": 1.0, + "content": "loss. We train using gradient flow (details in", + "type": "text" + }, + { + "bbox": [ + 293, + 306, + 312, + 317 + ], + "score": 0.56, + "content": "\\ S 1 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 307, + 505, + 318 + ], + "score": 1.0, + "content": ". We consider both NTK parameterization and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 317, + 286, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 286, + 329 + ], + "score": 1.0, + "content": "standard parameterization (Lee et al., 2019).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "In Figure 12, for each width, we train NTK-parameterized networks from five different random", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "initializations, and plot the evolution of the sharpness during gradient flow. Observe that the max-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 357, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 368 + ], + "score": 1.0, + "content": "imum sharpness along the gradient flow trajectory is larger for narrow networks, and smaller for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 381 + ], + "score": 1.0, + "content": "wide networks. (As elsewhere in this paper, note that the sharpness drops at the end of training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "due to the cross-entropy loss.) In Figure 13, we plot summary statistics from these training runs.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 504, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 171, + 403 + ], + "score": 1.0, + "content": "Namely, define", + "type": "text" + }, + { + "bbox": [ + 171, + 389, + 194, + 400 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 388, + 492, + 403 + ], + "score": 1.0, + "content": "as the maximum sharpness over the gradient flow trajectory, and define", + "type": "text" + }, + { + "bbox": [ + 492, + 389, + 504, + 400 + ], + "score": 0.86, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "as the initial sharpness. In Figure 13(a), for each width we plot the mean and standard deviation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 409, + 504, + 425 + ], + "spans": [ + { + "bbox": [ + 104, + 409, + 218, + 425 + ], + "score": 1.0, + "content": "of the maximum sharpness", + "type": "text" + }, + { + "bbox": [ + 218, + 411, + 241, + 422 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 409, + 481, + 425 + ], + "score": 1.0, + "content": "over the five different random initializations. Observe that", + "type": "text" + }, + { + "bbox": [ + 482, + 411, + 504, + 422 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "score": 1.0, + "content": "becomes smaller, on average, as the width is made larger. In Figure 13(b), for each width we plot", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 375, + 446 + ], + "score": 1.0, + "content": "the mean and standard deviation of the maximum sharpness gain", + "type": "text" + }, + { + "bbox": [ + 375, + 433, + 412, + 444 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "over the five different", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 376, + 457 + ], + "score": 1.0, + "content": "random initializations. Observe that the maximum sharpness gain", + "type": "text" + }, + { + "bbox": [ + 377, + 444, + 414, + 455 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "also becomes smaller", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 328, + 468 + ], + "score": 1.0, + "content": "as the width is made larger. NTK theory suggests that", + "type": "text" + }, + { + "bbox": [ + 328, + 455, + 366, + 466 + ], + "score": 0.93, + "content": "\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 453, + 505, + 468 + ], + "score": 1.0, + "content": "should deterministically tend to 1", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 466, + 388, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 156, + 478 + ], + "score": 1.0, + "content": "as the width", + "type": "text" + }, + { + "bbox": [ + 156, + 466, + 181, + 476 + ], + "score": 0.87, + "content": " \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 466, + 388, + 478 + ], + "score": 1.0, + "content": ", and Figure 13(b) is consistent with this prediction.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 108, + 482, + 504, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "In Figures 14 and 15, we conduct similar experiments, but with standard parameterization rather", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "than NTK parameterization. Similar to NTK parameterization, we observe in Figure 14 that the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 505, + 372, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 372, + 516 + ], + "score": 1.0, + "content": "sharpness rises more for narrow networks than for wide networks.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 384, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 385, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 385, + 96 + ], + "score": 1.0, + "content": "D EMPIRICAL STUDY OF PROGRESSIVE SHARPENING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we empirically study how problem parameters such as network width, network", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "depth, and dataset size affect the degree to which progressive sharpening occurs. To study progres-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "sive sharpening on its own, without the confounding factor of instability, we train neural networks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "using gradient flow (Runge-Kutta) rather than gradient descent. Informally speaking, gradient flow", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 493, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 493, + 163 + ], + "score": 1.0, + "content": "does “what gradient descent would do if gradient descent didn’t have to worry about instability.”", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 106, + 505, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 504, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "We observe that progressive sharpening occurs to a greater degree: (1) for narrower networks than", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 189 + ], + "score": 1.0, + "content": "for wider networks (which is consistent with infinite-width NTK theory), (2) for deeper networks", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 188, + 430, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 430, + 201 + ], + "score": 1.0, + "content": "than for shallower networks, and (3) for larger datasets than for smaller datasets.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 166, + 506, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 212, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "The effect of width When networks parameterized in a certain way (the “NTK parameterization”)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "are made infinitely wide and trained using gradient flow, the Hessian moves a vanishingly small", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "score": 1.0, + "content": "amount during training, implying that no progressive sharpening occurs (Jacot et al., 2018; Lee", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "et al., 2019; Jacot et al., 2020; Li & Liang, 2018). Therefore, a natural hypothesis is that progressive", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "sharpening might attenuate as network width increases. We now run an experiment which supports", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 171, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 171, + 280 + ], + "score": 1.0, + "content": "this hypothesis.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 211, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "We consider fully-connected architectures with two hidden layers and tanh activations, with widths", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 107, + 295, + 225, + 307 + ], + "score": 0.64, + "content": "\\{ 3 2 , 6 4 , 1 2 8 , 2 5 6 , 5 1 2 , 1 0 2 4 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 294, + 505, + 308 + ], + "score": 1.0, + "content": ". We train on a size-5,000 subset of CIFAR-10 using the cross-entropy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 293, + 318 + ], + "score": 1.0, + "content": "loss. We train using gradient flow (details in", + "type": "text" + }, + { + "bbox": [ + 293, + 306, + 312, + 317 + ], + "score": 0.56, + "content": "\\ S 1 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 307, + 505, + 318 + ], + "score": 1.0, + "content": ". We consider both NTK parameterization and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 317, + 286, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 286, + 329 + ], + "score": 1.0, + "content": "standard parameterization (Lee et al., 2019).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 284, + 505, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "In Figure 12, for each width, we train NTK-parameterized networks from five different random", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "initializations, and plot the evolution of the sharpness during gradient flow. Observe that the max-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 357, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 368 + ], + "score": 1.0, + "content": "imum sharpness along the gradient flow trajectory is larger for narrow networks, and smaller for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 381 + ], + "score": 1.0, + "content": "wide networks. (As elsewhere in this paper, note that the sharpness drops at the end of training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "due to the cross-entropy loss.) In Figure 13, we plot summary statistics from these training runs.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 504, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 171, + 403 + ], + "score": 1.0, + "content": "Namely, define", + "type": "text" + }, + { + "bbox": [ + 171, + 389, + 194, + 400 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 388, + 492, + 403 + ], + "score": 1.0, + "content": "as the maximum sharpness over the gradient flow trajectory, and define", + "type": "text" + }, + { + "bbox": [ + 492, + 389, + 504, + 400 + ], + "score": 0.86, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "as the initial sharpness. In Figure 13(a), for each width we plot the mean and standard deviation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 409, + 504, + 425 + ], + "spans": [ + { + "bbox": [ + 104, + 409, + 218, + 425 + ], + "score": 1.0, + "content": "of the maximum sharpness", + "type": "text" + }, + { + "bbox": [ + 218, + 411, + 241, + 422 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 409, + 481, + 425 + ], + "score": 1.0, + "content": "over the five different random initializations. Observe that", + "type": "text" + }, + { + "bbox": [ + 482, + 411, + 504, + 422 + ], + "score": 0.88, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "score": 1.0, + "content": "becomes smaller, on average, as the width is made larger. In Figure 13(b), for each width we plot", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 375, + 446 + ], + "score": 1.0, + "content": "the mean and standard deviation of the maximum sharpness gain", + "type": "text" + }, + { + "bbox": [ + 375, + 433, + 412, + 444 + ], + "score": 0.9, + "content": "\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "over the five different", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 376, + 457 + ], + "score": 1.0, + "content": "random initializations. Observe that the maximum sharpness gain", + "type": "text" + }, + { + "bbox": [ + 377, + 444, + 414, + 455 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "also becomes smaller", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 328, + 468 + ], + "score": 1.0, + "content": "as the width is made larger. NTK theory suggests that", + "type": "text" + }, + { + "bbox": [ + 328, + 455, + 366, + 466 + ], + "score": 0.93, + "content": "\\lambda _ { \\mathrm { m a x } } / \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 453, + 505, + 468 + ], + "score": 1.0, + "content": "should deterministically tend to 1", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 466, + 388, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 156, + 478 + ], + "score": 1.0, + "content": "as the width", + "type": "text" + }, + { + "bbox": [ + 156, + 466, + 181, + 476 + ], + "score": 0.87, + "content": " \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 466, + 388, + 478 + ], + "score": 1.0, + "content": ", and Figure 13(b) is consistent with this prediction.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 333, + 506, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 482, + 504, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "In Figures 14 and 15, we conduct similar experiments, but with standard parameterization rather", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "than NTK parameterization. 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We use", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "gradient flow to train fully-connected tanh architectures of width 200 and varying depths — ranging", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 333, + 117 + ], + "score": 1.0, + "content": "from 1 hidden layer to 4 hidden layers. We train on a", + "type": "text" + }, + { + "bbox": [ + 333, + 105, + 345, + 115 + ], + "score": 0.34, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "subset of CIFAR-10 using both cross-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "entropy loss (in Figure 16) and square loss (in Figure 17). For each depth, we train from five", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "different random initializations (different colors). 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We use gradient flow to train networks of var-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "ious depths, ranging from 1 hidden layer to 4 hidden layers, using cross-entropy loss. We train", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "each network from five different random initializations (different colors). 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We use gradient flow to train networks of var-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "ious depths, ranging from 1 hidden layer to 4 hidden layers, using MSE loss. We train each network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 449, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 464 + ], + "score": 1.0, + "content": "from five different random initializations (different colors). Observe that progressive sharpening", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 462, + 295, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 295, + 473 + ], + "score": 1.0, + "content": "occurs to a greater degree for deeper networks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + } + ], + "index": 15.75 + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "Effect of dataset size We now explore the effect of dataset size on progressive sharpening. We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "use gradient flow to train a network on different-sized subsets of CIFAR-10. The network is a 2-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "hidden-layer, width-200 fully-connected tanh architecture, and we train using cross-entropy loss.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "The results are shown in Figure 18. Observe that progressive sharpening occurs to a greater degree", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 535, + 207, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 207, + 548 + ], + "score": 1.0, + "content": "as dataset size increases.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 491, + 505, + 548 + ] + }, + { + "type": "image", + "bbox": [ + 111, + 559, + 482, + 646 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 559, + 482, + 646 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 111, + 559, + 482, + 646 + ], + "spans": [ + { + "bbox": [ + 111, + 559, + 482, + 646 + ], + "score": 0.966, + "type": "image", + "image_path": "d880f1b51cb504affeeafd27c3459939af183c6cb9ef5d32bdd6757998401d0e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 111, + 559, + 482, + 588.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 111, + 588.0, + 482, + 617.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 111, + 617.0, + 482, + 646.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 659, + 505, + 692 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "Figure 18: The effect of dataset size. We use gradient flow to train a network on varying-sized", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "subsets of CIFAR-10. Observe that progressive sharpening occurs to a greater degree as the dataset", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 681, + 166, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 166, + 693 + ], + "score": 1.0, + "content": "size increases.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + } + ], + "index": 27.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 105, + 81, + 467, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 468, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 468, + 95 + ], + "score": 1.0, + "content": "E SPEED OF DIVERGENCE ON QUADRATIC TAYLOR APPROXIMATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 322, + 119 + ], + "score": 1.0, + "content": "If the sharpness at some iterate is strictly greater than", + "type": "text" + }, + { + "bbox": [ + 322, + 106, + 339, + 118 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 106, + 487, + 119 + ], + "score": 1.0, + "content": ", then gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 488, + 108, + 495, + 118 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "guaranteed to diverge if run on the quadratic Taylor approximation around that iterate. However, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "speed of this divergence could conceivably be slow — in particular, the train loss might continue to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "decrease for many iterations before it starts to increase. In this appendix we empirically demonstrate,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "to the contrary, that at the Edge of Stability, gradient descent diverges quickly if, at some iterate, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 460, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 460, + 173 + ], + "score": 1.0, + "content": "start running gradient descent on the quadratic Taylor approximation around that iterate.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "We consider the fully-connected tanh network from section 3, trained on a 5,000-sized subsample", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 427, + 203 + ], + "score": 1.0, + "content": "of CIFAR-10 using both cross-entropy loss and MSE loss. At some timestep", + "type": "text" + }, + { + "bbox": [ + 428, + 190, + 437, + 200 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 187, + 505, + 203 + ], + "score": 1.0, + "content": "during training,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "score": 1.0, + "content": "we suddenly switch from running gradient descent on the real neural training objective, to running", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 441, + 224 + ], + "score": 1.0, + "content": "gradient descent on the quadratic Taylor approximation around the iterate at step", + "type": "text" + }, + { + "bbox": [ + 441, + 212, + 451, + 222 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 210, + 506, + 224 + ], + "score": 1.0, + "content": ". We do this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "for three timesteps before gradient descent has entered the Edge of Stability, and three afterwards.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 245 + ], + "score": 1.0, + "content": "Figure 20 shows the results for cross-entropy loss, and Figure 21 shows the (similar) results for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "MSE loss. Before entering the Edge of Stability (top row), gradient descent on the quadratic Taylor", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "approximation behaves similar to gradient descent on the real neural training objective — that is,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 280 + ], + "score": 1.0, + "content": "the orange line almost overlaps the blue line. Yet after entering the Edge of Stability (bottom row),", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "gradient descent on the quadratic Taylor approximation quickly diverges, whereas gradient descent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 406, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 406, + 300 + ], + "score": 1.0, + "content": "on the real neural training objective makes consistent (if choppy) progress.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 506, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "In short, when gradient descent is not at the Edge of Stability, the quadratic Taylor approximation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "serves as a good model for the local progress of gradient descent. But when gradient descent is at", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "the Edge of Stability, the quadratic Taylor approximation is an extremely poor model for the local", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "progress of gradient descent. It is conceivable that there exists some simple modification to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "quadratic Taylor model which would fix this issue (e.g. perhaps if one ignores a certain direction,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "score": 1.0, + "content": "the quadratic Taylor model is accurate). Nevertheless, unless/until such a fix is discovered, it is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "unclear why quadratic Taylor approximations should yield any insight into the local behavior of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 382, + 177, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 177, + 393 + ], + "score": 1.0, + "content": "gradient descent.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "image", + "bbox": [ + 185, + 406, + 425, + 488 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 185, + 406, + 425, + 488 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 185, + 406, + 425, + 488 + ], + "spans": [ + { + "bbox": [ + 185, + 406, + 425, + 488 + ], + "score": 0.961, + "type": "image", + "image_path": "955346fd537d3f513f8126ca81891da22319daf5c8fd798c8c111382b7b6b357.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 406, + 425, + 419.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 185, + 419.6666666666667, + 425, + 433.33333333333337 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 185, + 433.33333333333337, + 425, + 447.00000000000006 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 185, + 447.00000000000006, + 425, + 460.66666666666674 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 185, + 460.66666666666674, + 425, + 474.3333333333334 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 185, + 474.3333333333334, + 425, + 488.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 502, + 505, + 547 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "Figure 19: We train a neural network using cross-entropy loss (left) and MSE loss (right). In Figure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "score": 1.0, + "content": "20 and 21, we show what happens when, at the iterations marked above by vertical dotted lines,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "we switch from gradient descent on the real neural training objective to gradient descent on the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 535, + 237, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 237, + 548 + ], + "score": 1.0, + "content": "quadratic Taylor approximation.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + } + ], + "index": 31.0 + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 105, + 81, + 467, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 468, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 468, + 95 + ], + "score": 1.0, + "content": "E SPEED OF DIVERGENCE ON QUADRATIC TAYLOR APPROXIMATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 322, + 119 + ], + "score": 1.0, + "content": "If the sharpness at some iterate is strictly greater than", + "type": "text" + }, + { + "bbox": [ + 322, + 106, + 339, + 118 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 106, + 487, + 119 + ], + "score": 1.0, + "content": ", then gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 488, + 108, + 495, + 118 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "guaranteed to diverge if run on the quadratic Taylor approximation around that iterate. However, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "speed of this divergence could conceivably be slow — in particular, the train loss might continue to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "decrease for many iterations before it starts to increase. In this appendix we empirically demonstrate,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "to the contrary, that at the Edge of Stability, gradient descent diverges quickly if, at some iterate, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 460, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 460, + 173 + ], + "score": 1.0, + "content": "start running gradient descent on the quadratic Taylor approximation around that iterate.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "We consider the fully-connected tanh network from section 3, trained on a 5,000-sized subsample", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 427, + 203 + ], + "score": 1.0, + "content": "of CIFAR-10 using both cross-entropy loss and MSE loss. At some timestep", + "type": "text" + }, + { + "bbox": [ + 428, + 190, + 437, + 200 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 187, + 505, + 203 + ], + "score": 1.0, + "content": "during training,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 214 + ], + "score": 1.0, + "content": "we suddenly switch from running gradient descent on the real neural training objective, to running", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 441, + 224 + ], + "score": 1.0, + "content": "gradient descent on the quadratic Taylor approximation around the iterate at step", + "type": "text" + }, + { + "bbox": [ + 441, + 212, + 451, + 222 + ], + "score": 0.85, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 210, + 506, + 224 + ], + "score": 1.0, + "content": ". We do this", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "for three timesteps before gradient descent has entered the Edge of Stability, and three afterwards.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 245 + ], + "score": 1.0, + "content": "Figure 20 shows the results for cross-entropy loss, and Figure 21 shows the (similar) results for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "score": 1.0, + "content": "MSE loss. Before entering the Edge of Stability (top row), gradient descent on the quadratic Taylor", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "approximation behaves similar to gradient descent on the real neural training objective — that is,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 280 + ], + "score": 1.0, + "content": "the orange line almost overlaps the blue line. Yet after entering the Edge of Stability (bottom row),", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 290 + ], + "score": 1.0, + "content": "gradient descent on the quadratic Taylor approximation quickly diverges, whereas gradient descent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 288, + 406, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 406, + 300 + ], + "score": 1.0, + "content": "on the real neural training objective makes consistent (if choppy) progress.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 178, + 506, + 300 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 506, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "In short, when gradient descent is not at the Edge of Stability, the quadratic Taylor approximation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "serves as a good model for the local progress of gradient descent. But when gradient descent is at", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "the Edge of Stability, the quadratic Taylor approximation is an extremely poor model for the local", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "progress of gradient descent. It is conceivable that there exists some simple modification to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 348, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 348, + 506, + 361 + ], + "score": 1.0, + "content": "quadratic Taylor model which would fix this issue (e.g. perhaps if one ignores a certain direction,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 371 + ], + "score": 1.0, + "content": "the quadratic Taylor model is accurate). 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In Figure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 526 + ], + "score": 1.0, + "content": "20 and 21, we show what happens when, at the iterations marked above by vertical dotted lines,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "we switch from gradient descent on the real neural training objective to gradient descent on the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 535, + 237, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 237, + 548 + ], + "score": 1.0, + "content": "quadratic Taylor approximation.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + } + ], + "index": 31.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 105, + 524, + 273 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 105, + 524, + 273 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 105, + 524, + 273 + ], + "spans": [ + { + "bbox": [ + 111, + 105, + 524, + 273 + ], + "score": 0.972, + "type": "image", + "image_path": "4614c000ab8bb8a0dc9aba4d60480907ab576f1310315b7f847bdf10fb234171.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 105, + 524, + 161.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 161.0, + 524, + 217.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 217.0, + 524, + 273.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 286, + 506, + 375 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 238, + 300 + ], + "score": 1.0, + "content": "Figure 20: Cross-entropy loss", + "type": "text" + }, + { + "bbox": [ + 239, + 287, + 285, + 299 + ], + "score": 0.89, + "content": "( \\eta = 2 / 6 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 286, + 505, + 300 + ], + "score": 1.0, + "content": ". At six different iterations during the training of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "network from Figure 19 (marked by the vertical dotted black lines), we switch from running gradient", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "descent on the real neural training objective (for which the train loss is plotted in blue) to running", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "gradient descent on the quadratic Taylor approximation around the current iterate (for which the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 326, + 343 + ], + "score": 1.0, + "content": "train loss is plotted in orange). Top row are timesteps", + "type": "text" + }, + { + "bbox": [ + 326, + 331, + 391, + 342 + ], + "score": 0.44, + "content": "\\{ 2 0 0 , 4 0 0 , 6 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "before gradient descent has", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "entered the Edge of Stability; observe that the orange line (Taylor approximation) closely tracks the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "blue line (real objective). 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While this heuristic is computationally impractical", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "due to the time required to compute the sharpness at each iteration, it is often regarded as an ideal,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "for instance in LeCun et al. (1998) (Eq. 39), LeCun et al. (1993), and Schaul et al. 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While this heuristic is computationally impractical", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "due to the time required to compute the sharpness at each iteration, it is often regarded as an ideal,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "for instance in LeCun et al. (1998) (Eq. 39), LeCun et al. (1993), and Schaul et al. 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The train accuracy (not pictured) is only", + "type": "text" + }, + { + "bbox": [ + 484, + 234, + 504, + 244 + ], + "score": 0.85, + "content": "6 6 \\%", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "when the sharpness starts to decrease, which suggests that the cause of this decrease is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 142, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "unrelated to the effect described in Appendix C, whereby the sharpness decreases at the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 142, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "end of training. Figure 5(a) of Jastrz˛ebski et al. 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Consistent with prior reports, we find that the smaller the batch size, the lower the sharpness", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 208, + 374, + 220 + ], + "spans": [ + { + "bbox": [ + 142, + 208, + 374, + 220 + ], + "score": 1.0, + "content": "(Keskar et al., 2016; Jastrz˛ebski et al., 2017; 2019; 2020).", + "type": "text" + } + ], + "index": 9, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 222, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 128, + 222, + 505, + 236 + ], + "score": 1.0, + "content": "3. Notice that when training with cross-entropy loss at batch size 8 (the blue line), the sharp-", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 234, + 504, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 484, + 245 + ], + "score": 1.0, + "content": "ness decreases throughout most of training. The train accuracy (not pictured) is only", + "type": "text" + }, + { + "bbox": [ + 484, + 234, + 504, + 244 + ], + "score": 0.85, + "content": "6 6 \\%", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "when the sharpness starts to decrease, which suggests that the cause of this decrease is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 142, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "unrelated to the effect described in Appendix C, whereby the sharpness decreases at the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 266, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 142, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "end of training. Figure 5(a) of Jastrz˛ebski et al. (2020) also depicts a network where the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 277, + 311, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 311, + 291 + ], + "score": 1.0, + "content": "sharpness decreases during SGD training.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + } + ], + "index": 10, + "bbox_fs": [ + 127, + 160, + 505, + 291 + ] + }, + { + "type": "image", + "bbox": [ + 111, + 305, + 505, + 501 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 305, + 505, + 501 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 305, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 111, + 305, + 505, + 501 + ], + "score": 0.934, + "type": "image", + "image_path": "43c9eb37df93a822a4ae22a49f7be8053d7b68ffd685c8aaa8216120ea5ec9e4.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 111, + 305, + 505, + 370.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 111, + 370.3333333333333, + 505, + 435.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 111, + 435.66666666666663, + 505, + 500.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 514, + 505, + 571 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "Figure 24: Evolution of sharpness during SGD. We train a network using SGD at various batch", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 539 + ], + "score": 1.0, + "content": "sizes, and plot the evolution of the sharpness. In the top row, we train with cross-entropy loss and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 536, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 144, + 551 + ], + "score": 1.0, + "content": "step size", + "type": "text" + }, + { + "bbox": [ + 144, + 537, + 185, + 549 + ], + "score": 0.9, + "content": "\\eta = 0 . 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 536, + 461, + 551 + ], + "score": 1.0, + "content": "; in the bottom row, we train with mean squared error and step size", + "type": "text" + }, + { + "bbox": [ + 461, + 537, + 501, + 549 + ], + "score": 0.9, + "content": "\\eta = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 536, + 506, + 551 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "The leftmost column shows all batch sizes; the center column focuses on just the smallest; and the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 559, + 468, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 447, + 573 + ], + "score": 1.0, + "content": "rightmost column focuses on just the largest. The black dashed horizontal line marks", + "type": "text" + }, + { + "bbox": [ + 447, + 559, + 464, + 571 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 559, + 468, + 573 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + } + ], + "index": 19.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 369, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 370, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 370, + 97 + ], + "score": 1.0, + "content": "H SGD ACCLIMATES TO THE HYPERPARAMETERS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 503, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we conduct an experiment which suggests that some version of “Edge of Stability”", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 186, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 186, + 129 + ], + "score": 1.0, + "content": "may hold for SGD.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 505, + 146 + ], + "score": 1.0, + "content": "One way to interpret our main findings is that gradient descent “acclimates” to the step size in such", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 145, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 506, + 158 + ], + "score": 1.0, + "content": "a way that each training update sometimes increases, and sometimes decreases, the training loss, yet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 157, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 505, + 168 + ], + "score": 1.0, + "content": "an update with a smaller step size would always decrease the training loss. We now demonstrate", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 167, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "that this interpretation may generalize to SGD. In particular, we will demonstrate that SGD seems to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "“acclimate” to the step size and batch size in such a way that an actual update sometimes increases", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "and sometimes decreases the loss in expectation, yet a update with a larger step size or smaller batch", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "score": 1.0, + "content": "size would almost always increase the loss in expectation, and a step with a smaller step size or a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 211, + 391, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 391, + 223 + ], + "score": 1.0, + "content": "larger batch size would almost always decrease the loss in expectation.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 288, + 239 + ], + "score": 1.0, + "content": "In Figure 25, we train the tanh network from", + "type": "text" + }, + { + "bbox": [ + 288, + 228, + 299, + 239 + ], + "score": 0.57, + "content": "\\ S 3", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 228, + 505, + 239 + ], + "score": 1.0, + "content": "with MSE loss, using SGD with step size 0.01 and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "batch size 32. We periodically compute the training loss (over the full dataset) and plot these on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "the left pane of Figure 25. Observe that the training loss does not decrease monotonically, but of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 506, + 275 + ], + "score": 1.0, + "content": "course this is not surprising — SGD is a random algorithm. However, what may be more surprising", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "is that SGD is not even decreasing the training loss in expectation. On the right pane of Figure", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "25, every 500 steps during training, we use the Monte Carlo method to approximately compute the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "expected change in training loss that would result from an SGD step (the expectation here is over", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 319 + ], + "score": 1.0, + "content": "the randomness involved in selecting the minibatch). Observe that at many points during training,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 330 + ], + "score": 1.0, + "content": "an SGD step would decrease the loss (as desired) in expectation, but at other points, and SGD step", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 326, + 262, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 262, + 339 + ], + "score": 1.0, + "content": "would increase the loss in expectation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 343, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "In Figure 26(a), while training that network, we compute the expected change in training loss that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "would result from taking an SGD step with the same step size used during training (i.e. 0.01), but", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "half the batch size used during training (i.e. 16). We observe that an SGD step with half the batch", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "score": 1.0, + "content": "size would consistently cause an increase in the training loss in expectation. In Figure 26(b) we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "repeat this experiment, but with twice the batch size used during training (i.e. 64). Notice that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "an SGD step with twice the batch size would consistently cause a decrease in the training loss in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "score": 1.0, + "content": "expectation. In Figure 26(c) and (d) repeat this experiment with the step size; we observe that an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "SGD step with a larger step size (0.02) would consistently increase the training loss in expectation,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "while an SGD step with a smaller step size (0.005) would consistently decrease the training loss in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 442, + 157, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 157, + 455 + ], + "score": 1.0, + "content": "expectation.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "In Figure 27, as a “control” experiment, we both train and measure the expected loss change under", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "the following four hyperparameter settings: (step size 0.01, batch size 16), (step size 0.01, batch size", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 494 + ], + "score": 1.0, + "content": "64), (step size 0.02, batch size 32), and (step size 0.005, batch size 32). In each case, we observe", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "that, after a brief period at the beginning of training, each SGD update sometime increases and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 502, + 318, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 318, + 515 + ], + "score": 1.0, + "content": "sometimes decreases the training loss in expectation.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Therefore, at least for this single network, we can conclude that no matter the hyperparameters,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "SGD quickly navigates to, and then lingers in, regions of the loss landscape in which an SGD", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "update with those hyperparameters sometimes increases, and sometimes decreases the training loss", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "in expectation, yet an SGD update with a smaller step size or larger batch size would consistently", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "decrease the loss in expectation, and an SGD update with a larger step size or smaller batch size", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 574, + 312, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 312, + 587 + ], + "score": 1.0, + "content": "would consistently increase the loss in expectation.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + } + ], + "page_idx": 28, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "29", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 369, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 370, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 370, + 97 + ], + "score": 1.0, + "content": "H SGD ACCLIMATES TO THE HYPERPARAMETERS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 503, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we conduct an experiment which suggests that some version of “Edge of Stability”", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 186, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 186, + 129 + ], + "score": 1.0, + "content": "may hold for SGD.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 105, + 505, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 505, + 146 + ], + "score": 1.0, + "content": "One way to interpret our main findings is that gradient descent “acclimates” to the step size in such", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 145, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 506, + 158 + ], + "score": 1.0, + "content": "a way that each training update sometimes increases, and sometimes decreases, the training loss, yet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 157, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 505, + 168 + ], + "score": 1.0, + "content": "an update with a smaller step size would always decrease the training loss. We now demonstrate", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 167, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 506, + 180 + ], + "score": 1.0, + "content": "that this interpretation may generalize to SGD. In particular, we will demonstrate that SGD seems to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "“acclimate” to the step size and batch size in such a way that an actual update sometimes increases", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "and sometimes decreases the loss in expectation, yet a update with a larger step size or smaller batch", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 212 + ], + "score": 1.0, + "content": "size would almost always increase the loss in expectation, and a step with a smaller step size or a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 211, + 391, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 391, + 223 + ], + "score": 1.0, + "content": "larger batch size would almost always decrease the loss in expectation.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 134, + 506, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 288, + 239 + ], + "score": 1.0, + "content": "In Figure 25, we train the tanh network from", + "type": "text" + }, + { + "bbox": [ + 288, + 228, + 299, + 239 + ], + "score": 0.57, + "content": "\\ S 3", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 228, + 505, + 239 + ], + "score": 1.0, + "content": "with MSE loss, using SGD with step size 0.01 and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "batch size 32. We periodically compute the training loss (over the full dataset) and plot these on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "the left pane of Figure 25. Observe that the training loss does not decrease monotonically, but of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 506, + 275 + ], + "score": 1.0, + "content": "course this is not surprising — SGD is a random algorithm. However, what may be more surprising", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "is that SGD is not even decreasing the training loss in expectation. On the right pane of Figure", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "25, every 500 steps during training, we use the Monte Carlo method to approximately compute the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "expected change in training loss that would result from an SGD step (the expectation here is over", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 319 + ], + "score": 1.0, + "content": "the randomness involved in selecting the minibatch). Observe that at many points during training,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 314, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 330 + ], + "score": 1.0, + "content": "an SGD step would decrease the loss (as desired) in expectation, but at other points, and SGD step", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 326, + 262, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 262, + 339 + ], + "score": 1.0, + "content": "would increase the loss in expectation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 228, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 343, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "In Figure 26(a), while training that network, we compute the expected change in training loss that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "would result from taking an SGD step with the same step size used during training (i.e. 0.01), but", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "half the batch size used during training (i.e. 16). We observe that an SGD step with half the batch", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "score": 1.0, + "content": "size would consistently cause an increase in the training loss in expectation. In Figure 26(b) we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "repeat this experiment, but with twice the batch size used during training (i.e. 64). Notice that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "an SGD step with twice the batch size would consistently cause a decrease in the training loss in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "score": 1.0, + "content": "expectation. In Figure 26(c) and (d) repeat this experiment with the step size; we observe that an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "SGD step with a larger step size (0.02) would consistently increase the training loss in expectation,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "while an SGD step with a smaller step size (0.005) would consistently decrease the training loss in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 442, + 157, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 157, + 455 + ], + "score": 1.0, + "content": "expectation.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 342, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 471 + ], + "score": 1.0, + "content": "In Figure 27, as a “control” experiment, we both train and measure the expected loss change under", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "the following four hyperparameter settings: (step size 0.01, batch size 16), (step size 0.01, batch size", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 494 + ], + "score": 1.0, + "content": "64), (step size 0.02, batch size 32), and (step size 0.005, batch size 32). In each case, we observe", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "that, after a brief period at the beginning of training, each SGD update sometime increases and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 502, + 318, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 318, + 515 + ], + "score": 1.0, + "content": "sometimes decreases the training loss in expectation.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 458, + 505, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 519, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Therefore, at least for this single network, we can conclude that no matter the hyperparameters,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "SGD quickly navigates to, and then lingers in, regions of the loss landscape in which an SGD", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "update with those hyperparameters sometimes increases, and sometimes decreases the training loss", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "in expectation, yet an SGD update with a smaller step size or larger batch size would consistently", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "decrease the loss in expectation, and an SGD update with a larger step size or smaller batch size", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 574, + 312, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 312, + 587 + ], + "score": 1.0, + "content": "would consistently increase the loss in expectation.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 519, + 505, + 587 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 104, + 522, + 197 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 104, + 522, + 197 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 104, + 522, + 197 + ], + "spans": [ + { + "bbox": [ + 109, + 104, + 522, + 197 + ], + "score": 0.947, + "type": "image", + "image_path": "4e018c957930a5474c4334e97cd3b044261c25dc2f6b1ba664e7b8e5382ded7d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 104, + 522, + 135.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 135.0, + 522, + 166.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 166.0, + 522, + 197.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 213, + 506, + 279 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "Figure 25: SGD does not consistently decrease the training loss in expectation. We train the tanh", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "network from section 3 with MSE loss, using SGD with step size 0.01 and batch size 32. Periodically", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "during training, we compute (left) the full-batch training loss, and (right) the expected change in the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "full-batch training loss that would result from taking an SGD step (where the expectation is over the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 254, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 505, + 271 + ], + "score": 1.0, + "content": "randomness in sampling the minibatch). Strikingly, note that after the very beginning of training,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 267, + 504, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 504, + 280 + ], + "score": 1.0, + "content": "the expected loss change is sometimes negative (as desired) but oftentimes positive. See Figure 26.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "image", + "bbox": [ + 107, + 327, + 523, + 399 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 327, + 523, + 399 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 327, + 523, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 327, + 523, + 399 + ], + "score": 0.965, + "type": "image", + "image_path": "44ca4a3ebd286c2444ccd3e72ea0ff8569c9de959a3b0a9610b17cae9cd771cc.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 107, + 327, + 523, + 351.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 107, + 351.0, + 523, + 375.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 107, + 375.0, + 523, + 399.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 410, + 506, + 488 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "Figure 26: An SGD step with a smaller learning rate or a larger batch size than the ones used", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "during training would consistently decrease the loss in expectation. 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We train the tanh", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "network from section 3 with MSE loss, using SGD with step size 0.01 and batch size 32. 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We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 674, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 505, + 686 + ], + "score": 1.0, + "content": "observe that an SGD step using the same step size and batch size that are used during training would", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "sometimes increase and sometimes decrease the training loss in expectation. 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The dataset consists of the first 5,000 examples from CIFAR-10. To preprocess the dataset,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "we subtracted the mean from each channel, and then divided each channel by the standard deviation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 160 + ], + "score": 1.0, + "content": "(where both the mean and stddev were computed over the full CIFAR-10 dataset, not the 5k subset).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 106, + 166, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 504, + 178 + ], + "score": 1.0, + "content": "Architectures. 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Therefore, in an approximation, we instead computed the sharpness over just the first 5,000", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 441, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 441, + 227 + ], + "score": 1.0, + "content": "examples in the dataset (or, for the BN networks, over the first 5 batches out of 50).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 240, + 297, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 298, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 298, + 253 + ], + "score": 1.0, + "content": "I.3 BATCH NORMALIZATION EXPERIMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 260, + 501, + 283 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 503, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 258, + 273 + ], + "score": 1.0, + "content": "We used the CNN 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We set bptt", + "type": "text" + }, + { + "bbox": [ + 475, + 378, + 501, + 389 + ], + "score": 0.76, + "content": "\\qquad = \\ 3 5", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 378, + 505, + 390 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "which means that we divided the corpus into chunks of 35 tokens, and trained the network, us-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "ing the negative log likelihood loss, to predict each token from the preceding tokens in the same", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "chunk. Since computing the sharpness over the full dataset would not have been computationally", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 458, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 458, + 434 + ], + "score": 1.0, + "content": "practical, we computed the sharpness over a subset comprising 2500 training examples.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 447, + 194, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 196, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 196, + 460 + ], + "score": 1.0, + "content": "I.5 RUNGE-KUTTA", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 107, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 107, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "We used the “RK4” fourth-order Runge-Kutta algorithm (Press et al., 1992) to numerically integrate", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "the gradient flow ODE. The Runge-Kutta algorithm requires a step size. Rather than use a sophisti-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "cated algorithm for adaptive step size control, we decided to take advantage of the fact that we were", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 437, + 513 + ], + "score": 1.0, + "content": "already periodically computing the sharpness: at each step, we set the step size to", + "type": "text" + }, + { + "bbox": [ + 437, + 500, + 456, + 512 + ], + "score": 0.9, + "content": "\\alpha / \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 500, + 486, + 513 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 487, + 502, + 495, + 510 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 487, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 204, + 523 + ], + "score": 1.0, + "content": "a tunable parameter and", + "type": "text" + }, + { + "bbox": [ + 204, + 511, + 212, + 521 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 511, + 411, + 523 + ], + "score": 1.0, + "content": "is the most recent value for the sharpness. We set", + "type": "text" + }, + { + "bbox": [ + 411, + 511, + 437, + 521 + ], + "score": 0.9, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 511, + 449, + 523 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 450, + 511, + 483, + 521 + ], + "score": 0.88, + "content": "\\alpha = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 511, + 487, + 523 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 536, + 230, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 231, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 231, + 549 + ], + "score": 1.0, + "content": "I.6 RANDOM PROJECTIONS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 556, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 349, + 569 + ], + "score": 1.0, + "content": "In order to ascertain whether gradient descent at step size", + "type": "text" + }, + { + "bbox": [ + 350, + 559, + 357, + 568 + ], + "score": 0.82, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "initially followed the gradient flow", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 336, + 580 + ], + "score": 1.0, + "content": "trajectory (and, if so, for how long), we monitored the", + "type": "text" + }, + { + "bbox": [ + 336, + 568, + 347, + 578 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 568, + 506, + 580 + ], + "score": 1.0, + "content": "distance in weight space between the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 225, + 592 + ], + "score": 1.0, + "content": "gradient flow solution at time", + "type": "text" + }, + { + "bbox": [ + 226, + 580, + 231, + 588 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 578, + 385, + 592 + ], + "score": 1.0, + "content": "and the gradient descent iterate at step", + "type": "text" + }, + { + "bbox": [ + 386, + 579, + 401, + 591 + ], + "score": 0.89, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 578, + 506, + 592 + ], + "score": 1.0, + "content": ". 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(Note that this approach requires", + "type": "text" + }, + { + "bbox": [ + 405, + 614, + 417, + 623 + ], + "score": 0.84, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 612, + 489, + 626 + ], + "score": 1.0, + "content": "to be divisible by", + "type": "text" + }, + { + "bbox": [ + 490, + 615, + 496, + 625 + ], + "score": 0.69, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 612, + 504, + 626 + ], + "score": 1.0, + "content": ".)", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 505, + 729 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 643 + ], + "score": 1.0, + "content": "We essentially used this approach, but with one modification: regularly saving the entire net-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "work weight vector would have consumed a large amount of disk space, so we instead saved low-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 405, + 664 + ], + "score": 1.0, + "content": "dimensional random projections of the network weights. To be clear, let", + "type": "text" + }, + { + "bbox": [ + 405, + 652, + 412, + 662 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "be the number of net-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 199, + 676 + ], + "score": 1.0, + "content": "work weights, and let", + "type": "text" + }, + { + "bbox": [ + 199, + 663, + 207, + 673 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "be the number of random projections (a tunable parameter chosen such", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 671, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 124, + 687 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 674, + 156, + 684 + ], + "score": 0.89, + "content": "k \\ll d ,", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 671, + 300, + 687 + ], + "score": 1.0, + "content": "). Then we first generated a matrix", + "type": "text" + }, + { + "bbox": [ + 300, + 673, + 349, + 684 + ], + "score": 0.92, + "content": "\\mathbf { M } \\in \\breve { \\mathbb { R } } ^ { k \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 671, + 506, + 687 + ], + "score": 1.0, + "content": "by sampling each entry i.i.d from the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "standard normal distribution. 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Then we plotted the", + "type": "text" + }, + { + "bbox": [ + 409, + 707, + 419, + 718 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 707, + 505, + 719 + ], + "score": 1.0, + "content": "distance between the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 718, + 505, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 505, + 730 + ], + "score": 1.0, + "content": "low-dimensional vectors from gradient flow, and the low-dimensional vectors from gradient descent.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43 + } + ], + "page_idx": 31, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "32", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 504, + 107 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "For the two networks with batch normalization, running gradient descent with full-dataset batch", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "normalization would not have been feasible under our GPU memory constraints. Therefore, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 133, + 504, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 504, + 144 + ], + "score": 1.0, + "content": "instead used ghost batch normalization (Hoffer et al., 2017) with 50 ghost batches of size 1,000.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "This means that we divided the 50,000 examples in CIFAR-10 into 50 fixed groups of size 1,000", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "each, and defined the overall objective function to be the average of 50 fixed batch-wise objectives.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "To correctly compute the overall gradient of this training objective, we can just run backprop 50", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 361, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 361, + 189 + ], + "score": 1.0, + "content": "times (once on each group) and average the resulting gradients.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 110, + 505, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 193, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "score": 1.0, + "content": "To compute the sharpness over the full CIFAR-10 dataset would have been computationally expen-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "sive. 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We set bptt", + "type": "text" + }, + { + "bbox": [ + 475, + 378, + 501, + 389 + ], + "score": 0.76, + "content": "\\qquad = \\ 3 5", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 378, + 505, + 390 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "which means that we divided the corpus into chunks of 35 tokens, and trained the network, us-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "ing the negative log likelihood loss, to predict each token from the preceding tokens in the same", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "chunk. 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The Runge-Kutta algorithm requires a step size. 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One way to do would be", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 396, + 602 + ], + "score": 1.0, + "content": "as follows: (a) when running gradient flow, save the weights after every", + "type": "text" + }, + { + "bbox": [ + 397, + 590, + 410, + 599 + ], + "score": 0.85, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "units of time, for some", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 149, + 613 + ], + "score": 1.0, + "content": "parameter", + "type": "text" + }, + { + "bbox": [ + 149, + 601, + 162, + 611 + ], + "score": 0.76, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 600, + 406, + 613 + ], + "score": 1.0, + "content": "; (b) when running gradient descent, save the weights at each", + "type": "text" + }, + { + "bbox": [ + 406, + 600, + 425, + 614 + ], + "score": 0.88, + "content": "\\big ( \\frac { \\Delta t } { \\eta } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "-th step; (c) plot the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 612, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 404, + 626 + ], + "score": 1.0, + "content": "difference between these two sequences. (Note that this approach requires", + "type": "text" + }, + { + "bbox": [ + 405, + 614, + 417, + 623 + ], + "score": 0.84, + "content": "\\Delta t", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 612, + 489, + 626 + ], + "score": 1.0, + "content": "to be divisible by", + "type": "text" + }, + { + "bbox": [ + 490, + 615, + 496, + 625 + ], + "score": 0.69, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 612, + 504, + 626 + ], + "score": 1.0, + "content": ".)", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 556, + 506, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 505, + 729 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 643 + ], + "score": 1.0, + "content": "We essentially used this approach, but with one modification: regularly saving the entire net-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "work weight vector would have consumed a large amount of disk space, so we instead saved low-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 405, + 664 + ], + "score": 1.0, + "content": "dimensional random projections of the network weights. To be clear, let", + "type": "text" + }, + { + "bbox": [ + 405, + 652, + 412, + 662 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "be the number of net-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 199, + 676 + ], + "score": 1.0, + "content": "work weights, and let", + "type": "text" + }, + { + "bbox": [ + 199, + 663, + 207, + 673 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "be the number of random projections (a tunable parameter chosen such", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 671, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 124, + 687 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 674, + 156, + 684 + ], + "score": 0.89, + "content": "k \\ll d ,", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 671, + 300, + 687 + ], + "score": 1.0, + "content": "). Then we first generated a matrix", + "type": "text" + }, + { + "bbox": [ + 300, + 673, + 349, + 684 + ], + "score": 0.92, + "content": "\\mathbf { M } \\in \\breve { \\mathbb { R } } ^ { k \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 671, + 506, + 687 + ], + "score": 1.0, + "content": "by sampling each entry i.i.d from the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 697 + ], + "score": 1.0, + "content": "standard normal distribution. During training, rather than periodically save the whole weight vector", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 695, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 117, + 708 + ], + "score": 1.0, + "content": "(a", + "type": "text" + }, + { + "bbox": [ + 117, + 696, + 124, + 706 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 695, + 384, + 708 + ], + "score": 1.0, + "content": "-dimensional vector), we premultiplied this vector by the matrix", + "type": "text" + }, + { + "bbox": [ + 385, + 696, + 397, + 706 + ], + "score": 0.27, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 695, + 444, + 708 + ], + "score": 1.0, + "content": "to obtain a", + "type": "text" + }, + { + "bbox": [ + 445, + 696, + 452, + 706 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 695, + 505, + 708 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 707, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 408, + 719 + ], + "score": 1.0, + "content": "vector, and we periodically saved these vectors instead. Then we plotted the", + "type": "text" + }, + { + "bbox": [ + 409, + 707, + 419, + 718 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 707, + 505, + 719 + ], + "score": 1.0, + "content": "distance between the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 718, + 505, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 505, + 730 + ], + "score": 1.0, + "content": "low-dimensional vectors from gradient flow, and the low-dimensional vectors from gradient descent.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 630, + 506, + 730 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 323, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 324, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 324, + 96 + ], + "score": 1.0, + "content": "J EXPERIMENTS: VARY ARCHITECTURES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 503, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we fix the task as that of fitting a 5,000-sized subset of CIFAR-10, and we verify", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 330, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 330, + 129 + ], + "score": 1.0, + "content": "our main findings across a broad range of architectures.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 505, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "Procedure We consider fully-connected networks and convolutional networks, the latter with both", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "max-pooling and average pooling. For all of these, we consider tanh, ReLU, and ELU activations,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "and for fully-connected networks we moreover consider softplus and hardtanh activations. We train", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 174, + 471, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 337, + 186 + ], + "score": 1.0, + "content": "each network with both cross-entropy and MSE loss. See", + "type": "text" + }, + { + "bbox": [ + 337, + 174, + 354, + 185 + ], + "score": 0.8, + "content": "\\ S \\mathrm { I . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 174, + 471, + 186 + ], + "score": 1.0, + "content": "for full experimental details.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "score": 1.0, + "content": "In each case, we first use the Runge-Kutta method to numerically integrate the gradient flow ODE", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 125, + 214 + ], + "score": 1.0, + "content": "(see", + "type": "text" + }, + { + "bbox": [ + 126, + 202, + 143, + 213 + ], + "score": 0.82, + "content": "\\ S 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "for details). For architectures that give rise to continuously differentiable training objec-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "tives, the gradient flow ODE is guaranteed to have a unique solution (which we call the gradient flow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "trajectory), and Runge-Kutta will return a numerical approximation to this solution. On the other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "hand, for architectures with ReLU, hardtanh, or max-pooling, the training objective is not continu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "ously differentiable, so the gradient flow ODE does not necessarily have a unique solution, and there", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "are no guarantees a priori on what Runge-Kutta will return (more on this below under the “findings”", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 268, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 279 + ], + "score": 1.0, + "content": "heading). Still, in both cases, since our implementation of Runge-Kutta automatically adjusts the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "step size based on the local sharpness in order to remain stable, the Runge-Kutta trajectory can be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 438, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 438, + 302 + ], + "score": 1.0, + "content": "roughly viewed as “what gradient descent would do if instability was not an issue.”", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 108, + 306, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 383, + 319 + ], + "score": 1.0, + "content": "We then run gradient descent at a range of step sizes. These step sizes", + "type": "text" + }, + { + "bbox": [ + 384, + 308, + 390, + 318 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "were chosen by hand so that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 157, + 331 + ], + "score": 1.0, + "content": "the quantity", + "type": "text" + }, + { + "bbox": [ + 157, + 317, + 174, + 329 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 316, + 324, + 331 + ], + "score": 1.0, + "content": "would be spaced uniformly between", + "type": "text" + }, + { + "bbox": [ + 324, + 317, + 336, + 329 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 316, + 481, + 331 + ], + "score": 1.0, + "content": "(the sharpness at initialization) and", + "type": "text" + }, + { + "bbox": [ + 482, + 317, + 504, + 329 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 348, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 348, + 341 + ], + "score": 1.0, + "content": "(the maximum sharpness along the Runge-Kutta trajectory).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "score": 1.0, + "content": "Results During Runge-Kutta, we observe that the sharpness tends to continually increase dur-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "ing training (progressive sharpening), with the exception that when cross-entropy loss is used, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 412, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 412, + 388 + ], + "score": 1.0, + "content": "sharpness decreases at the very end of training, as explained in Appendix C.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 262, + 403 + ], + "score": 1.0, + "content": "During gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 262, + 393, + 269, + 402 + ], + "score": 0.72, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 390, + 448, + 403 + ], + "score": 1.0, + "content": ", we observe that once the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 448, + 390, + 465, + 402 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 390, + 505, + 403 + ], + "score": 1.0, + "content": ", it ceases", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 430, + 414 + ], + "score": 1.0, + "content": "to increase much further, and instead hovers right at, or just above, the value", + "type": "text" + }, + { + "bbox": [ + 430, + 402, + 447, + 413 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 401, + 505, + 414 + ], + "score": 1.0, + "content": ". For reasons", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "unknown, it tends to be true that for MSE loss, the sharpness hovers just a tiny bit above the value", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 492, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 123, + 436 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 424, + 418, + 437 + ], + "score": 1.0, + "content": ", while for cross-entropy loss the gap between the sharpness and the value", + "type": "text" + }, + { + "bbox": [ + 419, + 424, + 435, + 435 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 424, + 492, + 437 + ], + "score": 1.0, + "content": "is a bit larger.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 454 + ], + "score": 1.0, + "content": "For each step size, we monitor the distance between the gradient descent trajectory and the Runge-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 480, + 464 + ], + "score": 1.0, + "content": "Kutta trajectory — that is, we monitor the distance between the Runge-Kutta iterate at time", + "type": "text" + }, + { + "bbox": [ + 481, + 452, + 487, + 461 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 252, + 475 + ], + "score": 1.0, + "content": "the gradient descent iterate at step", + "type": "text" + }, + { + "bbox": [ + 252, + 462, + 268, + 474 + ], + "score": 0.89, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 462, + 290, + 475 + ], + "score": 1.0, + "content": "(see", + "type": "text" + }, + { + "bbox": [ + 290, + 462, + 308, + 474 + ], + "score": 0.79, + "content": "\\ S 1 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "for details). Empirically, for architectures that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "give rise to continuously differentiable training objectives, we observe that this distance is nearly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 229, + 497 + ], + "score": 1.0, + "content": "zero before the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 229, + 484, + 246, + 496 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 484, + 506, + 497 + ], + "score": 1.0, + "content": ", and it starts to climb immediately afterwards. This means that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "gradient descent closely tracks the gradient flow trajectory so long as the sharpness remains less", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 127, + 519 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 127, + 506, + 144, + 518 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 507, + 505, + 519 + ], + "score": 1.0, + "content": ". Note that this finding was not a foregone conclusion: gradient descent is guaranteed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "to track the gradient flow trajectory in the limit of infinitesimal step sizes (since gradient descent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "is the forward Euler discretization of the gradient flow ODE), but for non-infinitesimal step sizes,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "score": 1.0, + "content": "there is discretization error, which is studied in Barrett & Dherin (2021). Our empirical finding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "is essentially that this discretization error is small compared to the difference between trajectories", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 561, + 192, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 192, + 574 + ], + "score": 1.0, + "content": "caused by instability.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "On the other hand, for architectures with non-differentiable components such as ReLU or max-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "pooling, we sometimes observe that gradient descent tracks the Runge-Kutta trajectory so long as the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 217, + 612 + ], + "score": 1.0, + "content": "sharpness remains less than", + "type": "text" + }, + { + "bbox": [ + 217, + 600, + 234, + 612 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 600, + 505, + 612 + ], + "score": 1.0, + "content": ", but we also sometimes observe that the gradient descent trajectories", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "differ from one another (and from Runge-Kutta) from the beginning of training. In the former case,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "we can infer that the gradient flow trajectory apparently does exist, and is returned by Runge-Kutta;", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "in the latter case, we can infer that either (a) the gradient flow trajectory does not exist, or (b) that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "it does exist (and is returned by Runge-Kutta), but the step sizes we used for gradient descent were", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 187, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 187, + 667 + ], + "score": 1.0, + "content": "too large to track it.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5 + } + ], + "page_idx": 32, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "33", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 323, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 324, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 324, + 96 + ], + "score": 1.0, + "content": "J EXPERIMENTS: VARY ARCHITECTURES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 503, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "In this appendix, we fix the task as that of fitting a 5,000-sized subset of CIFAR-10, and we verify", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 330, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 330, + 129 + ], + "score": 1.0, + "content": "our main findings across a broad range of architectures.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 105, + 505, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 505, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "Procedure We consider fully-connected networks and convolutional networks, the latter with both", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "max-pooling and average pooling. For all of these, we consider tanh, ReLU, and ELU activations,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "and for fully-connected networks we moreover consider softplus and hardtanh activations. We train", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 174, + 471, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 337, + 186 + ], + "score": 1.0, + "content": "each network with both cross-entropy and MSE loss. See", + "type": "text" + }, + { + "bbox": [ + 337, + 174, + 354, + 185 + ], + "score": 0.8, + "content": "\\ S \\mathrm { I . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 174, + 471, + 186 + ], + "score": 1.0, + "content": "for full experimental details.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 140, + 506, + 186 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 190, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 203 + ], + "score": 1.0, + "content": "In each case, we first use the Runge-Kutta method to numerically integrate the gradient flow ODE", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 125, + 214 + ], + "score": 1.0, + "content": "(see", + "type": "text" + }, + { + "bbox": [ + 126, + 202, + 143, + 213 + ], + "score": 0.82, + "content": "\\ S 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "for details). For architectures that give rise to continuously differentiable training objec-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "tives, the gradient flow ODE is guaranteed to have a unique solution (which we call the gradient flow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "trajectory), and Runge-Kutta will return a numerical approximation to this solution. On the other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 247 + ], + "score": 1.0, + "content": "hand, for architectures with ReLU, hardtanh, or max-pooling, the training objective is not continu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "ously differentiable, so the gradient flow ODE does not necessarily have a unique solution, and there", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "are no guarantees a priori on what Runge-Kutta will return (more on this below under the “findings”", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 268, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 279 + ], + "score": 1.0, + "content": "heading). Still, in both cases, since our implementation of Runge-Kutta automatically adjusts the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "step size based on the local sharpness in order to remain stable, the Runge-Kutta trajectory can be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 438, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 438, + 302 + ], + "score": 1.0, + "content": "roughly viewed as “what gradient descent would do if instability was not an issue.”", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 190, + 505, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 306, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 383, + 319 + ], + "score": 1.0, + "content": "We then run gradient descent at a range of step sizes. These step sizes", + "type": "text" + }, + { + "bbox": [ + 384, + 308, + 390, + 318 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "were chosen by hand so that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 504, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 157, + 331 + ], + "score": 1.0, + "content": "the quantity", + "type": "text" + }, + { + "bbox": [ + 157, + 317, + 174, + 329 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 316, + 324, + 331 + ], + "score": 1.0, + "content": "would be spaced uniformly between", + "type": "text" + }, + { + "bbox": [ + 324, + 317, + 336, + 329 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 316, + 481, + 331 + ], + "score": 1.0, + "content": "(the sharpness at initialization) and", + "type": "text" + }, + { + "bbox": [ + 482, + 317, + 504, + 329 + ], + "score": 0.89, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 328, + 348, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 348, + 341 + ], + "score": 1.0, + "content": "(the maximum sharpness along the Runge-Kutta trajectory).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 305, + 506, + 341 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 364 + ], + "score": 1.0, + "content": "Results During Runge-Kutta, we observe that the sharpness tends to continually increase dur-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "ing training (progressive sharpening), with the exception that when cross-entropy loss is used, the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 412, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 412, + 388 + ], + "score": 1.0, + "content": "sharpness decreases at the very end of training, as explained in Appendix C.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 352, + 505, + 388 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 262, + 403 + ], + "score": 1.0, + "content": "During gradient descent with step size", + "type": "text" + }, + { + "bbox": [ + 262, + 393, + 269, + 402 + ], + "score": 0.72, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 390, + 448, + 403 + ], + "score": 1.0, + "content": ", we observe that once the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 448, + 390, + 465, + 402 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 390, + 505, + 403 + ], + "score": 1.0, + "content": ", it ceases", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 430, + 414 + ], + "score": 1.0, + "content": "to increase much further, and instead hovers right at, or just above, the value", + "type": "text" + }, + { + "bbox": [ + 430, + 402, + 447, + 413 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 401, + 505, + 414 + ], + "score": 1.0, + "content": ". For reasons", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "unknown, it tends to be true that for MSE loss, the sharpness hovers just a tiny bit above the value", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 492, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 123, + 436 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 424, + 418, + 437 + ], + "score": 1.0, + "content": ", while for cross-entropy loss the gap between the sharpness and the value", + "type": "text" + }, + { + "bbox": [ + 419, + 424, + 435, + 435 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 424, + 492, + 437 + ], + "score": 1.0, + "content": "is a bit larger.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 390, + 505, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 454 + ], + "score": 1.0, + "content": "For each step size, we monitor the distance between the gradient descent trajectory and the Runge-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 480, + 464 + ], + "score": 1.0, + "content": "Kutta trajectory — that is, we monitor the distance between the Runge-Kutta iterate at time", + "type": "text" + }, + { + "bbox": [ + 481, + 452, + 487, + 461 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 252, + 475 + ], + "score": 1.0, + "content": "the gradient descent iterate at step", + "type": "text" + }, + { + "bbox": [ + 252, + 462, + 268, + 474 + ], + "score": 0.89, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 462, + 290, + 475 + ], + "score": 1.0, + "content": "(see", + "type": "text" + }, + { + "bbox": [ + 290, + 462, + 308, + 474 + ], + "score": 0.79, + "content": "\\ S 1 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "for details). Empirically, for architectures that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "give rise to continuously differentiable training objectives, we observe that this distance is nearly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 229, + 497 + ], + "score": 1.0, + "content": "zero before the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 229, + 484, + 246, + 496 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 484, + 506, + 497 + ], + "score": 1.0, + "content": ", and it starts to climb immediately afterwards. This means that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "gradient descent closely tracks the gradient flow trajectory so long as the sharpness remains less", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 127, + 519 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 127, + 506, + 144, + 518 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 507, + 505, + 519 + ], + "score": 1.0, + "content": ". Note that this finding was not a foregone conclusion: gradient descent is guaranteed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "to track the gradient flow trajectory in the limit of infinitesimal step sizes (since gradient descent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "is the forward Euler discretization of the gradient flow ODE), but for non-infinitesimal step sizes,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "score": 1.0, + "content": "there is discretization error, which is studied in Barrett & Dherin (2021). Our empirical finding", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "is essentially that this discretization error is small compared to the difference between trajectories", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 561, + 192, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 192, + 574 + ], + "score": 1.0, + "content": "caused by instability.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 439, + 506, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "On the other hand, for architectures with non-differentiable components such as ReLU or max-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "pooling, we sometimes observe that gradient descent tracks the Runge-Kutta trajectory so long as the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 217, + 612 + ], + "score": 1.0, + "content": "sharpness remains less than", + "type": "text" + }, + { + "bbox": [ + 217, + 600, + 234, + 612 + ], + "score": 0.91, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 600, + 505, + 612 + ], + "score": 1.0, + "content": ", but we also sometimes observe that the gradient descent trajectories", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "differ from one another (and from Runge-Kutta) from the beginning of training. In the former case,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "we can infer that the gradient flow trajectory apparently does exist, and is returned by Runge-Kutta;", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "in the latter case, we can infer that either (a) the gradient flow trajectory does not exist, or (b) that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "it does exist (and is returned by Runge-Kutta), but the step sizes we used for gradient descent were", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 187, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 187, + 667 + ], + "score": 1.0, + "content": "too large to track it.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 577, + 506, + 667 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 286, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 288, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 288, + 95 + ], + "score": 1.0, + "content": "J.1 FULLY-CONNECTED TANH NETWORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 103, + 199, + 114 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 200, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 200, + 116 + ], + "score": 1.0, + "content": "J.1.1 SQUARE LOSS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "image", + "bbox": [ + 110, + 130, + 524, + 211 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 130, + 524, + 211 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 130, + 524, + 211 + ], + "spans": [ + { + "bbox": [ + 110, + 130, + 524, + 211 + ], + "score": 0.968, + "type": "image", + "image_path": "6d0728c745c52a1364ffe39ce6e8f8dd29a4e209e89c23debe9bab6652a7db62.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 110, + 130, + 524, + 157.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 110, + 157.0, + 524, + 184.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 110, + 184.0, + 524, + 211.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 224, + 506, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 315, + 236 + ], + "score": 1.0, + "content": "Figure 28: Gradient flow. We train the network to", + "type": "text" + }, + { + "bbox": [ + 315, + 224, + 335, + 234 + ], + "score": 0.88, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "accuracy with gradient flow, by using the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 387, + 247 + ], + "score": 1.0, + "content": "Runge-Kutta method to discretize the gradient flow ODE (details in", + "type": "text" + }, + { + "bbox": [ + 387, + 235, + 406, + 246 + ], + "score": 0.72, + "content": "\\ S 1 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "). We plot the train loss", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "(left), sharpness (center), and train accuracy (right) over time. Observe that the sharpness tends to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 258, + 371, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 371, + 269 + ], + "score": 1.0, + "content": "continually increase (except for a slight decrease at initialization).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + } + ], + "index": 4.75 + }, + { + "type": "image", + "bbox": [ + 111, + 318, + 525, + 487 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 318, + 525, + 487 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 318, + 525, + 487 + ], + "spans": [ + { + "bbox": [ + 111, + 318, + 525, + 487 + ], + "score": 0.917, + "type": "image", + "image_path": "35dd58b8cdeebf43d0600dd08fe120012208937c690e830d55713a80255dffe1.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 111, + 318, + 525, + 374.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 374.3333333333333, + 525, + 430.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 430.66666666666663, + 525, + 486.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 501, + 506, + 676 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 333, + 514 + ], + "score": 1.0, + "content": "Figure 29: Gradient descent. We train the network to", + "type": "text" + }, + { + "bbox": [ + 334, + 501, + 354, + 511 + ], + "score": 0.86, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "accuracy using gradient descent at a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 180, + 524 + ], + "score": 1.0, + "content": "range of step sizes", + "type": "text" + }, + { + "bbox": [ + 181, + 513, + 187, + 523 + ], + "score": 0.72, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ". Top left: we plot the train loss curves, with a vertical line marking the iteration", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 236, + 535 + ], + "score": 1.0, + "content": "where the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 236, + 523, + 253, + 534 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 522, + 505, + 535 + ], + "score": 1.0, + "content": ". Observe that the train loss monotonically decreases before this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "point, but behaves non-monotonically afterwards. Top middle: we plot the sharpness (measured at", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "regular intervals during training). For each step size, the horizontal dashed line of the appropriate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 554, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 554, + 279, + 569 + ], + "score": 1.0, + "content": "color marks the maximum stable sharpness", + "type": "text" + }, + { + "bbox": [ + 280, + 555, + 296, + 568 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 554, + 506, + 569 + ], + "score": 1.0, + "content": ". Observe that the sharpness tends to increase during", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 567, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 238, + 579 + ], + "score": 1.0, + "content": "training until reaching the value", + "type": "text" + }, + { + "bbox": [ + 238, + 567, + 255, + 578 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 567, + 506, + 579 + ], + "score": 1.0, + "content": ", and then hovers just a bit above that value. Bottom left: we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 145, + 590 + ], + "score": 1.0, + "content": "track the", + "type": "text" + }, + { + "bbox": [ + 145, + 577, + 156, + 588 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 577, + 464, + 590 + ], + "score": 1.0, + "content": "distance between (random projections of) the gradient flow iterate at time", + "type": "text" + }, + { + "bbox": [ + 465, + 578, + 470, + 587 + ], + "score": 0.67, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 251, + 600 + ], + "score": 1.0, + "content": "gradient descent iterate at iteration", + "type": "text" + }, + { + "bbox": [ + 252, + 588, + 267, + 600 + ], + "score": 0.93, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 589, + 313, + 600 + ], + "score": 1.0, + "content": "(details in", + "type": "text" + }, + { + "bbox": [ + 313, + 588, + 333, + 599 + ], + "score": 0.61, + "content": "\\ S 1 . 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 589, + 417, + 600 + ], + "score": 1.0, + "content": ". For each step size", + "type": "text" + }, + { + "bbox": [ + 417, + 590, + 424, + 600 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 589, + 506, + 600 + ], + "score": 1.0, + "content": ", the vertical dotted", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 321, + 613 + ], + "score": 1.0, + "content": "line marks the time when the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 321, + 600, + 338, + 611 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 599, + 506, + 613 + ], + "score": 1.0, + "content": ". Observe that the distance is essentially", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "zero until this time, and starts to grow afterwards. From this, we can conclude that gradient descent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 360, + 635 + ], + "score": 1.0, + "content": "closely tracks the gradient flow trajectory (moving at a speed", + "type": "text" + }, + { + "bbox": [ + 360, + 623, + 367, + 632 + ], + "score": 0.59, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 620, + 506, + 635 + ], + "score": 1.0, + "content": ") until the point on that trajectory", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 223, + 645 + ], + "score": 1.0, + "content": "where the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 223, + 632, + 240, + 644 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 632, + 506, + 645 + ], + "score": 1.0, + "content": ". Bottom middle: to further visualize the previous point, we plot", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 377, + 656 + ], + "score": 1.0, + "content": "the evolution of sharpness during gradient descent, but with time", + "type": "text" + }, + { + "bbox": [ + 378, + 644, + 388, + 653 + ], + "score": 0.68, + "content": "\\underline { { \\underline { { \\mathbf { \\Pi } } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 642, + 425, + 656 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 425, + 644, + 435, + 654 + ], + "score": 0.75, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 642, + 506, + 656 + ], + "score": 1.0, + "content": "step size) on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 113, + 664 + ], + "score": 0.61, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "-axis rather than iteration. We plot the gradient flow sharpness in black. Observe that the gradient", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 665, + 443, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 422, + 678 + ], + "score": 1.0, + "content": "descent sharpness matches the gradient flow sharpness until reaching the value", + "type": "text" + }, + { + "bbox": [ + 422, + 665, + 439, + 677 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 665, + 443, + 678 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 19.5 + } + ], + "index": 14.75 + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 286, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 288, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 288, + 95 + ], + "score": 1.0, + "content": "J.1 FULLY-CONNECTED TANH NETWORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 103, + 199, + 114 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 200, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 200, + 116 + ], + "score": 1.0, + "content": "J.1.1 SQUARE LOSS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "image", + "bbox": [ + 110, + 130, + 524, + 211 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 130, + 524, + 211 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 130, + 524, + 211 + ], + "spans": [ + { + "bbox": [ + 110, + 130, + 524, + 211 + ], + "score": 0.968, + "type": "image", + "image_path": "6d0728c745c52a1364ffe39ce6e8f8dd29a4e209e89c23debe9bab6652a7db62.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 110, + 130, + 524, + 157.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 110, + 157.0, + 524, + 184.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 110, + 184.0, + 524, + 211.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 224, + 506, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 315, + 236 + ], + "score": 1.0, + "content": "Figure 28: Gradient flow. We train the network to", + "type": "text" + }, + { + "bbox": [ + 315, + 224, + 335, + 234 + ], + "score": 0.88, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "accuracy with gradient flow, by using the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 387, + 247 + ], + "score": 1.0, + "content": "Runge-Kutta method to discretize the gradient flow ODE (details in", + "type": "text" + }, + { + "bbox": [ + 387, + 235, + 406, + 246 + ], + "score": 0.72, + "content": "\\ S 1 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "). We plot the train loss", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "(left), sharpness (center), and train accuracy (right) over time. Observe that the sharpness tends to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 258, + 371, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 371, + 269 + ], + "score": 1.0, + "content": "continually increase (except for a slight decrease at initialization).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + } + ], + "index": 4.75 + }, + { + "type": "image", + "bbox": [ + 111, + 318, + 525, + 487 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 318, + 525, + 487 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 318, + 525, + 487 + ], + "spans": [ + { + "bbox": [ + 111, + 318, + 525, + 487 + ], + "score": 0.917, + "type": "image", + "image_path": "35dd58b8cdeebf43d0600dd08fe120012208937c690e830d55713a80255dffe1.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 111, + 318, + 525, + 374.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 374.3333333333333, + 525, + 430.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 430.66666666666663, + 525, + 486.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 501, + 506, + 676 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 333, + 514 + ], + "score": 1.0, + "content": "Figure 29: Gradient descent. We train the network to", + "type": "text" + }, + { + "bbox": [ + 334, + 501, + 354, + 511 + ], + "score": 0.86, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "accuracy using gradient descent at a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 180, + 524 + ], + "score": 1.0, + "content": "range of step sizes", + "type": "text" + }, + { + "bbox": [ + 181, + 513, + 187, + 523 + ], + "score": 0.72, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ". Top left: we plot the train loss curves, with a vertical line marking the iteration", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 236, + 535 + ], + "score": 1.0, + "content": "where the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 236, + 523, + 253, + 534 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 522, + 505, + 535 + ], + "score": 1.0, + "content": ". Observe that the train loss monotonically decreases before this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 546 + ], + "score": 1.0, + "content": "point, but behaves non-monotonically afterwards. Top middle: we plot the sharpness (measured at", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "regular intervals during training). For each step size, the horizontal dashed line of the appropriate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 554, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 554, + 279, + 569 + ], + "score": 1.0, + "content": "color marks the maximum stable sharpness", + "type": "text" + }, + { + "bbox": [ + 280, + 555, + 296, + 568 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 554, + 506, + 569 + ], + "score": 1.0, + "content": ". Observe that the sharpness tends to increase during", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 567, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 238, + 579 + ], + "score": 1.0, + "content": "training until reaching the value", + "type": "text" + }, + { + "bbox": [ + 238, + 567, + 255, + 578 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 567, + 506, + 579 + ], + "score": 1.0, + "content": ", and then hovers just a bit above that value. Bottom left: we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 145, + 590 + ], + "score": 1.0, + "content": "track the", + "type": "text" + }, + { + "bbox": [ + 145, + 577, + 156, + 588 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 577, + 464, + 590 + ], + "score": 1.0, + "content": "distance between (random projections of) the gradient flow iterate at time", + "type": "text" + }, + { + "bbox": [ + 465, + 578, + 470, + 587 + ], + "score": 0.67, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 251, + 600 + ], + "score": 1.0, + "content": "gradient descent iterate at iteration", + "type": "text" + }, + { + "bbox": [ + 252, + 588, + 267, + 600 + ], + "score": 0.93, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 589, + 313, + 600 + ], + "score": 1.0, + "content": "(details in", + "type": "text" + }, + { + "bbox": [ + 313, + 588, + 333, + 599 + ], + "score": 0.61, + "content": "\\ S 1 . 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 589, + 417, + 600 + ], + "score": 1.0, + "content": ". For each step size", + "type": "text" + }, + { + "bbox": [ + 417, + 590, + 424, + 600 + ], + "score": 0.74, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 589, + 506, + 600 + ], + "score": 1.0, + "content": ", the vertical dotted", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 321, + 613 + ], + "score": 1.0, + "content": "line marks the time when the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 321, + 600, + 338, + 611 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 599, + 506, + 613 + ], + "score": 1.0, + "content": ". Observe that the distance is essentially", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "zero until this time, and starts to grow afterwards. From this, we can conclude that gradient descent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 360, + 635 + ], + "score": 1.0, + "content": "closely tracks the gradient flow trajectory (moving at a speed", + "type": "text" + }, + { + "bbox": [ + 360, + 623, + 367, + 632 + ], + "score": 0.59, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 620, + 506, + 635 + ], + "score": 1.0, + "content": ") until the point on that trajectory", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 223, + 645 + ], + "score": 1.0, + "content": "where the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 223, + 632, + 240, + 644 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 632, + 506, + 645 + ], + "score": 1.0, + "content": ". Bottom middle: to further visualize the previous point, we plot", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 377, + 656 + ], + "score": 1.0, + "content": "the evolution of sharpness during gradient descent, but with time", + "type": "text" + }, + { + "bbox": [ + 378, + 644, + 388, + 653 + ], + "score": 0.68, + "content": "\\underline { { \\underline { { \\mathbf { \\Pi } } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 642, + 425, + 656 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 425, + 644, + 435, + 654 + ], + "score": 0.75, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 642, + 506, + 656 + ], + "score": 1.0, + "content": "step size) on the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 113, + 664 + ], + "score": 0.61, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "-axis rather than iteration. We plot the gradient flow sharpness in black. Observe that the gradient", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 665, + 443, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 422, + 678 + ], + "score": 1.0, + "content": "descent sharpness matches the gradient flow sharpness until reaching the value", + "type": "text" + }, + { + "bbox": [ + 422, + 665, + 439, + 677 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 665, + 443, + 678 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 19.5 + } + ], + "index": 14.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 109, + 525, + 189 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 109, + 525, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 109, + 525, + 189 + ], + "spans": [ + { + "bbox": [ + 110, + 109, + 525, + 189 + ], + "score": 0.967, + "type": "image", + "image_path": "9f1176dc6f69dfd4fa5b6d6db6c53ba62bde8c94190927b8a35edf279a89486b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 109, + 525, + 135.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 135.66666666666666, + 525, + 162.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 162.33333333333331, + 525, + 188.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 201, + 505, + 225 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "Figure 30: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 212, + 417, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 417, + 225 + ], + "score": 1.0, + "content": "figure the sharpness drops at the end of training due to the cross-entropy loss.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 111, + 244, + 525, + 414 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 244, + 525, + 414 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 244, + 525, + 414 + ], + "spans": [ + { + "bbox": [ + 111, + 244, + 525, + 414 + ], + "score": 0.971, + "type": "image", + "image_path": "b81380c85e577848398396434eefa62bd8bb599a82a59eb78c6a957b33272662.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 111, + 244, + 525, + 300.6666666666667 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 300.6666666666667, + 525, + 357.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 357.33333333333337, + 525, + 414.00000000000006 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 138, + 426, + 471, + 439 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 137, + 426, + 473, + 440 + ], + "spans": [ + { + "bbox": [ + 137, + 426, + 473, + 440 + ], + "score": 1.0, + "content": "Figure 31: Gradient descent. Refer to the Figure 29 caption for more information.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + } + ], + "index": 7.0 + } + ], + "page_idx": 34, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 83, + 236, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 236, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 236, + 94 + ], + "score": 1.0, + "content": "J.1.2 CROSS-ENTROPY LOSS", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 109, + 525, + 189 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 109, + 525, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 109, + 525, + 189 + ], + "spans": [ + { + "bbox": [ + 110, + 109, + 525, + 189 + ], + "score": 0.967, + "type": "image", + "image_path": "9f1176dc6f69dfd4fa5b6d6db6c53ba62bde8c94190927b8a35edf279a89486b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 109, + 525, + 135.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 135.66666666666666, + 525, + 162.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 162.33333333333331, + 525, + 188.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 201, + 505, + 225 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "Figure 30: Gradient flow. Refer to the Figure 28 caption for more information. Additionally, in this", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 212, + 417, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 417, + 225 + ], + "score": 1.0, + "content": "figure the sharpness drops at the end of training due to the cross-entropy loss.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 111, + 244, + 525, + 414 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 244, + 525, + 414 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 244, + 525, + 414 + ], + "spans": [ + { + "bbox": [ + 111, + 244, + 525, + 414 + ], + "score": 0.971, + "type": "image", + "image_path": "b81380c85e577848398396434eefa62bd8bb599a82a59eb78c6a957b33272662.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 111, + 244, + 525, + 300.6666666666667 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 300.6666666666667, + 525, + 357.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 357.33333333333337, + 525, + 414.00000000000006 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 138, + 426, + 471, + 439 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 137, + 426, + 473, + 440 + ], + "spans": [ + { + "bbox": [ + 137, + 426, + 473, + 440 + ], + "score": 1.0, + "content": "Figure 31: Gradient descent. 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Refer to the Figure 29 caption for more information.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + } + ], + "index": 7.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 289, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 291, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 291, + 95 + ], + "score": 1.0, + "content": "J.4 FULLY-CONNECTED RELU NETWORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "Note that since the ReLU activation function is not continuously differentiable, the training objective", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 137, + 199, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 136, + 200, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 200, + 151 + ], + "score": 1.0, + "content": "J.4.1 SQUARE LOSS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 111, + 164, + 525, + 245 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 164, + 525, + 245 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 164, + 525, + 245 + ], + "spans": [ + { + "bbox": [ + 111, + 164, + 525, + 245 + ], + "score": 0.963, + "type": "image", + "image_path": "b65f18aee6331aef07845d67a2af73b7ba1cbdc19cfcdb0b6b9b48beab0e5596.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 111, + 164, + 525, + 191.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 191.0, + 525, + 218.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 218.0, + 525, + 245.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 258, + 506, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 259, + 504, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 309, + 270 + ], + "score": 1.0, + "content": "Figure 40: Runge-Kutta. We train the network to", + "type": "text" + }, + { + "bbox": [ + 309, + 259, + 329, + 269 + ], + "score": 0.88, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 259, + 504, + 270 + ], + "score": 1.0, + "content": "accuracy using the Runge-Kutta algorithm.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "(Since a unique gradient flow trajectory is not guaranted to exist, we hesitate to call this “gradient", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "flow”; Runge-Kutta should essentially be viewed as gradient descent with a very small step size.)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 291, + 334, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 334, + 304 + ], + "score": 1.0, + "content": "Observe that the sharpness tends to continually increase.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + } + ], + "index": 6.75 + }, + { + "type": "image", + "bbox": [ + 110, + 322, + 525, + 492 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 322, + 525, + 492 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 322, + 525, + 492 + ], + "spans": [ + { + "bbox": [ + 110, + 322, + 525, + 492 + ], + "score": 0.972, + "type": "image", + "image_path": "b1b8344d483a24a4f72f14784ebc40d2c8d2c9bde3f1f512b731dbdf4b9a4abb.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 110, + 322, + 525, + 378.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 110, + 378.6666666666667, + 525, + 435.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 110, + 435.33333333333337, + 525, + 492.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 505, + 506, + 615 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "Figure 41: Gradient descent. All panes except bottom left: refer to the Figure 29 caption for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 302, + 528 + ], + "score": 1.0, + "content": "more information. Bottom left: We track the", + "type": "text" + }, + { + "bbox": [ + 302, + 517, + 313, + 527 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "distance between (random projections of) the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 525, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 218, + 541 + ], + "score": 1.0, + "content": "Runge-Kutta iterate at time", + "type": "text" + }, + { + "bbox": [ + 219, + 528, + 224, + 537 + ], + "score": 0.62, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 525, + 399, + 541 + ], + "score": 1.0, + "content": "and the gradient descent iterate at iteration", + "type": "text" + }, + { + "bbox": [ + 399, + 527, + 415, + 539 + ], + "score": 0.88, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 525, + 495, + 541 + ], + "score": 1.0, + "content": ". For each step size", + "type": "text" + }, + { + "bbox": [ + 495, + 529, + 501, + 539 + ], + "score": 0.63, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 525, + 506, + 541 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 397, + 551 + ], + "score": 1.0, + "content": "the vertical dotted loss marks the time when the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 397, + 539, + 414, + 550 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 537, + 506, + 551 + ], + "score": 1.0, + "content": ". In contrast to Figure", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "29, here the distance between gradient descent and gradient flow starts to noticeably grow from the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "beginning of training. From this we conclude that for this architecture, gradient descent does not", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 569, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 506, + 585 + ], + "score": 1.0, + "content": "track the Runge-Kutta trajectory at first. Furthermore, for this architecture, observe that the training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "loss starts behaving non-monotonically (and the sharpness begins to plateau) before the sharpness", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 123, + 605 + ], + "score": 1.0, + "content": "hits", + "type": "text" + }, + { + "bbox": [ + 124, + 593, + 140, + 605 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 592, + 506, + 605 + ], + "score": 1.0, + "content": ". As discussed in Appendix A (“Caveats”), we believe that this is also due to the fact that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 604, + 219, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 219, + 615 + ], + "score": 1.0, + "content": "ReLU is non-differentiable.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + } + ], + "index": 15.25 + } + ], + "page_idx": 39, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "40", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 289, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 291, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 291, + 95 + ], + "score": 1.0, + "content": "J.4 FULLY-CONNECTED RELU NETWORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "Note that since the ReLU activation function is not continuously differentiable, the training objective", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "is not continuously differentiable, and so a unique gradient flow trajectory is not guaranteed to exist.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 102, + 505, + 127 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 137, + 199, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 136, + 200, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 200, + 151 + ], + "score": 1.0, + "content": "J.4.1 SQUARE LOSS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 111, + 164, + 525, + 245 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 164, + 525, + 245 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 164, + 525, + 245 + ], + "spans": [ + { + "bbox": [ + 111, + 164, + 525, + 245 + ], + "score": 0.963, + "type": "image", + "image_path": "b65f18aee6331aef07845d67a2af73b7ba1cbdc19cfcdb0b6b9b48beab0e5596.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 111, + 164, + 525, + 191.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 111, + 191.0, + 525, + 218.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 218.0, + 525, + 245.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 258, + 506, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 259, + 504, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 309, + 270 + ], + "score": 1.0, + "content": "Figure 40: Runge-Kutta. We train the network to", + "type": "text" + }, + { + "bbox": [ + 309, + 259, + 329, + 269 + ], + "score": 0.88, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 259, + 504, + 270 + ], + "score": 1.0, + "content": "accuracy using the Runge-Kutta algorithm.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "(Since a unique gradient flow trajectory is not guaranted to exist, we hesitate to call this “gradient", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "flow”; Runge-Kutta should essentially be viewed as gradient descent with a very small step size.)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 291, + 334, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 334, + 304 + ], + "score": 1.0, + "content": "Observe that the sharpness tends to continually increase.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + } + ], + "index": 6.75 + }, + { + "type": "image", + "bbox": [ + 110, + 322, + 525, + 492 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 322, + 525, + 492 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 322, + 525, + 492 + ], + "spans": [ + { + "bbox": [ + 110, + 322, + 525, + 492 + ], + "score": 0.972, + "type": "image", + "image_path": "b1b8344d483a24a4f72f14784ebc40d2c8d2c9bde3f1f512b731dbdf4b9a4abb.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 110, + 322, + 525, + 378.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 110, + 378.6666666666667, + 525, + 435.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 110, + 435.33333333333337, + 525, + 492.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 505, + 506, + 615 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "Figure 41: Gradient descent. All panes except bottom left: refer to the Figure 29 caption for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 302, + 528 + ], + "score": 1.0, + "content": "more information. Bottom left: We track the", + "type": "text" + }, + { + "bbox": [ + 302, + 517, + 313, + 527 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "distance between (random projections of) the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 525, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 218, + 541 + ], + "score": 1.0, + "content": "Runge-Kutta iterate at time", + "type": "text" + }, + { + "bbox": [ + 219, + 528, + 224, + 537 + ], + "score": 0.62, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 525, + 399, + 541 + ], + "score": 1.0, + "content": "and the gradient descent iterate at iteration", + "type": "text" + }, + { + "bbox": [ + 399, + 527, + 415, + 539 + ], + "score": 0.88, + "content": "t / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 525, + 495, + 541 + ], + "score": 1.0, + "content": ". For each step size", + "type": "text" + }, + { + "bbox": [ + 495, + 529, + 501, + 539 + ], + "score": 0.63, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 525, + 506, + 541 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 397, + 551 + ], + "score": 1.0, + "content": "the vertical dotted loss marks the time when the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 397, + 539, + 414, + 550 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 537, + 506, + 551 + ], + "score": 1.0, + "content": ". In contrast to Figure", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "29, here the distance between gradient descent and gradient flow starts to noticeably grow from the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "beginning of training. From this we conclude that for this architecture, gradient descent does not", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 569, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 506, + 585 + ], + "score": 1.0, + "content": "track the Runge-Kutta trajectory at first. Furthermore, for this architecture, observe that the training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "score": 1.0, + "content": "loss starts behaving non-monotonically (and the sharpness begins to plateau) before the sharpness", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 123, + 605 + ], + "score": 1.0, + "content": "hits", + "type": "text" + }, + { + "bbox": [ + 124, + 593, + 140, + 605 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 592, + 506, + 605 + ], + "score": 1.0, + "content": ". As discussed in Appendix A (“Caveats”), we believe that this is also due to the fact that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 604, + 219, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 219, + 615 + ], + "score": 1.0, + "content": "ReLU is non-differentiable.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + } + ], + "index": 15.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 108, + 524, + 189 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 108, + 524, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 108, + 524, + 189 + ], + "spans": [ + { + "bbox": [ + 110, + 108, + 524, + 189 + ], + "score": 0.967, + "type": "image", + "image_path": "080e0e96ecf6c9e65d4b58a5e5a0f60aabcee34aaba1510ebe924b9b16f9d32b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 108, + 524, + 135.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 135.0, + 524, + 162.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 162.0, + 524, + 189.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 201, + 505, + 225 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 201, + 504, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 504, + 214 + ], + "score": 1.0, + "content": "Figure 42: Runge-Kutta. 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See", + "type": "text" + }, + { + "bbox": [ + 343, + 143, + 360, + 154 + ], + "score": 0.82, + "content": "\\ S \\mathrm { I } . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 142, + 460, + 155 + ], + "score": 1.0, + "content": "for experimental details.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "Empirically, our findings hold for batch-normalized networks. The one catch is that when training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 169, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 505, + 185 + ], + "score": 1.0, + "content": "batch-normalized networks at very small step sizes, it is apparently inadequate to measure the sharp-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "ness directly at the iterates themselves, as we do elsewhere in the paper. Namely, observe that in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "Figure 71(e), when we run gradient descent at the red step size, the sharpness (measured directly at", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 283, + 217 + ], + "score": 1.0, + "content": "the iterates) plateaus a bit beneath the value", + "type": "text" + }, + { + "bbox": [ + 283, + 204, + 300, + 216 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 203, + 506, + 217 + ], + "score": 1.0, + "content": ". 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The explanation is that the sharp-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 321, + 237 + ], + "score": 1.0, + "content": "ness in between successive iterates does in fact cross", + "type": "text" + }, + { + "bbox": [ + 321, + 225, + 338, + 237 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 226, + 505, + 237 + ], + "score": 1.0, + "content": ". In Figure 71(b), we track the maximum", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 236, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 504, + 248 + ], + "score": 1.0, + "content": "sharpness on the path “in between” successive iterates. (To estimate the maximum sharpness be-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 504, + 260 + ], + "score": 1.0, + "content": "tween a pair of successive iterates, we compute the sharpness at a grid of eight points spaced evenly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "between them, and then take the maximum of these values.) Observe that this quantity does rise to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 123, + 281 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "and hover there. We do not know why measuring the sharpness between iterates is necessary", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "for batch-normalized networks, whereas for non-BN networks it suffices to measure the sharpness", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 231, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 231, + 303 + ], + "score": 1.0, + "content": "only at the iterates themselves.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 366, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 367, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 117, + 321 + ], + "score": 1.0, + "content": "In", + "type": "text" + }, + { + "bbox": [ + 117, + 308, + 138, + 319 + ], + "score": 0.86, + "content": "\\ S \\mathrm { K } . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 306, + 367, + 321 + ], + "score": 1.0, + "content": ", we reconcile these findings with Santurkar et al. (2018).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "image", + "bbox": [ + 149, + 337, + 466, + 401 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 149, + 337, + 466, + 401 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 337, + 466, + 401 + ], + "spans": [ + { + "bbox": [ + 149, + 337, + 466, + 401 + ], + "score": 0.955, + "type": "image", + "image_path": "404db4afb0e9177c92c54850c7f0dc79184ef00be99f37df5201e730d36bdd39.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 149, + 337, + 466, + 358.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 149, + 358.3333333333333, + 466, + 379.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 149, + 379.66666666666663, + 466, + 400.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 179, + 412, + 430, + 425 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 179, + 411, + 431, + 427 + ], + "spans": [ + { + "bbox": [ + 179, + 411, + 315, + 427 + ], + "score": 1.0, + "content": "Figure 70: We train a ELU CNN", + "type": "text" + }, + { + "bbox": [ + 315, + 413, + 344, + 424 + ], + "score": 0.72, + "content": "\\left( + \\mathbf { B } \\mathbf { N } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 411, + 431, + 427 + ], + "score": 1.0, + "content": "using gradient flow.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + } + ], + "index": 21.0 + }, + { + "type": "image", + "bbox": [ + 147, + 458, + 462, + 590 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 458, + 462, + 590 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 147, + 458, + 462, + 590 + ], + "spans": [ + { + "bbox": [ + 147, + 458, + 462, + 590 + ], + "score": 0.966, + "type": "image", + "image_path": "96afe39b6a9023f038910844c477d8fff9a2289d8886f7c8e921094afd86297d.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 147, + 458, + 462, + 502.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 147, + 502.0, + 462, + 546.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 147, + 546.0, + 462, + 590.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 601, + 505, + 712 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 250, + 615 + ], + "score": 1.0, + "content": "Figure 71: We train an ELU CNN", + "type": "text" + }, + { + "bbox": [ + 251, + 603, + 280, + 613 + ], + "score": 0.62, + "content": "\\mathbf { \\left( + \\delta B N \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "using gradient decent at a range of step sizes. (a) we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "plot the train loss, with a vertical dotted line marking the iteration where the sharpness on the path", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 156, + 637 + ], + "score": 1.0, + "content": "first crosses", + "type": "text" + }, + { + "bbox": [ + 156, + 624, + 173, + 636 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 624, + 505, + 637 + ], + "score": 1.0, + "content": ". The inset shows that the red curve is indeed behaving non-monotonically. (b) we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "track the sharpness “between iterates.” This means that instead of computing the sharpness right", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "at the iterates themselves (as we do elsewhere in the paper, and in pane (e) here), we compute the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "maximum sharpness on the line between between successive iterates. Observe that this quantity", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 668, + 504, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 137, + 681 + ], + "score": 1.0, + "content": "rises to", + "type": "text" + }, + { + "bbox": [ + 137, + 668, + 154, + 680 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 668, + 504, + 681 + ], + "score": 1.0, + "content": "(marked by the horizontal dashed line) and then hovers right at, or just above that value.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 268, + 692 + ], + "score": 1.0, + "content": "(c) we plot the same quantity by “time”", + "type": "text" + }, + { + "bbox": [ + 268, + 681, + 276, + 689 + ], + "score": 0.65, + "content": "\\mathit { \\Psi } = \\mathit { \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 679, + 311, + 692 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 312, + 680, + 321, + 690 + ], + "score": 0.77, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "step size) rather than iteration. (e) we plot the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 691, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 505, + 702 + ], + "score": 1.0, + "content": "sharpness at the iterates themselves. Note that for the red step size, this quantity plateaus at a value", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 700, + 369, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 166, + 715 + ], + "score": 1.0, + "content": "that is beneath", + "type": "text" + }, + { + "bbox": [ + 167, + 701, + 183, + 713 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 700, + 369, + 715 + ], + "score": 1.0, + "content": ". (f) distance from the gradient flow trajectory.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + } + ], + "index": 27.25 + } + ], + "page_idx": 54, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "55", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 331, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 333, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 333, + 96 + ], + "score": 1.0, + "content": "K BATCH NORMALIZATION EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 122 + ], + "score": 1.0, + "content": "In this appendix, we demonstrate that our findings hold for networks that are trained with batch", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "score": 1.0, + "content": "normalization (BN) (Ioffe & Szegedy, 2015). We experiment on a size-5,000 subset of CIFAR-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "10, and we consider convolutional networks with three different activation functions: ELU (Figure", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 460, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 342, + 155 + ], + "score": 1.0, + "content": "70-71, tanh (Figure 72-73), and ReLU (Figure 74-75). See", + "type": "text" + }, + { + "bbox": [ + 343, + 143, + 360, + 154 + ], + "score": 0.82, + "content": "\\ S \\mathrm { I } . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 142, + 460, + 155 + ], + "score": 1.0, + "content": "for experimental details.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 109, + 506, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 159, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "Empirically, our findings hold for batch-normalized networks. The one catch is that when training", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 169, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 104, + 169, + 505, + 185 + ], + "score": 1.0, + "content": "batch-normalized networks at very small step sizes, it is apparently inadequate to measure the sharp-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "ness directly at the iterates themselves, as we do elsewhere in the paper. Namely, observe that in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "Figure 71(e), when we run gradient descent at the red step size, the sharpness (measured directly at", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 283, + 217 + ], + "score": 1.0, + "content": "the iterates) plateaus a bit beneath the value", + "type": "text" + }, + { + "bbox": [ + 283, + 204, + 300, + 216 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 203, + 506, + 217 + ], + "score": 1.0, + "content": ". At first, this might sound puzzling: after all, if the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 504, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 194, + 228 + ], + "score": 1.0, + "content": "sharpness is less than", + "type": "text" + }, + { + "bbox": [ + 194, + 214, + 211, + 226 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 213, + 504, + 228 + ], + "score": 1.0, + "content": "then gradient descent should be stable. The explanation is that the sharp-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 321, + 237 + ], + "score": 1.0, + "content": "ness in between successive iterates does in fact cross", + "type": "text" + }, + { + "bbox": [ + 321, + 225, + 338, + 237 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 226, + 505, + 237 + ], + "score": 1.0, + "content": ". In Figure 71(b), we track the maximum", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 236, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 504, + 248 + ], + "score": 1.0, + "content": "sharpness on the path “in between” successive iterates. (To estimate the maximum sharpness be-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 504, + 260 + ], + "score": 1.0, + "content": "tween a pair of successive iterates, we compute the sharpness at a grid of eight points spaced evenly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "between them, and then take the maximum of these values.) Observe that this quantity does rise to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 123, + 281 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "and hover there. We do not know why measuring the sharpness between iterates is necessary", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "for batch-normalized networks, whereas for non-BN networks it suffices to measure the sharpness", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 231, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 231, + 303 + ], + "score": 1.0, + "content": "only at the iterates themselves.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 158, + 506, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 366, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 367, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 117, + 321 + ], + "score": 1.0, + "content": "In", + "type": "text" + }, + { + "bbox": [ + 117, + 308, + 138, + 319 + ], + "score": 0.86, + "content": "\\ S \\mathrm { K } . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 306, + 367, + 321 + ], + "score": 1.0, + "content": ", we reconcile these findings with Santurkar et al. 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(b) we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "track the sharpness “between iterates.” This means that instead of computing the sharpness right", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "at the iterates themselves (as we do elsewhere in the paper, and in pane (e) here), we compute the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "maximum sharpness on the line between between successive iterates. Observe that this quantity", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 668, + 504, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 137, + 681 + ], + "score": 1.0, + "content": "rises to", + "type": "text" + }, + { + "bbox": [ + 137, + 668, + 154, + 680 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 668, + 504, + 681 + ], + "score": 1.0, + "content": "(marked by the horizontal dashed line) and then hovers right at, or just above that value.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 268, + 692 + ], + "score": 1.0, + "content": "(c) we plot the same quantity by “time”", + "type": "text" + }, + { + "bbox": [ + 268, + 681, + 276, + 689 + ], + "score": 0.65, + "content": "\\mathit { \\Psi } = \\mathit { \\Psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 679, + 311, + 692 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 312, + 680, + 321, + 690 + ], + "score": 0.77, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "step size) rather than iteration. 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(2018)", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 441, + 700 + ], + "score": 1.0, + "content": "We have demonstrated that the sharpness hovers right at (or just above) the value", + "type": "text" + }, + { + "bbox": [ + 441, + 687, + 459, + 700 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "when both", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "BN and non-BN networks are trained using gradient descent at reasonable step sizes. 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(2018)", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 441, + 700 + ], + "score": 1.0, + "content": "We have demonstrated that the sharpness hovers right at (or just above) the value", + "type": "text" + }, + { + "bbox": [ + 441, + 687, + 459, + 700 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "when both", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "BN and non-BN networks are trained using gradient descent at reasonable step sizes. Therefore, at", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "least in the case of full-batch gradient descent, it cannot be said that batch normalization decreases", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 720, + 454, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 257, + 734 + ], + "score": 1.0, + "content": "the sharpness (i.e. improves the local", + "type": "text" + }, + { + "bbox": [ + 257, + 721, + 265, + 730 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 720, + 454, + 734 + ], + "score": 1.0, + "content": "-smoothness) along the optimization trajectory.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 687, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "Santurkar et al. (2018) argued that batch normalization improves the effective smoothness along", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "the optimization trajectory, where effective smoothness is defined as the Lipschitz constant of the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "gradient in the update direction (i.e. the negative gradient direction, for full-batch GD). That is, given", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 504, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 193, + 128 + ], + "score": 1.0, + "content": "an objective function", + "type": "text" + }, + { + "bbox": [ + 193, + 116, + 200, + 127 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 115, + 244, + 128 + ], + "score": 1.0, + "content": ", an iterate", + "type": "text" + }, + { + "bbox": [ + 244, + 116, + 250, + 125 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 115, + 313, + 128 + ], + "score": 1.0, + "content": ", and a distance", + "type": "text" + }, + { + "bbox": [ + 313, + 117, + 321, + 125 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 115, + 437, + 128 + ], + "score": 1.0, + "content": ", the effective smoothness of", + "type": "text" + }, + { + "bbox": [ + 437, + 116, + 444, + 127 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 115, + 497, + 128 + ], + "score": 1.0, + "content": "at parameter", + "type": "text" + }, + { + "bbox": [ + 498, + 116, + 504, + 125 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 322, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 158, + 138 + ], + "score": 1.0, + "content": "and distance", + "type": "text" + }, + { + "bbox": [ + 159, + 128, + 166, + 136 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 127, + 322, + 138 + ], + "score": 1.0, + "content": "is defined in Santurkar et al. 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When", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 325, + 243 + ], + "score": 1.0, + "content": "computing the effective smoothness, we use a distance", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 352, + 241 + ], + "score": 0.91, + "content": "\\alpha = \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 229, + 506, + 243 + ], + "score": 1.0, + "content": ". Observe that for both the BN and the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "score": 1.0, + "content": "non-BN network, the effective smoothness initially hovers around zero, but once gradient descent", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 392, + 265 + ], + "score": 1.0, + "content": "enters the Edge of Stability, the effective smoothness jumps to the value", + "type": "text" + }, + { + "bbox": [ + 393, + 252, + 409, + 263 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "and then remains there.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "Thus, at least for full-batch gradient descent on this particular architecture, batch normalization does", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 504, + 285 + ], + "score": 1.0, + "content": "not improve the effective smoothness along the optimization trajectory. (Despite this, note that for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "each step size, the BN network trains faster than the non-BN network, confirming that BN does", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 296, + 188, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 188, + 308 + ], + "score": 1.0, + "content": "accelerate training.)", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "image", + "bbox": [ + 112, + 321, + 505, + 485 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 321, + 505, + 485 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 321, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 112, + 321, + 505, + 485 + ], + "score": 0.971, + "type": "image", + "image_path": "eae8d2d243df8c1253b39e48a1ddcded8efeab7593d115d544011a9caeda745d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 112, + 321, + 505, + 375.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 112, + 375.6666666666667, + 505, + 430.33333333333337 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 112, + 430.33333333333337, + 505, + 485.00000000000006 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 497, + 505, + 553 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "Figure 76: On a 5,000-size subset of CIFAR-10, we train a ReLU CNN both with BN (top row) and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "without BN (bottom row) at the same grid of step sizes. We plot the sharpness/smoothness (center", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "column) as well as the effective smoothness (Santurkar et al., 2018) (right column). Observe that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 465, + 543 + ], + "score": 1.0, + "content": "for both networks, the effective smoothness hovers around zero initially, and jumps up to", + "type": "text" + }, + { + "bbox": [ + 466, + 530, + 483, + 542 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 529, + 505, + 543 + ], + "score": 1.0, + "content": "once", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 286, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 286, + 555 + ], + "score": 1.0, + "content": "gradient descent enters the Edge of Stability.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + } + ], + "index": 21.0 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Note that this finding is actually consistent with Figure 4(c) in Santurkar et al. (2018), which is meant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "to show that BN improves effective smoothness when training a VGG network using SGD. Their", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 307, + 607 + ], + "score": 1.0, + "content": "Figure 4(c) shows that during SGD with step size", + "type": "text" + }, + { + "bbox": [ + 308, + 594, + 341, + 605 + ], + "score": 0.91, + "content": "\\eta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 594, + 505, + 607 + ], + "score": 1.0, + "content": ", the effective smoothness hovers around", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 354, + 617 + ], + "score": 1.0, + "content": "the value 20 for both the BN and the non-BN network. Since", + "type": "text" + }, + { + "bbox": [ + 355, + 605, + 410, + 617 + ], + "score": 0.9, + "content": "2 0 = 2 / ( 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ", this is fully consistent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "with our findings (though they use SGD rather than full-batch GD). Figure 4(c) does show that the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "effective smoothness behaves more regularly for the BN network than for the non-BN network. But", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "we disagree with their interpretation of this figure as demonstrating that BN improves the effective", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 220, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 220, + 662 + ], + "score": 1.0, + "content": "smoothness during training.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "The other piece of evidence in Santurkar et al. (2018) in support of the argument that batch normal-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "ization improves the effective smoothness during training is their Figure 9(c). This figure shows that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "a deep linear network (DLN) trained without BN has a much larger (i.e. worse) effective smooth-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 464, + 711 + ], + "score": 1.0, + "content": "ness during training than a DLN trained with BN. 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We have verified that", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + } + ], + "page_idx": 56, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "Santurkar et al. 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When", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 325, + 243 + ], + "score": 1.0, + "content": "computing the effective smoothness, we use a distance", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 352, + 241 + ], + "score": 0.91, + "content": "\\alpha = \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 229, + 506, + 243 + ], + "score": 1.0, + "content": ". Observe that for both the BN and the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "score": 1.0, + "content": "non-BN network, the effective smoothness initially hovers around zero, but once gradient descent", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 392, + 265 + ], + "score": 1.0, + "content": "enters the Edge of Stability, the effective smoothness jumps to the value", + "type": "text" + }, + { + "bbox": [ + 393, + 252, + 409, + 263 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "and then remains there.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "Thus, at least for full-batch gradient descent on this particular architecture, batch normalization does", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 504, + 285 + ], + "score": 1.0, + "content": "not improve the effective smoothness along the optimization trajectory. (Despite this, note that for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "each step size, the BN network trains faster than the non-BN network, confirming that BN does", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 296, + 188, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 188, + 308 + ], + "score": 1.0, + "content": "accelerate training.)", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 207, + 506, + 308 + ] + }, + { + "type": "image", + "bbox": [ + 112, + 321, + 505, + 485 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 321, + 505, + 485 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 321, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 112, + 321, + 505, + 485 + ], + "score": 0.971, + "type": "image", + "image_path": "eae8d2d243df8c1253b39e48a1ddcded8efeab7593d115d544011a9caeda745d.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 112, + 321, + 505, + 375.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 112, + 375.6666666666667, + 505, + 430.33333333333337 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 112, + 430.33333333333337, + 505, + 485.00000000000006 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 497, + 505, + 553 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "Figure 76: On a 5,000-size subset of CIFAR-10, we train a ReLU CNN both with BN (top row) and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "without BN (bottom row) at the same grid of step sizes. We plot the sharpness/smoothness (center", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "column) as well as the effective smoothness (Santurkar et al., 2018) (right column). Observe that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 529, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 465, + 543 + ], + "score": 1.0, + "content": "for both networks, the effective smoothness hovers around zero initially, and jumps up to", + "type": "text" + }, + { + "bbox": [ + 466, + 530, + 483, + 542 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 529, + 505, + 543 + ], + "score": 1.0, + "content": "once", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 286, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 286, + 555 + ], + "score": 1.0, + "content": "gradient descent enters the Edge of Stability.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + } + ], + "index": 21.0 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Note that this finding is actually consistent with Figure 4(c) in Santurkar et al. (2018), which is meant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "to show that BN improves effective smoothness when training a VGG network using SGD. Their", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 307, + 607 + ], + "score": 1.0, + "content": "Figure 4(c) shows that during SGD with step size", + "type": "text" + }, + { + "bbox": [ + 308, + 594, + 341, + 605 + ], + "score": 0.91, + "content": "\\eta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 594, + 505, + 607 + ], + "score": 1.0, + "content": ", the effective smoothness hovers around", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 354, + 617 + ], + "score": 1.0, + "content": "the value 20 for both the BN and the non-BN network. Since", + "type": "text" + }, + { + "bbox": [ + 355, + 605, + 410, + 617 + ], + "score": 0.9, + "content": "2 0 = 2 / ( 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ", this is fully consistent", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "with our findings (though they use SGD rather than full-batch GD). Figure 4(c) does show that the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "effective smoothness behaves more regularly for the BN network than for the non-BN network. But", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "we disagree with their interpretation of this figure as demonstrating that BN improves the effective", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 220, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 220, + 662 + ], + "score": 1.0, + "content": "smoothness during training.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 572, + 505, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "The other piece of evidence in Santurkar et al. (2018) in support of the argument that batch normal-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "ization improves the effective smoothness during training is their Figure 9(c). This figure shows that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "a deep linear network (DLN) trained without BN has a much larger (i.e. worse) effective smooth-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 464, + 711 + ], + "score": 1.0, + "content": "ness during training than a DLN trained with BN. 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See", + "type": "text" + }, + { + "bbox": [ + 246, + 196, + 264, + 207 + ], + "score": 0.81, + "content": "\\ S 1 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 194, + 506, + 209 + ], + "score": 1.0, + "content": "for full experimental details. In Figure 81, we train using", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "gradient flow (only partially, not to completion). Observe that the sharpness continually rises. In", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "Figure 82, we train using gradient descent at a range of step sizes. 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However, for this Transformer, we do not observe that gradient descent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 387, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 387, + 264 + ], + "score": 1.0, + "content": "closely tracks the gradient flow trajectory at the beginning of training.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "image", + "bbox": [ + 176, + 276, + 435, + 352 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 176, + 276, + 435, + 352 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 176, + 276, + 435, + 352 + ], + "spans": [ + { + "bbox": [ + 176, + 276, + 435, + 352 + ], + "score": 0.964, + "type": "image", + "image_path": "12cad1f71acba6d23954d772c7367242e0c306c559e7b0d15a5a6edc67b72da6.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 176, + 276, + 435, + 301.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 176, + 301.3333333333333, + 435, + 326.66666666666663 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 176, + 326.66666666666663, + 435, + 351.99999999999994 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 188, + 365, + 422, + 377 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 188, + 364, + 423, + 379 + ], + "spans": [ + { + "bbox": [ + 188, + 364, + 423, + 379 + ], + "score": 1.0, + "content": "Figure 81: Training a Transformer using gradient flow.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + } + ], + "index": 15.0 + }, + { + "type": "image", + "bbox": [ + 176, + 402, + 435, + 561 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 176, + 402, + 435, + 561 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 176, + 402, + 435, + 561 + ], + "spans": [ + { + "bbox": [ + 176, + 402, + 435, + 561 + ], + "score": 0.971, + "type": "image", + "image_path": "bffbb195c94c724c5b9c7619674a20d6367ab03d9eb252d20206a69b8297a98d.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 176, + 402, + 435, + 455.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 176, + 455.0, + 435, + 508.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 176, + 508.0, + 435, + 561.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 574, + 506, + 718 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "Figure 82: Training a Transformer using gradient descent. We train a Transformer for 15,000", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "iterations on the WikiText-2 language modeling dataset. (For this problem, training to completion", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "would not be computationally practical.) 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Observe that for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 253, + 653 + ], + "score": 1.0, + "content": "each step size, the sharpness rises to", + "type": "text" + }, + { + "bbox": [ + 253, + 640, + 270, + 653 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "and then hovers right at, or just above, that value. Bottom", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 650, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 665 + ], + "score": 1.0, + "content": "left: for the initial phase of training, we plot the distance between the gradient descent trajectory and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "the gradient flow trajectory, with a vertical dotted line marking the gradient descent iteration where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 672, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 214, + 688 + ], + "score": 1.0, + "content": "the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 214, + 674, + 231, + 686 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 672, + 506, + 688 + ], + "score": 1.0, + "content": ". Observe that the distance begins to rise from the start of training,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 683, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 506, + 699 + ], + "score": 1.0, + "content": "indicating that for this architecture, gradient descent does not track the gradient flow trajectory", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 694, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 505, + 709 + ], + "score": 1.0, + "content": "initially. Bottom right: for the initial phase of training, we plot the evolution of the sharpness by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 706, + 498, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 706, + 134, + 719 + ], + "score": 1.0, + "content": "“time”", + "type": "text" + }, + { + "bbox": [ + 135, + 708, + 144, + 716 + ], + "score": 0.8, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 706, + 178, + 719 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 179, + 707, + 196, + 718 + ], + "score": 0.8, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 706, + 498, + 719 + ], + "score": 1.0, + "content": "rather than iteration. The black dots are the sharpness during gradient flow.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + } + ], + "index": 22.0 + } + ], + "page_idx": 59, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 762 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 762 + ], + "score": 1.0, + "content": "60", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 231, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 232, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 232, + 96 + ], + "score": 1.0, + "content": "L ADDITIONAL TASKS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "So far, we have verified our findings on image classification and language modeling. In this ap-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "pendix, we verify our findings on three additional tasks: training a Transformer on the WikiText-2", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "language modeling dataset (L.1), training a one-hidden-layer network on a one-dimensional toy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 357, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 357, + 152 + ], + "score": 1.0, + "content": "regression task (L.2), and training a deep linear network (L.3).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 105, + 505, + 152 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 163, + 270, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 271, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 271, + 176 + ], + "score": 1.0, + "content": "L.1 TRANSFORMER ON WIKITEXT-2", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 184, + 506, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 183, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 198 + ], + "score": 1.0, + "content": "We consider the problem of training a Transformer on the WikText-2 word-level language modeling", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 194, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 104, + 194, + 246, + 209 + ], + "score": 1.0, + "content": "dataset (Merity et al., 2016). See", + "type": "text" + }, + { + "bbox": [ + 246, + 196, + 264, + 207 + ], + "score": 0.81, + "content": "\\ S 1 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 194, + 506, + 209 + ], + "score": 1.0, + "content": "for full experimental details. In Figure 81, we train using", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "gradient flow (only partially, not to completion). Observe that the sharpness continually rises. In", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "Figure 82, we train using gradient descent at a range of step sizes. Consistent with our general", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 227, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 218, + 242 + ], + "score": 1.0, + "content": "findings, for each step size", + "type": "text" + }, + { + "bbox": [ + 218, + 231, + 225, + 240 + ], + "score": 0.65, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 227, + 385, + 242 + ], + "score": 1.0, + "content": ", we observe that the sharpness rises to", + "type": "text" + }, + { + "bbox": [ + 385, + 228, + 402, + 240 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 227, + 506, + 242 + ], + "score": 1.0, + "content": "and then hovers right at,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 252 + ], + "score": 1.0, + "content": "or just above, that value. However, for this Transformer, we do not observe that gradient descent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 387, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 387, + 264 + ], + "score": 1.0, + "content": "closely tracks the gradient flow trajectory at the beginning of training.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 183, + 506, + 264 + ] + }, + { + "type": "image", + "bbox": [ + 176, + 276, + 435, + 352 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 176, + 276, + 435, + 352 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 176, + 276, + 435, + 352 + ], + "spans": [ + { + "bbox": [ + 176, + 276, + 435, + 352 + ], + "score": 0.964, + "type": "image", + "image_path": "12cad1f71acba6d23954d772c7367242e0c306c559e7b0d15a5a6edc67b72da6.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 176, + 276, + 435, + 301.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 176, + 301.3333333333333, + 435, + 326.66666666666663 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 176, + 326.66666666666663, + 435, + 351.99999999999994 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 188, + 365, + 422, + 377 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 188, + 364, + 423, + 379 + ], + "spans": [ + { + "bbox": [ + 188, + 364, + 423, + 379 + ], + "score": 1.0, + "content": "Figure 81: Training a Transformer using gradient flow.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + } + ], + "index": 15.0 + }, + { + "type": "image", + "bbox": [ + 176, + 402, + 435, + 561 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 176, + 402, + 435, + 561 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 176, + 402, + 435, + 561 + ], + "spans": [ + { + "bbox": [ + 176, + 402, + 435, + 561 + ], + "score": 0.971, + "type": "image", + "image_path": "bffbb195c94c724c5b9c7619674a20d6367ab03d9eb252d20206a69b8297a98d.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 176, + 402, + 435, + 455.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 176, + 455.0, + 435, + 508.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 176, + 508.0, + 435, + 561.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 574, + 506, + 718 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "Figure 82: Training a Transformer using gradient descent. We train a Transformer for 15,000", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "iterations on the WikiText-2 language modeling dataset. (For this problem, training to completion", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "would not be computationally practical.) Top left: we plot the train loss curves, with a vertical", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 380, + 621 + ], + "score": 1.0, + "content": "dotted line marking the iteration where the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 380, + 608, + 397, + 620 + ], + "score": 0.87, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 608, + 506, + 621 + ], + "score": 1.0, + "content": ". The train loss decreases", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "monotonically before this line, but behaves non-monotonically afterwards. Top right: we plot the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 630, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 414, + 642 + ], + "score": 1.0, + "content": "evolution of the sharpness, with a horizontal dashed line marking the value", + "type": "text" + }, + { + "bbox": [ + 414, + 630, + 431, + 642 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 630, + 506, + 642 + ], + "score": 1.0, + "content": ". Observe that for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 253, + 653 + ], + "score": 1.0, + "content": "each step size, the sharpness rises to", + "type": "text" + }, + { + "bbox": [ + 253, + 640, + 270, + 653 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "and then hovers right at, or just above, that value. Bottom", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 650, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 665 + ], + "score": 1.0, + "content": "left: for the initial phase of training, we plot the distance between the gradient descent trajectory and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "the gradient flow trajectory, with a vertical dotted line marking the gradient descent iteration where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 672, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 214, + 688 + ], + "score": 1.0, + "content": "the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 214, + 674, + 231, + 686 + ], + "score": 0.9, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 672, + 506, + 688 + ], + "score": 1.0, + "content": ". Observe that the distance begins to rise from the start of training,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 683, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 506, + 699 + ], + "score": 1.0, + "content": "indicating that for this architecture, gradient descent does not track the gradient flow trajectory", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 694, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 505, + 709 + ], + "score": 1.0, + "content": "initially. Bottom right: for the initial phase of training, we plot the evolution of the sharpness by", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 706, + 498, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 706, + 134, + 719 + ], + "score": 1.0, + "content": "“time”", + "type": "text" + }, + { + "bbox": [ + 135, + 708, + 144, + 716 + ], + "score": 0.8, + "content": "=", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 706, + 178, + 719 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 179, + 707, + 196, + 718 + ], + "score": 0.8, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 706, + 498, + 719 + ], + "score": 1.0, + "content": "rather than iteration. The black dots are the sharpness during gradient flow.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + } + ], + "index": 22.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "Task Our toy regression problem is to approximate a Chebyshev polynomial using a neural net-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 441, + 126 + ], + "score": 1.0, + "content": "work. To generate a toy dataset, we take 20 points spaced uniformly on the interval", + "type": "text" + }, + { + "bbox": [ + 442, + 114, + 470, + 126 + ], + "score": 0.91, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 114, + 505, + 126 + ], + "score": 1.0, + "content": ", and we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 393, + 138 + ], + "score": 1.0, + "content": "label them noiselessly using the Chebyshev polynomial of some degree", + "type": "text" + }, + { + "bbox": [ + 393, + 126, + 400, + 135 + ], + "score": 0.65, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 124, + 505, + 138 + ], + "score": 1.0, + "content": ". 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We train using the MSE loss until the loss reaches 0.05.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 350, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 504, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 504, + 362 + ], + "score": 1.0, + "content": "Results In Figure 84, we fit the Chebyshev degree 3, 4, and 5 datasets using gradient flow. 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The only", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "difference is that in Figure 88, the train loss does not really behave non-monotonically — for each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 614, + 504, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 143, + 625 + ], + "score": 1.0, + "content": "step size", + "type": "text" + }, + { + "bbox": [ + 143, + 614, + 150, + 624 + ], + "score": 0.76, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 614, + 504, + 625 + ], + "score": 1.0, + "content": ", there is a brief blip at some point, but otherwise, the train loss decreases monotonically.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 557, + 506, + 625 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 159, + 151, + 453, + 237 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 159, + 151, + 453, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 159, + 151, + 453, + 237 + ], + "spans": [ + { + "bbox": [ + 159, + 151, + 453, + 237 + ], + "score": 0.961, + "type": "image", + "image_path": "e326bc0690fd1050067b147d39b711c643bccb982cd01f9353f4e60775546f3a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 159, + 151, + 453, + 179.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 159, + 179.66666666666666, + 453, + 208.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 159, + 208.33333333333331, + 453, + 236.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 250, + 504, + 285 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "Figure 87: Training a deep linear network using gradient flow. 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We train a deep linear network", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 633, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 649 + ], + "score": 1.0, + "content": "using gradient descent at a range of step sizes (see legend). (Note that in the top left pane, the blue,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 645, + 367, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 367, + 658 + ], + "score": 1.0, + "content": "orange and green dotted lines are directly on top of one another.)", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + } + ], + "index": 8.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 433, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 434, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 434, + 95 + ], + "score": 1.0, + "content": "M EXPERIMENTS: STANDARD ARCHITECTURES ON CIFAR-10", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 503, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 122 + ], + "score": 1.0, + "content": "In this appendix, we demonstrate that our findings hold for three standard architectures on the stan-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "dard dataset CIFAR-10. The three architectures are: a VGG with batch normalization (Figures", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "89-90), a VGG without batch normalization (Figures 91-92), and a ResNet with batch normalization", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 143, + 328, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 194, + 155 + ], + "score": 1.0, + "content": "(Figures 93 -94). See", + "type": "text" + }, + { + "bbox": [ + 194, + 143, + 211, + 154 + ], + "score": 0.81, + "content": "\\ S 1 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 143, + 328, + 155 + ], + "score": 1.0, + "content": "for full experimental details.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "For each of these three architectures, we confirm our main points: (1) so long as the sharpness is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 143, + 183 + ], + "score": 1.0, + "content": "less than", + "type": "text" + }, + { + "bbox": [ + 144, + 171, + 160, + 183 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 171, + 417, + 183 + ], + "score": 1.0, + "content": ", the sharpness tends to increase; and (2) if the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 417, + 171, + 434, + 183 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 171, + 505, + 183 + ], + "score": 1.0, + "content": ", gradient descent", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "enters a regime (the Edge of Stability) in which (a) the training loss behaves non-monotonically,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "score": 1.0, + "content": "yet consistently decreases over long timescales, and (b) the sharpness hovers right at, or just above,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 145, + 216 + ], + "score": 1.0, + "content": "the value", + "type": "text" + }, + { + "bbox": [ + 145, + 204, + 162, + 216 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 204, + 505, + 216 + ], + "score": 1.0, + "content": ". Moreover, we observe that even though these architectures use the ReLU activation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "function (which is not continuously differentiable), gradient descent closely tracks the Runge-Kutta", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 433, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 412, + 239 + ], + "score": 1.0, + "content": "trajectory until reaching the point on that trajectory where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 413, + 226, + 429, + 238 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 226, + 433, + 239 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 504, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "Furthermore, we observe that for these three standard architectures, the following additional points", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "hold: (1) progressive sharpening occurs to a dramatic degree, and (2) stable step sizes are so small", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 244, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 244, + 276 + ], + "score": 1.0, + "content": "as to be completely unreasonable.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 294, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "score": 1.0, + "content": "Progressive sharpening occurs to a dramatic degree To assess the degree of progressive sharp-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "score": 1.0, + "content": "ening, we train these networks using Runge-Kutta / gradient flow, which can be viewed as gradient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "descent with an infinitesimally small step size. (In practice, the Runge-Kutta algorithm does have a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "step size parameter, and throughout training, we periodically adjust this step size in order to ensure", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "score": 1.0, + "content": "that the algorithm remains stable.) Intuitively, training with gradient flow tell us how far the sharp-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "ness would rise “if it didn’t have to worry about” instability caused by nonzero step sizes. In Figure", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 291, + 373 + ], + "score": 1.0, + "content": "89, we train the VGG with BN to completion", + "type": "text" + }, + { + "bbox": [ + 291, + 361, + 311, + 371 + ], + "score": 0.84, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "training accuracy) using Runge-Kutta / gradient", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "flow, and find that the sharpness rises from its initial value of 6.38 to a peak value of 2227.6. For the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 382, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 504, + 396 + ], + "score": 1.0, + "content": "other two architectures, progressive sharpening occurs to such a degree that it is not computation-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "ally feasible for to train using Runge-Kutta / gradient flow all the way to completion. (The reason", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 504, + 416 + ], + "score": 1.0, + "content": "is that in regions where the sharpness is high, the Runge-Kutta step size must be made small, so", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "Runge-Kutta requires very many iterations.) Therefore, we instead train these two networks only", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "partially. In Figure 91, for the VGG without BN, we find that the sharpness rises from its initial", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 256, + 450 + ], + "score": 1.0, + "content": "value of 0.64 to the value 2461.78 at", + "type": "text" + }, + { + "bbox": [ + 257, + 437, + 284, + 448 + ], + "score": 0.87, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "accuracy, when we stop training. In Figure 93, for the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 448, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 476, + 461 + ], + "score": 1.0, + "content": "ResNet, we find that the sharpness rises from its initial value of 1.07 to the value 760.6 at", + "type": "text" + }, + { + "bbox": [ + 477, + 448, + 504, + 459 + ], + "score": 0.86, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "accuracy, when we stop training. Thus, even though we observed in Appendix D that progressive", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 471, + 504, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 504, + 482 + ], + "score": 1.0, + "content": "sharpening attenuates as the width of fully-connected networks is made larger, it appears that either:", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "(1) this does not happen for modern families of architectures such as ResNet and VGG, or (2) this", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "does happen for modern families of architectures, but practical network widths lie on the narrow end", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 503, + 157, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 157, + 515 + ], + "score": 1.0, + "content": "of the scale.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 416, + 547 + ], + "score": 1.0, + "content": "Stable step sizes are so small as to be unreasonable Recall from 3.3 that if", + "type": "text" + }, + { + "bbox": [ + 416, + 534, + 438, + 545 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "is the maximum", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 469, + 558 + ], + "score": 1.0, + "content": "sharpness along the gradient flow trajectory, then any stable step size must be less than", + "type": "text" + }, + { + "bbox": [ + 469, + 544, + 501, + 556 + ], + "score": 0.9, + "content": "2 / \\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 543, + 505, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Therefore, for these three architectures, because progressive sharpening occurs to a dramatic degree", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 125, + 579 + ], + "score": 1.0, + "content": "(i.e.", + "type": "text" + }, + { + "bbox": [ + 125, + 566, + 148, + 578 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "is extraordinarily large), any stable step size must be extraordinarily small, which means", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "that training will require many iterations. Yet, at the same time, we find that these three networks", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "can be successfully trained in far fewer iterations by using larger step step size. This means that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 600, + 448, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 448, + 612 + ], + "score": 1.0, + "content": "training at a stable step size is extremely suboptimal. We now elaborate on this point:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "VGG with BN. For this network, gradient flow terminates at time 15.66, and the maximum", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "sharpness along the gradient flow trajectory is 2227.6. Therefore, the largest stable step size is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 202, + 649 + ], + "score": 0.87, + "content": "2 / 2 2 2 7 . 5 9 = 0 . 0 0 0 8 9 7", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 637, + 426, + 650 + ], + "score": 1.0, + "content": ", and training to completion at this step size would take", + "type": "text" + }, + { + "bbox": [ + 427, + 638, + 505, + 650 + ], + "score": 0.81, + "content": "1 4 . 9 1 / 0 . 0 0 0 8 9 7 =", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "16622 iterations. Meanwhile, we empirically observe that the network can also be trained to com-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 259, + 673 + ], + "score": 1.0, + "content": "pletion at the much larger step size of", + "type": "text" + }, + { + "bbox": [ + 259, + 660, + 297, + 671 + ], + "score": 0.91, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "in just 329 iterations. Therefore, using a stable step", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 348, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 267, + 683 + ], + "score": 1.0, + "content": "size is suboptimal by a factor of at least", + "type": "text" + }, + { + "bbox": [ + 267, + 671, + 344, + 683 + ], + "score": 0.82, + "content": "1 6 6 2 2 / 3 2 9 = 5 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 671, + 348, + 683 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "For the other two architectures, since we are unable to train to completion using gradient flow, we", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "are unable to obtain a tight lower bound for the number of iterations required to run gradient descent", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "to completion at a stable step size. Therefore, by extension, we are unable to compute a tight lower", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "bound for the suboptimality factor of stable step sizes. As a substitute, we will instead compute", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49.5 + } + ], + "page_idx": 64, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "65", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 433, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 434, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 434, + 95 + ], + "score": 1.0, + "content": "M EXPERIMENTS: STANDARD ARCHITECTURES ON CIFAR-10", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 503, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 505, + 122 + ], + "score": 1.0, + "content": "In this appendix, we demonstrate that our findings hold for three standard architectures on the stan-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "score": 1.0, + "content": "dard dataset CIFAR-10. The three architectures are: a VGG with batch normalization (Figures", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "89-90), a VGG without batch normalization (Figures 91-92), and a ResNet with batch normalization", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 143, + 328, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 194, + 155 + ], + "score": 1.0, + "content": "(Figures 93 -94). See", + "type": "text" + }, + { + "bbox": [ + 194, + 143, + 211, + 154 + ], + "score": 0.81, + "content": "\\ S 1 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 143, + 328, + 155 + ], + "score": 1.0, + "content": "for full experimental details.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 110, + 505, + 155 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "For each of these three architectures, we confirm our main points: (1) so long as the sharpness is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 143, + 183 + ], + "score": 1.0, + "content": "less than", + "type": "text" + }, + { + "bbox": [ + 144, + 171, + 160, + 183 + ], + "score": 0.88, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 171, + 417, + 183 + ], + "score": 1.0, + "content": ", the sharpness tends to increase; and (2) if the sharpness reaches", + "type": "text" + }, + { + "bbox": [ + 417, + 171, + 434, + 183 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 171, + 505, + 183 + ], + "score": 1.0, + "content": ", gradient descent", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "enters a regime (the Edge of Stability) in which (a) the training loss behaves non-monotonically,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 506, + 206 + ], + "score": 1.0, + "content": "yet consistently decreases over long timescales, and (b) the sharpness hovers right at, or just above,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 145, + 216 + ], + "score": 1.0, + "content": "the value", + "type": "text" + }, + { + "bbox": [ + 145, + 204, + 162, + 216 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 204, + 505, + 216 + ], + "score": 1.0, + "content": ". Moreover, we observe that even though these architectures use the ReLU activation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "function (which is not continuously differentiable), gradient descent closely tracks the Runge-Kutta", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 433, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 412, + 239 + ], + "score": 1.0, + "content": "trajectory until reaching the point on that trajectory where the sharpness hits", + "type": "text" + }, + { + "bbox": [ + 413, + 226, + 429, + 238 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 226, + 433, + 239 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 159, + 506, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 504, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "Furthermore, we observe that for these three standard architectures, the following additional points", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "hold: (1) progressive sharpening occurs to a dramatic degree, and (2) stable step sizes are so small", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 244, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 244, + 276 + ], + "score": 1.0, + "content": "as to be completely unreasonable.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 241, + 506, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 294, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "score": 1.0, + "content": "Progressive sharpening occurs to a dramatic degree To assess the degree of progressive sharp-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 504, + 318 + ], + "score": 1.0, + "content": "ening, we train these networks using Runge-Kutta / gradient flow, which can be viewed as gradient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "descent with an infinitesimally small step size. (In practice, the Runge-Kutta algorithm does have a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "step size parameter, and throughout training, we periodically adjust this step size in order to ensure", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 352 + ], + "score": 1.0, + "content": "that the algorithm remains stable.) Intuitively, training with gradient flow tell us how far the sharp-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "ness would rise “if it didn’t have to worry about” instability caused by nonzero step sizes. In Figure", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 291, + 373 + ], + "score": 1.0, + "content": "89, we train the VGG with BN to completion", + "type": "text" + }, + { + "bbox": [ + 291, + 361, + 311, + 371 + ], + "score": 0.84, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "training accuracy) using Runge-Kutta / gradient", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "flow, and find that the sharpness rises from its initial value of 6.38 to a peak value of 2227.6. For the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 382, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 504, + 396 + ], + "score": 1.0, + "content": "other two architectures, progressive sharpening occurs to such a degree that it is not computation-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "ally feasible for to train using Runge-Kutta / gradient flow all the way to completion. (The reason", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 504, + 416 + ], + "score": 1.0, + "content": "is that in regions where the sharpness is high, the Runge-Kutta step size must be made small, so", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "Runge-Kutta requires very many iterations.) Therefore, we instead train these two networks only", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "partially. In Figure 91, for the VGG without BN, we find that the sharpness rises from its initial", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 256, + 450 + ], + "score": 1.0, + "content": "value of 0.64 to the value 2461.78 at", + "type": "text" + }, + { + "bbox": [ + 257, + 437, + 284, + 448 + ], + "score": 0.87, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "accuracy, when we stop training. In Figure 93, for the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 448, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 476, + 461 + ], + "score": 1.0, + "content": "ResNet, we find that the sharpness rises from its initial value of 1.07 to the value 760.6 at", + "type": "text" + }, + { + "bbox": [ + 477, + 448, + 504, + 459 + ], + "score": 0.86, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "accuracy, when we stop training. Thus, even though we observed in Appendix D that progressive", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 471, + 504, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 504, + 482 + ], + "score": 1.0, + "content": "sharpening attenuates as the width of fully-connected networks is made larger, it appears that either:", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "(1) this does not happen for modern families of architectures such as ResNet and VGG, or (2) this", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "does happen for modern families of architectures, but practical network widths lie on the narrow end", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 503, + 157, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 157, + 515 + ], + "score": 1.0, + "content": "of the scale.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 294, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 416, + 547 + ], + "score": 1.0, + "content": "Stable step sizes are so small as to be unreasonable Recall from 3.3 that if", + "type": "text" + }, + { + "bbox": [ + 416, + 534, + 438, + 545 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "is the maximum", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 469, + 558 + ], + "score": 1.0, + "content": "sharpness along the gradient flow trajectory, then any stable step size must be less than", + "type": "text" + }, + { + "bbox": [ + 469, + 544, + 501, + 556 + ], + "score": 0.9, + "content": "2 / \\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 543, + 505, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Therefore, for these three architectures, because progressive sharpening occurs to a dramatic degree", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 125, + 579 + ], + "score": 1.0, + "content": "(i.e.", + "type": "text" + }, + { + "bbox": [ + 125, + 566, + 148, + 578 + ], + "score": 0.91, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "is extraordinarily large), any stable step size must be extraordinarily small, which means", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "that training will require many iterations. Yet, at the same time, we find that these three networks", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "can be successfully trained in far fewer iterations by using larger step step size. This means that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 600, + 448, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 448, + 612 + ], + "score": 1.0, + "content": "training at a stable step size is extremely suboptimal. We now elaborate on this point:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 532, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "VGG with BN. For this network, gradient flow terminates at time 15.66, and the maximum", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "sharpness along the gradient flow trajectory is 2227.6. Therefore, the largest stable step size is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 202, + 649 + ], + "score": 0.87, + "content": "2 / 2 2 2 7 . 5 9 = 0 . 0 0 0 8 9 7", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 637, + 426, + 650 + ], + "score": 1.0, + "content": ", and training to completion at this step size would take", + "type": "text" + }, + { + "bbox": [ + 427, + 638, + 505, + 650 + ], + "score": 0.81, + "content": "1 4 . 9 1 / 0 . 0 0 0 8 9 7 =", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "16622 iterations. Meanwhile, we empirically observe that the network can also be trained to com-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 259, + 673 + ], + "score": 1.0, + "content": "pletion at the much larger step size of", + "type": "text" + }, + { + "bbox": [ + 259, + 660, + 297, + 671 + ], + "score": 0.91, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "in just 329 iterations. Therefore, using a stable step", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 671, + 348, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 267, + 683 + ], + "score": 1.0, + "content": "size is suboptimal by a factor of at least", + "type": "text" + }, + { + "bbox": [ + 267, + 671, + 344, + 683 + ], + "score": 0.82, + "content": "1 6 6 2 2 / 3 2 9 = 5 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 671, + 348, + 683 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 615, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "For the other two architectures, since we are unable to train to completion using gradient flow, we", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "are unable to obtain a tight lower bound for the number of iterations required to run gradient descent", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "to completion at a stable step size. Therefore, by extension, we are unable to compute a tight lower", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "bound for the suboptimality factor of stable step sizes. As a substitute, we will instead compute", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "both: (1) a tight lower bound on the suboptimality of training partially at a stable step size, and (2)", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 477, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 477, + 107 + ], + "score": 1.0, + "content": "a very loose lower bound on the suboptimality of training to completion at a stable step size.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "both: (1) a tight lower bound on the suboptimality of training partially at a stable step size, and (2)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 477, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 477, + 107 + ], + "score": 1.0, + "content": "a very loose lower bound on the suboptimality of training to completion at a stable step size.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 342, + 123 + ], + "score": 1.0, + "content": "VGG without BN. For this network, gradient flow reaches", + "type": "text" + }, + { + "bbox": [ + 343, + 111, + 370, + 121 + ], + "score": 0.88, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "accuracy at time 8, and the maxi-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "mum sharpness up through this point on the gradient flow trajectory is 2461.8. Therefore, the largest", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 180, + 144 + ], + "score": 1.0, + "content": "stable step size is", + "type": "text" + }, + { + "bbox": [ + 180, + 132, + 266, + 143 + ], + "score": 0.89, + "content": "2 / 2 4 6 1 . 8 = 0 . 0 0 0 8 1", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 132, + 331, + 144 + ], + "score": 1.0, + "content": ", and training to", + "type": "text" + }, + { + "bbox": [ + 332, + 132, + 359, + 143 + ], + "score": 0.87, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "accuracy at this step size would re-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 129, + 156 + ], + "score": 1.0, + "content": "quire", + "type": "text" + }, + { + "bbox": [ + 130, + 144, + 196, + 155 + ], + "score": 0.85, + "content": "8 / 0 . 0 0 0 8 1 = 9 , \\quad", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "876 iterations. Meanwhile, we empirically observe that the network can also", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 160, + 167 + ], + "score": 1.0, + "content": "be trained to", + "type": "text" + }, + { + "bbox": [ + 160, + 155, + 188, + 165 + ], + "score": 0.84, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 154, + 328, + 167 + ], + "score": 1.0, + "content": "accuracy at the larger step size of", + "type": "text" + }, + { + "bbox": [ + 329, + 155, + 369, + 166 + ], + "score": 0.9, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "in just 355 iterations. Therefore,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 227, + 178 + ], + "score": 1.0, + "content": "when training this network to", + "type": "text" + }, + { + "bbox": [ + 227, + 165, + 247, + 176 + ], + "score": 0.86, + "content": "37 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "accuracy, stable step sizes are suboptimal by a factor of at least", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 180, + 188 + ], + "score": 0.85, + "content": "9 8 7 6 / 3 5 5 = \\mathrm { \\bar { 2 7 } } . 8 .", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 175, + 429, + 189 + ], + "score": 1.0, + "content": ". This is a tight lower bound on the suboptimality of training to", + "type": "text" + }, + { + "bbox": [ + 430, + 176, + 457, + 187 + ], + "score": 0.86, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "accuracy at", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 178, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 178, + 200 + ], + "score": 1.0, + "content": "a stable step size.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "To obtain a loose lower bound on the suboptimality of training the VGG without BN to completion", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "at a stable step size, we note that (a) since the maximum sharpness up through time 8 is 2461.8,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "score": 1.0, + "content": "the maximum sharpness along the entire gradient flow trajectory must be at least 2461.8; and (b)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 504, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 305, + 250 + ], + "score": 1.0, + "content": "since by time 8 gradient flow has only attained", + "type": "text" + }, + { + "bbox": [ + 305, + 237, + 333, + 248 + ], + "score": 0.87, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 237, + 484, + 250 + ], + "score": 1.0, + "content": "training accuracy, the time to reach", + "type": "text" + }, + { + "bbox": [ + 484, + 237, + 504, + 248 + ], + "score": 0.85, + "content": "9 9 \\%", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "accuracy (i.e. completion) must be at least 8. (Note that both of these lower bounds are extremely", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "loose.) Therefore, training this network to completion at a stable step size would require at least", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 269, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 206, + 282 + ], + "score": 0.88, + "content": "8 / ( 2 / 2 4 6 1 . 8 ) \\ = \\ 9 , 8 7 6", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 269, + 476, + 282 + ], + "score": 1.0, + "content": "iterations (which is the same number of iterations as training to", + "type": "text" + }, + { + "bbox": [ + 477, + 270, + 504, + 281 + ], + "score": 0.85, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "accuracy at a stable step size). Meanwhile, we find that the network can be trained to completion at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 194, + 304 + ], + "score": 1.0, + "content": "the larger step size of", + "type": "text" + }, + { + "bbox": [ + 195, + 292, + 233, + 303 + ], + "score": 0.91, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "in just 1782 iterations. Therefore, training to completion at a stable", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 371, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 285, + 315 + ], + "score": 1.0, + "content": "step size is suboptimal by a factor of at least", + "type": "text" + }, + { + "bbox": [ + 285, + 303, + 367, + 315 + ], + "score": 0.88, + "content": "9 , 8 7 6 / 1 7 8 2 = 5 . 5 4", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 303, + 371, + 315 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 311, + 332 + ], + "score": 1.0, + "content": "ResNet. For this network, gradient flow reaches", + "type": "text" + }, + { + "bbox": [ + 311, + 320, + 339, + 330 + ], + "score": 0.87, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "accuracy at time 70, and the maximum", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "sharpness up through this point on the gradient flow trajectory is 760.6. Therefore, the largest sta-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 169, + 354 + ], + "score": 1.0, + "content": "ble step size is", + "type": "text" + }, + { + "bbox": [ + 169, + 342, + 247, + 353 + ], + "score": 0.89, + "content": "2 / 7 6 0 . 6 = 0 . 0 0 2 6", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 342, + 313, + 354 + ], + "score": 1.0, + "content": ", and training to", + "type": "text" + }, + { + "bbox": [ + 313, + 342, + 341, + 352 + ], + "score": 0.88, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "accuracy at this step size would require", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 178, + 364 + ], + "score": 0.81, + "content": "7 0 / 0 . 0 0 2 6 = 2 6 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "923 iterations. Meanwhile, we empirically observe that the network can also be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 147, + 376 + ], + "score": 1.0, + "content": "trained to", + "type": "text" + }, + { + "bbox": [ + 147, + 364, + 175, + 374 + ], + "score": 0.85, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 363, + 315, + 376 + ], + "score": 1.0, + "content": "accuracy at the larger step size of", + "type": "text" + }, + { + "bbox": [ + 315, + 364, + 349, + 375 + ], + "score": 0.92, + "content": "\\eta = 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "in just 99 iterations. Therefore, when", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 207, + 388 + ], + "score": 1.0, + "content": "training this network to", + "type": "text" + }, + { + "bbox": [ + 207, + 375, + 235, + 385 + ], + "score": 0.88, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "accuracy, stable step sizes are suboptimal by a factor of at least", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 385, + 189, + 397 + ], + "score": 0.9, + "content": "2 6 , 9 2 { \\bar { 3 } } / 9 9 = 2 7 1 . 9 .", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 384, + 439, + 399 + ], + "score": 1.0, + "content": ". This is a tight lower bound on the suboptimality of training to", + "type": "text" + }, + { + "bbox": [ + 439, + 385, + 466, + 396 + ], + "score": 0.88, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "at a stable step size. For the loose lower bound on the suboptimality of training to completion at a sta-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "ble step size, note that (by similar reasoning as the VGG-without-BN above), training to completion", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "at a stable step size must require at least 26, 923 iterations. Meanwhile, we find that the network can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 304, + 443 + ], + "score": 1.0, + "content": "be trained to completion at the larger step size of", + "type": "text" + }, + { + "bbox": [ + 305, + 430, + 338, + 441 + ], + "score": 0.9, + "content": "\\eta = 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "in just 807 iterations. Therefore, training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 440, + 470, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 385, + 452 + ], + "score": 1.0, + "content": "to completion at a stable step size is suboptimal by a factor of at least", + "type": "text" + }, + { + "bbox": [ + 385, + 440, + 467, + 452 + ], + "score": 0.82, + "content": "2 6 , 9 2 3 / 8 0 7 = 3 3 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 440, + 470, + 452 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5 + } + ], + "page_idx": 65, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 104, + 82, + 505, + 107 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 342, + 123 + ], + "score": 1.0, + "content": "VGG without BN. For this network, gradient flow reaches", + "type": "text" + }, + { + "bbox": [ + 343, + 111, + 370, + 121 + ], + "score": 0.88, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "accuracy at time 8, and the maxi-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "mum sharpness up through this point on the gradient flow trajectory is 2461.8. Therefore, the largest", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 180, + 144 + ], + "score": 1.0, + "content": "stable step size is", + "type": "text" + }, + { + "bbox": [ + 180, + 132, + 266, + 143 + ], + "score": 0.89, + "content": "2 / 2 4 6 1 . 8 = 0 . 0 0 0 8 1", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 132, + 331, + 144 + ], + "score": 1.0, + "content": ", and training to", + "type": "text" + }, + { + "bbox": [ + 332, + 132, + 359, + 143 + ], + "score": 0.87, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "accuracy at this step size would re-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 129, + 156 + ], + "score": 1.0, + "content": "quire", + "type": "text" + }, + { + "bbox": [ + 130, + 144, + 196, + 155 + ], + "score": 0.85, + "content": "8 / 0 . 0 0 0 8 1 = 9 , \\quad", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "876 iterations. Meanwhile, we empirically observe that the network can also", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 160, + 167 + ], + "score": 1.0, + "content": "be trained to", + "type": "text" + }, + { + "bbox": [ + 160, + 155, + 188, + 165 + ], + "score": 0.84, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 154, + 328, + 167 + ], + "score": 1.0, + "content": "accuracy at the larger step size of", + "type": "text" + }, + { + "bbox": [ + 329, + 155, + 369, + 166 + ], + "score": 0.9, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "in just 355 iterations. Therefore,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 227, + 178 + ], + "score": 1.0, + "content": "when training this network to", + "type": "text" + }, + { + "bbox": [ + 227, + 165, + 247, + 176 + ], + "score": 0.86, + "content": "37 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "accuracy, stable step sizes are suboptimal by a factor of at least", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 180, + 188 + ], + "score": 0.85, + "content": "9 8 7 6 / 3 5 5 = \\mathrm { \\bar { 2 7 } } . 8 .", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 175, + 429, + 189 + ], + "score": 1.0, + "content": ". This is a tight lower bound on the suboptimality of training to", + "type": "text" + }, + { + "bbox": [ + 430, + 176, + 457, + 187 + ], + "score": 0.86, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "accuracy at", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 188, + 178, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 178, + 200 + ], + "score": 1.0, + "content": "a stable step size.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 110, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 204, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "To obtain a loose lower bound on the suboptimality of training the VGG without BN to completion", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 228 + ], + "score": 1.0, + "content": "at a stable step size, we note that (a) since the maximum sharpness up through time 8 is 2461.8,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "score": 1.0, + "content": "the maximum sharpness along the entire gradient flow trajectory must be at least 2461.8; and (b)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 504, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 305, + 250 + ], + "score": 1.0, + "content": "since by time 8 gradient flow has only attained", + "type": "text" + }, + { + "bbox": [ + 305, + 237, + 333, + 248 + ], + "score": 0.87, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 237, + 484, + 250 + ], + "score": 1.0, + "content": "training accuracy, the time to reach", + "type": "text" + }, + { + "bbox": [ + 484, + 237, + 504, + 248 + ], + "score": 0.85, + "content": "9 9 \\%", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "accuracy (i.e. completion) must be at least 8. (Note that both of these lower bounds are extremely", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "loose.) Therefore, training this network to completion at a stable step size would require at least", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 269, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 206, + 282 + ], + "score": 0.88, + "content": "8 / ( 2 / 2 4 6 1 . 8 ) \\ = \\ 9 , 8 7 6", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 269, + 476, + 282 + ], + "score": 1.0, + "content": "iterations (which is the same number of iterations as training to", + "type": "text" + }, + { + "bbox": [ + 477, + 270, + 504, + 281 + ], + "score": 0.85, + "content": "3 7 . 1 \\%", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "accuracy at a stable step size). Meanwhile, we find that the network can be trained to completion at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 194, + 304 + ], + "score": 1.0, + "content": "the larger step size of", + "type": "text" + }, + { + "bbox": [ + 195, + 292, + 233, + 303 + ], + "score": 0.91, + "content": "\\eta = 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "in just 1782 iterations. Therefore, training to completion at a stable", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 371, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 285, + 315 + ], + "score": 1.0, + "content": "step size is suboptimal by a factor of at least", + "type": "text" + }, + { + "bbox": [ + 285, + 303, + 367, + 315 + ], + "score": 0.88, + "content": "9 , 8 7 6 / 1 7 8 2 = 5 . 5 4", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 303, + 371, + 315 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 204, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 311, + 332 + ], + "score": 1.0, + "content": "ResNet. For this network, gradient flow reaches", + "type": "text" + }, + { + "bbox": [ + 311, + 320, + 339, + 330 + ], + "score": 0.87, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "accuracy at time 70, and the maximum", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "sharpness up through this point on the gradient flow trajectory is 760.6. Therefore, the largest sta-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 169, + 354 + ], + "score": 1.0, + "content": "ble step size is", + "type": "text" + }, + { + "bbox": [ + 169, + 342, + 247, + 353 + ], + "score": 0.89, + "content": "2 / 7 6 0 . 6 = 0 . 0 0 2 6", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 342, + 313, + 354 + ], + "score": 1.0, + "content": ", and training to", + "type": "text" + }, + { + "bbox": [ + 313, + 342, + 341, + 352 + ], + "score": 0.88, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "accuracy at this step size would require", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 178, + 364 + ], + "score": 0.81, + "content": "7 0 / 0 . 0 0 2 6 = 2 6 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "923 iterations. Meanwhile, we empirically observe that the network can also be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 147, + 376 + ], + "score": 1.0, + "content": "trained to", + "type": "text" + }, + { + "bbox": [ + 147, + 364, + 175, + 374 + ], + "score": 0.85, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 363, + 315, + 376 + ], + "score": 1.0, + "content": "accuracy at the larger step size of", + "type": "text" + }, + { + "bbox": [ + 315, + 364, + 349, + 375 + ], + "score": 0.92, + "content": "\\eta = 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "in just 99 iterations. Therefore, when", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 207, + 388 + ], + "score": 1.0, + "content": "training this network to", + "type": "text" + }, + { + "bbox": [ + 207, + 375, + 235, + 385 + ], + "score": 0.88, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "accuracy, stable step sizes are suboptimal by a factor of at least", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 107, + 385, + 189, + 397 + ], + "score": 0.9, + "content": "2 6 , 9 2 { \\bar { 3 } } / 9 9 = 2 7 1 . 9 .", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 384, + 439, + 399 + ], + "score": 1.0, + "content": ". This is a tight lower bound on the suboptimality of training to", + "type": "text" + }, + { + "bbox": [ + 439, + 385, + 466, + 396 + ], + "score": 0.88, + "content": "4 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "at a stable step size. For the loose lower bound on the suboptimality of training to completion at a sta-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "ble step size, note that (by similar reasoning as the VGG-without-BN above), training to completion", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "at a stable step size must require at least 26, 923 iterations. Meanwhile, we find that the network can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 304, + 443 + ], + "score": 1.0, + "content": "be trained to completion at the larger step size of", + "type": "text" + }, + { + "bbox": [ + 305, + 430, + 338, + 441 + ], + "score": 0.9, + "content": "\\eta = 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "in just 807 iterations. 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Observe", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 186, + 443, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 443, + 197 + ], + "score": 1.0, + "content": "that the sharpness rises dramatically from 6.38 at initialization to a peak of 2227.59.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 110, + 221, + 505, + 384 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 221, + 505, + 384 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 221, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 110, + 221, + 505, + 384 + ], + "score": 0.946, + "type": "image", + "image_path": "4b37034eb812e5a6f0e13a18abb8eed6b97188c126c689072faf59d018799676.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 110, + 221, + 505, + 275.3333333333333 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 110, + 275.3333333333333, + 505, + 329.66666666666663 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 110, + 329.66666666666663, + 505, + 383.99999999999994 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 396, + 506, + 518 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Figure 90: We train a VGG with BN to completion using gradient descent at different step sizes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "(see legend in the top right pane). Top left: we plot the train loss, with a vertical dotted line", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 329, + 432 + ], + "score": 1.0, + "content": "marking the iteration where the sharpness first crosses", + "type": "text" + }, + { + "bbox": [ + 329, + 419, + 346, + 431 + ], + "score": 0.89, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 418, + 506, + 432 + ], + "score": 1.0, + "content": ". Top center: we plot the evolution of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 428, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 464, + 444 + ], + "score": 1.0, + "content": "the sharpness, with a horizontal dashed line (of the appropriate color) marking the value", + "type": "text" + }, + { + "bbox": [ + 464, + 430, + 481, + 442 + ], + "score": 0.86, + "content": "2 / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 428, + 506, + 444 + ], + "score": 1.0, + "content": ". Top", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 454 + ], + "score": 1.0, + "content": "right: we plot the train accuracy. 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Observe that the distance between gradient descent and gradient flow is almost zero before", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 496, + 497 + ], + "score": 1.0, + "content": "this instant, but starts to rise shortly afterwards. Bottom center: we plot the sharpness by time", + "type": "text" + }, + { + "bbox": [ + 496, + 486, + 505, + 495 + ], + "score": 0.57, + "content": "\\left( = \\right.", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 142, + 510 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 142, + 496, + 158, + 507 + ], + "score": 0.83, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 495, + 506, + 510 + ], + "score": 1.0, + "content": ") rather than iteration. 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Observe that the distance between gradient descent and gradient flow is almost zero before", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 496, + 497 + ], + "score": 1.0, + "content": "this instant, but starts to rise shortly afterwards. Bottom center: we plot the sharpness by time", + "type": "text" + }, + { + "bbox": [ + 496, + 486, + 505, + 495 + ], + "score": 0.57, + "content": "\\left( = \\right.", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 495, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 142, + 510 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 142, + 496, + 158, + 507 + ], + "score": 0.83, + "content": "\\times \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 495, + 506, + 510 + ], + "score": 1.0, + "content": ") rather than iteration. 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For Nets-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 240, + 502, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 231, + 265 + ], + "score": 1.0, + "content": "terov momentum with step size", + "type": "text" + }, + { + "bbox": [ + 231, + 251, + 239, + 261 + ], + "score": 0.78, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 249, + 344, + 265 + ], + "score": 1.0, + "content": "and momentum parameter", + "type": "text" + }, + { + "bbox": [ + 345, + 249, + 378, + 261 + ], + "score": 0.91, + "content": "\\beta = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 249, + 426, + 265 + ], + "score": 1.0, + "content": ", the MSS is", + "type": "text" + }, + { + "bbox": [ + 425, + 240, + 466, + 260 + ], + "score": 1.0, + "content": "η2+2 β", + "type": "text" + }, + { + "bbox": [ + 427, + 248, + 501, + 264 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 2 + 2 \\beta } { \\eta ( 1 + 2 \\beta ) } \\approx \\frac { 1 . 3 5 7 1 4 } { \\eta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 247, + 502, + 264 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 324, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 262, + 276 + ], + "score": 1.0, + "content": "We run gradient descent until reaching", + "type": "text" + }, + { + "bbox": [ + 263, + 262, + 282, + 273 + ], + "score": 0.82, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 261, + 324, + 276 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 156, + 506, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 278, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 292 + ], + "score": 1.0, + "content": "We find that the sharpness rises until reaching the maximum stable sharpness (MSS) given by equa-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "tion 1, and then either plateaus just above that value, or oscillates around that value. 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