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+# Variational Diffusion Models
+
+Diederik P. Kingma⇤, Tim Salimans⇤, Ben Poole, Jonathan Ho Google Research
+
+# Abstract
+
+Diffusion-based generative models have demonstrated a capacity for perceptually impressive synthesis, but can they also be great likelihood-based models? We answer this in the affirmative, and introduce a family of diffusion-based generative models that obtain state-of-the-art likelihoods on standard image density estimation benchmarks. Unlike other diffusion-based models, our method allows for efficient optimization of the noise schedule jointly with the rest of the model. We show that the variational lower bound (VLB) simplifies to a remarkably short expression in terms of the signal-to-noise ratio of the diffused data, thereby improving our theoretical understanding of this model class. Using this insight, we prove an equivalence between several models proposed in the literature. In addition, we show that the continuous-time VLB is invariant to the noise schedule, except for the signal-to-noise ratio at its endpoints. This enables us to learn a noise schedule that minimizes the variance of the resulting VLB estimator, leading to faster optimization. Combining these advances with architectural improvements, we obtain state-of-the-art likelihoods on image density estimation benchmarks, outperforming autoregressive models that have dominated these benchmarks for many years, with often significantly faster optimization. In addition, we show how to use the model as part of a bits-back compression scheme, and demonstrate lossless compression rates close to the theoretical optimum.
+
+# 1 Introduction
+
+Likelihood-based generative modeling is a central task in machine learning that is the basis for a wide range of applications ranging from speech synthesis [Oord et al., 2016], to translation [Sutskever et al., 2014], to compression [MacKay, 2003], to many others. Autoregressive models have long been the dominant model class on this task due to their tractable likelihood and expressivity, as shown in Figure 1. Diffusion models have recently shown impressive results in image [Ho et al., 2020, Song et al., 2021b, Nichol and Dhariwal, 2021] and audio generation [Kong et al., 2020, Chen et al., 2020] in terms of perceptual quality, but have yet to match autoregressive models on density estimation benchmarks. In this paper we make several technical contributions that allow diffusion models to challenge the dominance of autoregressive models in this domain. Our main contributions are as follows:
+
+• We introduce a flexible family of diffusion-based generative models that achieve new stateof-the-art log-likelihoods on standard image density estimation benchmarks (CIFAR-10 and ImageNet). This is enabled by incorporating Fourier features into the diffusion model and using a learnable specification of the diffusion process, among other modeling innovations. • We improve our theoretical understanding of density modeling using diffusion models by analyzing their variational lower bound (VLB), deriving a remarkably simple expression in terms of the signal-to-noise ratio of the diffusion process. This result delivers new insight into the model class: for the continuous-time (infinite-depth) setting we prove a novel invariance of the generative model and its VLB to the specification of the diffusion process, and we show that various diffusion models from the literature are equivalent up to a trivial time-dependent rescaling of the data.
+
+![](images/06e9c455ac61783c9981ff0abd0aa80c8f2cbcf08994ba403cd836adb96a768a.jpg)  
+Figure 1: Autoregressive generative models were long dominant in standard image density estimation benchmarks. In contrast, we propose a family of diffusion-based generative models, Variational Diffusion Models (VDMs), that outperforms contemporary autoregressive models in these benchmarks. See Table 1 for more results and comparisons.
+
+# 2 Related work
+
+Our work builds on diffusion probabilistic models (DPMs) [Sohl-Dickstein et al., 2015], or diffusion models in short. DPMs can be viewed as a type of variational autoencoder (VAE) [Kingma and Welling, 2013, Rezende et al., 2014], whose structure and loss function allows for efficient training of arbitrarily deep models. Interest in diffusion models has recently reignited due to their impressive image generation results [Ho et al., 2020, Song and Ermon, 2020].
+
+Ho et al. [2020] introduced a number of model innovations to the original DPM, with impressive results on image generation quality benchmarks. They showed that the VLB objective, for a diffusion model with discrete time and diffusion variances shared across input dimensions, is equivalent to multiscale denoising score matching, up to particular weightings per noise scale. Further improvements were proposed by Nichol and Dhariwal [2021], resulting in better log-likelihood scores. Gao et al. [2020] show how diffusion can also be used to efficiently optimize energy-based models (EBMs) towards a close approximation of the log-likelihood objective, resulting in high-fidelity samples even after long MCMC chains.
+
+Song and Ermon [2019] first proposed learning generative models through a multi-scale denoising score matching objective, with improved methods in Song and Ermon [2020]. This was later extended to continuous-time diffusion with novel sampling algorithms based on reversing the diffusion process [Song et al., 2021b].
+
+Concurrent to our work, Song et al. [2021a], Huang et al. [2021], and Vahdat et al. [2021] also derived variational lower bounds to the data likelihood under a continuous-time diffusion model. Where we consider the infinitely deep limit of a standard VAE, Song et al. [2021a] and Vahdat et al. [2021] present different derivations based on stochastic differential equations. Huang et al. [2021] considers both perspectives and discusses the similarities between the two approaches. An advantage of our analysis compared to these other works is that we present an intuitive expression of the VLB in terms of the signal-to-noise ratio of the diffused data, leading to much simplified expressions of the discrete-time and continuous-time loss, allowing for simple and numerically stable implementation. This also leads to new results on the invariance of the generative model and its VLB to the specification of the diffusion process. We empirically compare to these works, as well as others, in Table 1.
+
+Previous approaches to diffusion probabilistic models fixed the diffusion process; in contrast optimize the diffusion process parameters jointly with the rest of the model. This turns the model into a type of VAE [Kingma and Welling, 2013, Rezende et al., 2014]. This is enabled by directly parameterizing the mean and variance of the marginal $q ( \mathbf { z } _ { t } | \mathbf { z } _ { 0 } )$ , where previous approaches instead parameterized the individual diffusion steps $q ( \mathbf { z } _ { t + \epsilon } | \mathbf { z } _ { t } )$ . In addition, our denoising models include several architecture changes, the most important of which is the use of Fourier features, which enable us to reach much better likelihoods than previous diffusion probabilistic models.
+
+# 3 Model
+
+We will focus on the most basic case of generative modeling, where we have a dataset of observations of $\mathbf { x }$ , and the task is to estimate the marginal distribution $p ( \mathbf { x } )$ . As with most generative models, the described methods can be extended to the case of multiple observed variables, and/or the task of estimating conditional densities $p ( \mathbf { x } | \mathbf { y } )$ . The proposed latent-variable model consists of a diffusion process (Section 3.1) that we invert to obtain a hierarchical generative model (Section 3.3). As we will show, the model choices below result in a surprisingly simple variational lower bound (VLB) of the marginal likelihood, which we use for optimization of the parameters.
+
+# 3.1 Forward time diffusion process
+
+Our starting point is a Gaussian diffusion process that begins with the data $\mathbf { x }$ , and defines a sequence of increasingly noisy versions of $\mathbf { x }$ which we call the latent variables $\mathbf { z } _ { t }$ , where $t$ runs from $t = 0$ (least noisy) to $t = 1$ (most noisy). The distribution of latent variable $\mathbf { z } _ { t }$ conditioned on $\mathbf { x }$ , for any $t \in [ 0 , 1 ]$ is given by:
+
+$$
+q ( \mathbf { z } _ { t } | \mathbf { x } ) = \mathcal { N } \left( \alpha _ { t } \mathbf { x } , \sigma _ { t } ^ { 2 } \mathbf { I } \right) ,
+$$
+
+where $\alpha _ { t }$ and $\sigma _ { t } ^ { 2 }$ are strictly positive scalar-valued functions of $t$ . Furthermore, let us define the signal-to-noise ratio (SNR):
+
+$$
+\mathrm { S N R } ( t ) = \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 } .
+$$
+
+We assume that the $\mathrm { S N R } ( t )$ is strictly monotonically decreasing in time, i.e. that $\mathrm { S N R } ( t ) < \mathrm { S N R } ( s )$ for any $t > s$ . This formalizes the notion that the $\mathbf { z } _ { t }$ is increasingly noisy as we go forward in time. We also assume that both $\alpha _ { t }$ and $\sigma _ { t } ^ { 2 }$ are smooth, such that their derivatives with respect to time $t$ are finite. This diffusion process specification includes the variance-preserving diffusion process as used by [Sohl-Dickstein et al., 2015, Ho et al., 2020] as a special case, where $\alpha _ { t } = \sqrt { 1 - \sigma _ { t } ^ { 2 } }$ . Another special case is the variance-exploding diffusion process as used by [Song and Ermon, 2019, Song et al., 2021b], where $\alpha _ { t } ^ { 2 } = 1$ . In experiments, we use the variance-preserving version.
+
+The distributions $q ( \mathbf { z } _ { t } | \mathbf { z } _ { s } )$ for any $t > s$ are also Gaussian, and given in Appendix A. The joint distribution of latent variables $\left( { \bf z } _ { s } , { \bf z } _ { t } , { \bf z } _ { u } \right)$ at any subsequent timesteps $0 \leq s < t < u \leq 1$ is Markov: $q ( { \bf z } _ { u } | { \bf z } _ { t } , { \bf z } _ { s } ) = q ( { \bf z } _ { u } | { \bf z } _ { t } )$ . Given the distributions above, it is relatively straightforward to verify through Bayes rule that ${ \dot { q } } ( \mathbf { z } _ { s } | \mathbf { z } _ { t } , \mathbf { x } )$ , for any $0 \leq s < t \leq 1$ , is also Gaussian. This distribution is also given in Appendix A.
+
+# 3.2 Noise schedule
+
+In previous work, the noise schedule has a fixed form (see Appendix $_ \mathrm { H }$ , Fig. 4a). In contrast, we learn this schedule through the parameterization
+
+$$
+\sigma _ { t } ^ { 2 } = \mathrm { s i g m o i d } ( \gamma _ { \eta } ( t ) )
+$$
+
+where $\gamma _ { \eta } ( t )$ is a monotonic neural network with parameters $\eta$ , as detailed in Appendix H.
+
+Motivated by the equivalence discussed in Section 5.1, we use $\alpha _ { t } = \sqrt { 1 - \sigma _ { t } ^ { 2 } }$ in our experiments for both the discrete-time and continuous-time models, i.e. variance-preserving diffusion processes. It is straightforward to verify that $\alpha _ { t } ^ { 2 }$ and $\mathrm { S N R } ( t )$ , as a function of $\gamma _ { \eta } ( t )$ , then simplify to:
+
+$$
+\begin{array} { r } { \alpha _ { t } ^ { 2 } = \mathrm { s i g m o i d } ( - \gamma _ { \pm } ( t ) ) } \\ { \mathrm { S N R } ( t ) = \exp ( - \gamma _ { \pm } ( t ) ) } \end{array}
+$$
+
+# 3.3 Reverse time generative model
+
+We define our generative model by inverting the diffusion process of Section 3.1, yielding a hierarchical generative model that samples a sequence of latents $\mathbf { z } _ { t }$ , with time running backward from $t = 1$ to $t = 0$ . We consider both the case where this sequence consists of a finite number of steps $T$ , as well as a continuous time model corresponding to $T \to \infty$ . We start by presenting the discrete-time case.
+
+Given finite $T$ , we discretize time uniformly into $T$ timesteps (segments) of width $\tau = 1 / T$ . Defining $s ( i ) = ( i - 1 ) / T$ and $t ( i ) = i / T$ , our hierarchical generative model for data $\mathbf { x }$ is then given by:
+
+$$
+p ( \mathbf { x } ) = \int _ { \mathbf { z } } p ( \mathbf { z } _ { 1 } ) p ( \mathbf { x } | \mathbf { z } _ { 0 } ) \prod _ { i = 1 } ^ { T } p ( \mathbf { z } _ { s ( i ) } | \mathbf { z } _ { t ( i ) } ) .
+$$
+
+With the variance preserving diffusion specification and sufficiently small SNR(1), we have that $q ( \mathbf { z } _ { 1 } | \mathbf { x } ) \approx \mathcal { N } ( \mathbf { z } _ { 1 } ; \bar { 0 , } \mathbf { I } )$ . We therefore model the marginal distribution of $\mathbf { z } _ { 1 }$ as a spherical Gaussian:
+
+$$
+p ( \mathbf { z } _ { 1 } ) = \mathcal { N } ( \mathbf { z } _ { 1 } ; 0 , \mathbf { I } ) .
+$$
+
+We wish to choose a model $p ( \mathbf { x } | \mathbf { z } _ { 0 } )$ that is close to the unknown $q ( \mathbf { x } | \mathbf { z } _ { 0 } )$ . Let $x _ { i }$ and $z _ { 0 , i }$ be the $i$ -th elements of $\mathbf { x } , \mathbf { z } _ { 0 }$ , respectively. We then use a factorized distribution of the form:
+
+$$
+p ( \mathbf { x } | \mathbf { z } _ { 0 } ) = \prod _ { i } p ( x _ { i } | z _ { 0 , i } ) ,
+$$
+
+where we choose $p ( x _ { i } | z _ { 0 , i } ) \propto q ( z _ { 0 , i } | x _ { i } )$ , which is normalized by summing over all possible discrete values of $x _ { i }$ (256 in the case of 8-bit image data). With sufficiently large $\mathrm { S N R } ( 0 )$ , this becomes a very close approximation to the true $q ( \mathbf { x } | \mathbf { z } _ { 0 } )$ , as the influence of the unknown data distribution $q ( \mathbf { x } )$ is overwhelmed by the likelihood $q ( \mathbf { z } _ { 0 } | \mathbf { x } )$ . Finally, we choose the conditional model distributions as
+
+$$
+p ( \mathbf { z } _ { s } | \mathbf { z } _ { t } ) = q ( \mathbf { z } _ { s } | \mathbf { z } _ { t } , \mathbf { x } = \hat { \mathbf { x } } _ { \theta } \big ( \mathbf { z } _ { t } ; t \big ) \big ) ,
+$$
+
+i.e. the same as $q ( \mathbf { z } _ { s } | \mathbf { z } _ { t } , \mathbf { x } )$ , but with the original data $\mathbf { x }$ replaced by the output of a denoising model $\hat { \mathbf { x } } _ { \pmb { \theta } } ( \mathbf { z } _ { t } ; t )$ that predicts $\mathbf { x }$ from its noisy version $\mathbf { z } _ { t }$ . Note that in practice we parameterize the denoising model as a function of a noise prediction model (Section 3.4), bridging the gap with previous work on diffusion models [Ho et al., 2020]. The means and variances of $p ( \mathbf { z } _ { s } | \mathbf { z } _ { t } )$ simplify to a remarkable degree; see Appendix A.
+
+# 3.4 Noise prediction model and Fourier features
+
+We parameterize the denoising model in terms of a noise prediction model $\hat { \epsilon } _ { \theta } ( { \bf z } _ { t } ; t )$
+
+$$
+\hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ; t ) = ( \mathbf { z } _ { t } - \sigma _ { t } \hat { \epsilon } _ { \theta } ( \mathbf { z } _ { t } ; t ) ) / \alpha _ { t } ,
+$$
+
+where $\hat { \epsilon } _ { \theta } ( { \bf z } _ { t } ; t )$ is parameterized as a neural network. The noise prediction models we use in experiments closely follow Ho et al. [2020], except that they process the data solely at the original resolution. The exact parameterization of the noise prediction model and noise schedule is discussed in Appendix B.
+
+Prior work on diffusion models has mainly focused on the perceptual quality of generated samples, which emphasizes coarse scale patterns and global consistency of generated images. Here, we optimize for likelihood, which is sensitive to fine scale details and exact values of individual pixels. To capture the fine scale details of the data, we propose adding a set of Fourier features to the input of our noise prediction model. Let $\mathbf { x }$ be the original data, scaled to the range $[ - 1 , 1 ]$ , and let $\mathbf { z }$ be the resulting latent variable, with similar magnitudes. We then append channels $\sin ( 2 ^ { n } \pi \mathbf { z } )$ and $\cos ( 2 ^ { n } \pi \mathbf { z } )$ , where $n$ runs over a range of integers $\{ n _ { m i n } , . . . , n _ { m a x } \}$ . These features are high frequency periodic functions that amplify small changes in the input data $\mathbf { z } _ { t }$ ; see Appendix C for further details. Including these features in the input of our denoising model leads to large improvements in likelihood as demonstrated in Section 6 and Figure 5, especially when combined with a learnable SNR function. We did not observe such improvements when incorporating Fourier features into autoregressive models.
+
+# 3.5 Variational lower bound
+
+We optimize the parameters towards the variational lower bound (VLB) of the marginal likelihood, which is given by
+
+$$
+\begin{array} { r } { - \log p ( \mathbf { x } ) \leq - \mathrm { V L B } ( \mathbf { x } ) = \underbrace { D _ { K L } \big ( q ( \mathbf { z } _ { 1 } | \mathbf { x } ) | | p ( \mathbf { z } _ { 1 } ) \big ) } _ { \mathrm { P r i o r ~ l o s s } } + \underbrace { { \mathbb { E } _ { q ( \mathbf { z } _ { 0 } | \mathbf { x } ) } } \left[ - \log p ( \mathbf { x } | \mathbf { z } _ { 0 } ) \right] } _ { \mathrm { R e c o n s t u r c i t o n ~ l o s s } } + \underbrace { \mathcal { L } _ { T } ( \mathbf { x } ) . } _ { \mathrm { D i f f u s i o n ~ l o s s } } } \end{array}
+$$
+
+The prior loss and reconstruction loss can be (stochastically and differentiably) estimated using standard techniques; see [Kingma and Welling, 2013]. The diffusion loss, $\mathcal { L } _ { T } ( \mathbf { x } )$ , is more complicated, and depends on the hyperparameter $T$ that determines the depth of the generative model.
+
+# 4 Discrete-time model
+
+In the case of finite $T$ , using $s ( i ) = ( i - 1 ) / T , t ( i ) = i / T$ , the diffusion loss is:
+
+$$
+\mathcal { L } _ { T } ( \mathbf { x } ) = \sum _ { i = 1 } ^ { T } \mathbb { E } _ { q ( \mathbf { z } _ { t ( i ) } \mid \mathbf { x } ) } D _ { K L } \big [ q ( \mathbf { z } _ { s ( i ) } \vert \mathbf { z } _ { t ( i ) } , \mathbf { x } ) \vert \big \vert p ( \mathbf { z } _ { s ( i ) } \vert \mathbf { z } _ { t ( i ) } ) \big ] .
+$$
+
+In appendix $\mathrm { E }$ we show that this expression simplifies considerably, yielding:
+
+$$
+\mathcal { L } _ { T } ( \mathbf { x } ) = \frac { T } { 2 } \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) , i \sim U \{ 1 , T \} } \left[ \left( \mathrm { S N R } ( s ) - \mathrm { S N R } ( t ) \right) | | \mathbf { x } - \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ; t ) | | _ { 2 } ^ { 2 } \right] ,
+$$
+
+where $U \{ 1 , T \}$ is the uniform distribution on the integers $\{ 1 , \ldots , T \}$ , and ${ \bf z } _ { t } = \alpha _ { t } { \bf x } + \sigma _ { t } \epsilon$ . This is the general discrete-time loss for any choice of forward diffusion parameters $\left( \sigma _ { t } , \alpha _ { t } \right)$ . When plugging in the specifications of $\sigma _ { t } , \alpha _ { t }$ and $\hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ; t )$ that we use in experiments, given in Sections 3.2 and 3.4, the loss simplifies to:
+
+$$
+\mathcal { L } _ { T } ( \mathbf { x } ) = \frac { T } { 2 } \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) , i \sim U \lbrace 1 , T \rbrace } \left[ ( \exp ( \gamma _ { \eta } ( t ) - \gamma _ { \eta } ( s ) ) - 1 ) \| \epsilon - \hat { \epsilon } _ { \theta } ( \mathbf { z } _ { t } ; t ) \| _ { 2 } ^ { 2 } \right]
+$$
+
+where $\mathbf { z } _ { t } = \mathrm { s i g m o i d } ( - \gamma _ { \eta } ( t ) ) \mathbf { x } + \mathrm { s i g m o i d } ( \gamma _ { \eta } ( t ) ) \pmb { \epsilon }$ . In the discrete-time case, we simply jointly optimize $\eta$ and $\pmb \theta$ by maximizing the VLB through a Monte Carlo estimator of Equation 14.
+
+Note that $\exp ( . ) - 1$ has a numerically stable primitive $e x p m 1 ( . )$ in common numerical computing packages; see figure 6. Equation 14 allows for numerically stable implementation in 32-bit or lower-precision floating point, in contrast with previous implementations of discrete-time diffusion models (e.g. [Ho et al., 2020]), which had to resort to 64-bit floating point.
+
+# 4.1 More steps leads to a lower loss
+
+A natural question to ask is what the number of timesteps $T$ should be, and whether more timesteps is always better in terms of the VLB. In Appendix F we analyze the difference between the diffusion loss with $T$ timesteps, $\mathcal { L } _ { T } ( \mathbf { x } )$ , and the diffusion loss with double the timesteps, $\mathcal { L } _ { 2 T } ( \mathbf { x } )$ , while keeping the SNR function fixed. We then find that if our trained denoising model $\hat { \mathbf { x } } _ { \pmb { \theta } }$ is sufficiently good, we have that $\mathcal { L } _ { 2 T } ( \mathbf { x } ) < \mathcal { L } _ { T } ( \mathbf { x } )$ , i.e. that our VLB will be better for a larger number of timesteps. Intuitively, the discrete time diffusion loss is an upper Riemann sum approximation of an integral of a strictly decreasing function, meaning that a finer approximation yields a lower diffusion loss. This result is illustrated in Figure 2.
+
+# 5 Continuous-time model: $T \to \infty$
+
+Since taking more time steps leads to a better VLB, we now take $T \to \infty$ , effectively treating time $t$ as continuous rather than discrete. The model for $p ( \mathbf { z } _ { t } )$ can in this case be described as a continuous time diffusion process [Song et al., 2021b] governed by a stochastic differential equation; see Appendix D. In Appendix E we show that in this limit the diffusion loss $\mathcal { L } _ { T } ( \mathbf { x } )$ simplifies further. Letting $\mathrm { S N R } ^ { \prime } ( t ) = d \hat { \mathrm { S N R } } ( t ) / d t$ , we have, with ${ \bf z } _ { t } = \alpha _ { t } { \bf x } + \sigma _ { t } \epsilon$ :
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } _ { \infty } ( \mathbf { x } ) = - \frac { 1 } { 2 } \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) } \int _ { 0 } ^ { 1 } \mathrm { S N R } ^ { \prime } ( t ) \left\| \mathbf { x } - \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ; t ) \right\| _ { 2 } ^ { 2 } d t , } \\ { \displaystyle \quad \quad = - \frac { 1 } { 2 } \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) , t \sim \mathcal { U } ( 0 , 1 ) } \left[ \mathrm { S N R } ^ { \prime } ( t ) \left\| \mathbf { x } - \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ; t ) \right\| _ { 2 } ^ { 2 } \right] . } \end{array}
+$$
+
+![](images/a1bce51bea03c5d5f7bc9253564d5e084a5720fd426ee069732cd96a8af54609.jpg)  
+Figure 2: Illustration of the diffusion loss with few segments $T$ (left), more segments $T$ (middle), and infinite segments $T$ (continuous time, right). The continuous-time loss (Equation 18) is an integral of mean squared error (MSE) over SNR, here visualized as a black curve. The black curve is strictly decreasing when the model is sufficiently well trained, so the discrete-time loss (Equation 13) is an upper bound (an upper Riemann sum approximation) of this integral that becomes better when segments are added.
+
+This is the general continuous-time loss for any choice of forward diffusion parameters $\left( \sigma _ { t } , \alpha _ { t } \right)$ . When plugging in the specifications of $\sigma _ { t } , \alpha _ { t }$ and $\hat { \mathbf { x } } _ { \pmb { \theta } } ( \mathbf { z } _ { t } ; t )$ that we use in experiments, given in Sections 3.2 and 3.4, the loss simplifies to:
+
+$$
+\mathcal { L } _ { \infty } ( \mathbf { x } ) = \frac { 1 } { 2 } \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) , t \sim \mathcal { U } ( 0 , 1 ) } \left[ \gamma _ { \eta } ^ { \prime } ( t ) \left. \epsilon - \hat { \epsilon } _ { \theta } ( \mathbf { z } _ { t } ; t ) \right. _ { 2 } ^ { 2 } \right] ,
+$$
+
+where $\gamma _ { \pmb { \eta } } ^ { \prime } ( t ) = d \gamma _ { \pmb { \eta } } ( t ) / d t$ . We use the Monte Carlo estimator of this loss for evaluation and optimization.
+
+# 5.1 Equivalence of diffusion models in continuous time
+
+The signal-to-noise function $\mathrm { S N R } ( t )$ is invertible due to the monotonicity assumption in Section 3.1. Due to this invertibility, we can perform a change of variables, and make everything a function of $v \equiv \operatorname { S N R } ( t )$ instead of $t$ , such that $t = \mathrm { S N R } ^ { - \mathrm { \bar { 1 } } } ( v )$ . Let $\alpha _ { v }$ and $\sigma _ { v }$ be the functions $\alpha _ { t }$ and $\sigma _ { t }$ evaluated at $t = \mathrm { S N R } ^ { - 1 } ( v )$ , and correspondingly let $\mathbf { z } _ { v } = \alpha _ { v } \mathbf { x } + \sigma _ { v } \mathbf { \epsilon }$ . Similarly, we rewrite our noise prediction model as $\tilde { \mathbf { x } } _ { \pmb { \theta } } ( \mathbf { z } , v ) \equiv \hat { \mathbf { x } } _ { \pmb { \theta } } ( \mathbf { z } , \mathrm { S N R } ^ { - 1 } ( v ) )$ . With this change of variables, our continuous-time loss in Equation 15 can equivalently be written as:
+
+$$
+\mathcal { L } _ { \infty } ( \mathbf { x } ) = \frac { 1 } { 2 } \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) } \int _ { \mathrm { S N R } _ { \operatorname* { m i n } } } ^ { \mathrm { S N R } _ { \operatorname* { m a x } } } \left\| \mathbf { x } - \tilde { \mathbf { x } } _ { \pmb { \theta } } ( \mathbf { z } _ { v } , v ) \right\| _ { 2 } ^ { 2 } d v ,
+$$
+
+where instead of integrating w.r.t. time $t$ we now integrate w.r.t. the signal-to-noise ratio $v$ , and where $\mathrm { S N R } _ { \operatorname* { m i n } } = \mathrm { S N R } ( 1 )$ and $\mathrm { S N R } _ { \operatorname* { m a x } } = \mathrm { S N R } ( 0 )$ .
+
+What this equation shows us is that the only effect the functions $\alpha ( t )$ and $\sigma ( t )$ have on the diffusion loss is through the values $\mathrm { S N R } ( t ) = \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 }$ at endpoints $t = 0$ and $t = 1$ . Given these values $\operatorname { S N R } _ { \operatorname* { m a x } }$ and $\mathrm { { S N R } _ { \operatorname* { m i n } } }$ , the diffusion loss is invariant to the shape of function $\mathrm { S N R } ( t )$ between $t = 0$ and $t = 1$ . The VLB is thus only impacted by the function $\mathrm { S N R } ( t )$ through its endpoints $\mathrm { { S N R } _ { \operatorname* { m i n } } }$ and $\operatorname { S N R } _ { \operatorname* { m a x } }$ .
+
+Furthermore, we find that the distribution $p ( \mathbf { x } )$ defined by our generative model is also invariant to the specification of the diffusion process. Specifically, let $p ^ { A } ( \mathbf { \bar { x } } )$ denote the distribution defined by the combination of a diffusion specification and denoising function $\{ \alpha _ { v } ^ { A } , \sigma _ { v } ^ { A } , \tilde { \mathbf { x } } _ { \theta } ^ { A } \}$ , and similarly let $p ^ { B } ( \mathbf { x } )$ {be the distribution defined through a different specification $\{ \alpha _ { v } ^ { B } , \sigma _ { v } ^ { B } , \tilde { \mathbf { x } } _ { \theta } ^ { B } \}$ , where both specifications have equal $\mathrm { { S N R } _ { \operatorname* { m i n } } }$ , $\operatorname { S N R } _ { \operatorname* { m a x } }$ ; as shown in Appendix G, we then have that $p ^ { A } ( \mathbf { x } ) =$ $p ^ { \mathbf { \hat { \cal B } } } ( \mathbf { x } )$ if $\tilde { \mathbf { x } } _ { \pmb { \theta } } ^ { B } ( \mathbf { z } , v ) \equiv \hat { \tilde { \mathbf { x } } } _ { \pmb { \theta } } ^ { A } ( ( \alpha _ { v } ^ { A } / \alpha _ { v } ^ { B } ) \mathbf { z } , v )$ . The distribution on all latents $\mathbf { z } _ { v }$ is then also the same under both specifications, up to a trivial rescaling. This means that any two diffusion models satisfying the mild constraints set in 3.1 (which includes e.g. the variance exploding and variance preserving specifications considered by Song et al. [2021b]), can thus be seen as equivalent in continuous time.
+
+# 5.2 Weighted diffusion loss
+
+This equivalence between diffusion specifications continues to hold even if, instead of the VLB, these models optimize a weighted diffusion loss of the form:
+
+$$
+\mathcal { L } _ { \infty } ( \mathbf { x } , w ) = \frac { 1 } { 2 } \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) } \int _ { \mathrm { S N R } _ { \operatorname* { m i n } } } ^ { \mathrm { S N R } _ { \operatorname* { m a x } } } \ d w ( v ) \left. \mathbf { x } - \tilde { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { v } , v ) \right. _ { 2 } ^ { 2 } d v ,
+$$
+
+Table 1: Summary of our findings for density modeling tasks, in terms of bits per dimension (BPD) on the test set. Model types are autoregressive (AR), normalizing flows (Flow), variational autoencoders (VAE), or diffusion models (Diff). Our results were obtained using the continuous-time formulation of our model. CIFAR-10 data augmentations are: (A) extensive, (B) small translations, or (C) horizontal flips. The numbers for VDM are variational bounds, and can likely be improved by estimating the marginal likelihood through importance sampling, or through evaluation of the corresponding continuous normalizing flow as done by Song et al. [2021a].   
+
+<table><tr><td>Model (Bits per dim on test set)</td><td>Type</td><td>CIFAR10 no data aug.</td><td>CIFAR10 data aug.</td><td>ImageNet 32x32</td><td>ImageNet 64x64</td></tr><tr><td>Previous work</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ResNet VAE with IAF [Kingma et al., 2016]</td><td>VAE</td><td>3.11</td><td></td><td></td><td></td></tr><tr><td>Very Deep VAE [Child,2020]</td><td>VAE</td><td>2.87</td><td></td><td>3.80</td><td>3.52</td></tr><tr><td>NVAE [Vahdat and Kautz,2020]</td><td>VAE</td><td>2.91</td><td></td><td>3.92</td><td></td></tr><tr><td>Glow [Kingma and Dhariwal, 2018]</td><td>Flow</td><td></td><td>3.35(B)</td><td>4.09</td><td>3.81</td></tr><tr><td>Flow++ [Ho et al.,2019a]</td><td>Flow</td><td>3.08</td><td></td><td>3.86</td><td>3.69</td></tr><tr><td>PixelCNN [Van Oord et al., 2016]</td><td>AR</td><td>3.03</td><td></td><td>3.83</td><td>3.57</td></tr><tr><td>PixelCNN++ [Salimans et al.,2017]</td><td>AR</td><td>2.92</td><td></td><td></td><td></td></tr><tr><td>Image Transformer [Parmar et al., 2018]</td><td>AR</td><td>2.90</td><td></td><td>3.77</td><td></td></tr><tr><td>SPN [Menick and Kalchbrenner, 2018]</td><td>AR</td><td></td><td></td><td></td><td>3.52</td></tr><tr><td>Sparse Transformer [Child et al.,2019]</td><td>AR</td><td>2.80</td><td></td><td></td><td>3.44</td></tr><tr><td>Routing Transformer [Roy et al., 2021]</td><td>AR</td><td></td><td></td><td></td><td>3.43</td></tr><tr><td>Sparse Transformer + DistAug [Jun et al.,2020]</td><td>AR</td><td></td><td>2.53(4)</td><td></td><td></td></tr><tr><td>DDPM [Ho et al., 2020]</td><td>Diff</td><td></td><td>3.69(C)</td><td></td><td></td></tr><tr><td>EBM-DRL [Gao et al., 2020]</td><td>Diff</td><td></td><td>3.18(C)</td><td></td><td></td></tr><tr><td>Score SDE [Song et al., 2021b]</td><td>Diff</td><td>2.99</td><td></td><td></td><td></td></tr><tr><td>Improved DDPM [Nichol and Dhariwal, 2021]</td><td>Diff</td><td>2.94</td><td></td><td></td><td>3.54</td></tr><tr><td>Concurrent work</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CR-NVAE [Sinha and Dieng,2021]</td><td>VAE</td><td></td><td>2.51(4)</td><td></td><td></td></tr><tr><td>LSGM [Vahdat et al., 2021]</td><td>Diff</td><td>2.87</td><td></td><td></td><td></td></tr><tr><td>ScoreFlow [Song et al.,2O21a] (variational bound)</td><td>Diff</td><td></td><td>2.90(C)</td><td>3.86</td><td></td></tr><tr><td>ScoreFlow [Song et al.,202la] (cont. norm. flow)</td><td>Diff</td><td>2.83</td><td>2.80(C)</td><td>3.76</td><td></td></tr><tr><td>Our work</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>VDM (variational bound)</td><td>Diff</td><td>2.65</td><td>2.49(A)</td><td>3.72</td><td>3.40</td></tr></table>
+
+which e.g. captures all the different objectives discussed by Song et al. [2021b], see Appendix K. Here, $w ( v )$ is a weighting function that generally puts increased emphasis on the noisier data compared to the VLB, and which thereby can sometimes improve perceptual generation quality as measured by certain metrics like FID and Inception Score. For the models presented in this paper, we further use $w ( v ) = 1$ as corresponding to the (unweighted) VLB.
+
+# 5.3 Variance minimization
+
+Lowering the variance of the Monte Carlo estimator of the continuous-time loss generally improves the efficiency of optimization. We found that using a low-discrepancy sampler for $t$ , as explained in Appendix I.1, leads to a significant reduction in variance. In addition, due to the invariance shown in Section 5.1 for the continous-time case, we can optimize the schedule between its endpoints w.r.t. to minimize the variance of our estimator of loss, as detailed in Appendix I. The endpoints of the noise schedule are simply optimized w.r.t. the VLB.
+
+# 6 Experiments
+
+We demonstrate our proposed class of diffusion models, which we call Variational Diffusion Models (VDMs), on the CIFAR-10 [Krizhevsky et al., 2009] dataset, and the downsampled ImageNet [Van Oord et al., 2016, Deng et al., 2009] dataset, where we focus on maximizing likelihood. For our result with data augmentation we used random flips, 90-degree rotations, and color channel swapping. More details of our model specifications are in Appendix B.
+
+![](images/54d231c9ca10990f784c07d9d3cd51bc63cfe05ce3e48f5ad4ff0710f43df11c.jpg)  
+Figure 3: Non cherry-picked unconditional samples from our Imagenet $6 4 \mathrm { x } 6 4$ model, trained in continuous time and generated using $T = 1 0 0 0$ . The model’s hyper-parameters and parameters are optimized w.r.t. the likelihood bound, so the model is not optimized for synthesis quality.
+
+# 6.1 Likelihood and samples
+
+Table 1 shows our results on modeling the CIFAR-10 dataset, and the downsampled ImageNet dataset. We establish a new state-of-the-art in terms of test set likelihood on all considered benchmarks, by a significant margin. Our model for CIFAR-10 without data augmentation surpasses the previous best result of 2.80 about $1 0 \mathrm { x }$ faster than it takes the Sparse Transformer to reach this, in wall clock time on equivalent hardware. Our CIFAR-10 model, whose hyper-parameters were tuned for likelihood, results in a FID (perceptual quality) score of 7.41. This would have been state-of-the-art until recently, but is worse than recent diffusion models that specifically target FID scores [Nichol and Dhariwal, 2021, Song et al., 2021b, Ho et al., 2020]. By instead using a weighted diffusion loss, with the weighting function $w ( \mathbf { S N R } )$ used by Ho et al. [2020] and described in Appendix K, our FID score improves to 4.0. We did not pursue further tuning of the model to improve FID instead of likelihood. A random sample of generated images from our model is provided in Figure 3. We provide additional samples from this model, as well as our other models for the other datasets, in Appendix M.
+
+# 6.2 Ablations
+
+Next, we investigate the relative importance of our contributions. In Table 2 we compare our discretetime and continuous-time specifications of the diffusion model: When evaluating our model with a small number of steps, our discretely trained models perform better by learning the diffusion schedule to optimize the VLB. However, as argued theoretically in Section 4.1, we find experimentally that more steps $T$ indeed gives better likelihood. When $T$ grows large, our continuously trained model performs best, helped by training its diffusion schedule to minimize variance instead.
+
+Minimizing the variance also helps the continuous time model to train faster, as shown in Figure 5. This effect is further examined in Table 4b, where we find dramatic variance reductions compared to our baselines in continuous time. Figure 4a shows how this effect is achieved: Compared to the other schedules, our learned schedule spends much more time in the high $\mathrm { S N R } ( t )$ / low $\bar { \sigma } _ { t } ^ { 2 }$ range.
+
+In Figure 5 we further show training curves for our model including and excluding the Fourier features proposed in Appendix C: with Fourier features enabled our model achieves much better likelihood. For comparison we also implemented Fourier features in a PixelCNN++ model [Salimans et al., 2017], where we do not see a benefit. In addition, we find that learning the SNR is necessary to get the most out of including Fourier features: if we fix the SNR schedule to that used by Ho et al. [2020], the maximum log-SNR is fixed to approximately 8 (see figure 7), and test set negative likelihood stays above 4 bits per dim. When learning the SNR endpoints, our maximum log-SNR ends up at 13.3, which, combined with the inclusion of Fourier features, leads to the SOTA test set likelihoods reported in Table 1.
+
+# 6.3 Lossless compression
+
+For a fixed number of evaluation timesteps $T _ { e v a l }$ , our diffusion model in discrete time is a hierarchical latent variable model that can be turned into a lossless compression algorithm using bits-back coding [Hinton and Van Camp, 1993]. As a proof of concept of practical lossless compression, Table 2 reports net codelengths on the CIFAR10 test set for various settings of $T _ { e v a l }$ using BB
+
+<table><tr><td>SNR(t) schedule</td><td>Var(BPD)</td></tr><tr><td>Learned (ours)</td><td>0.53</td></tr><tr><td>log SNR-linear</td><td>6.35</td></tr><tr><td>β-Linear [1]</td><td>31.6</td></tr><tr><td>α-Cosine [2]</td><td>31.1</td></tr></table>
+
+(b) Variance of VLB estimate
+
+![](images/c24191f17672e99720cf2abbe47c7f69d11815f1effee8dfcf8e9806285e04b4.jpg)  
+Figure 4: Our learned continuous-time variance-minimizing noise schedule $\mathrm { S N R } ( t )$ for CIFAR-10, compared to its log-linear initialization and to schedules from the literature: [1] The $\beta$ -Linear schedule from Ho et al. [2020], [2] The $\alpha$ -Cosine schedule from Nichol and Dhariwal [2021]. All schedules were scaled and shifted on the log scale such that the resulting $\mathrm { { S N R } _ { \operatorname* { m i n } } }$ , $\operatorname { S N R } _ { \operatorname* { m a x } }$ were the equal to our learned endpoints, resulting in the same VLB estimate of 2.66. We report the variance of our VLB estimate per data point, computed on the test set, and conditional on the data: This does not include the noise due to sampling minibatches of data.
+
+Table 2: Discrete versus continuous-time training and evaluation with CIFAR-10, in terms of bits per dimension (BPD).   
+
+<table><tr><td>Ttrain</td><td>Teual</td><td>BPD</td><td>Bits-Back Net BPD</td></tr><tr><td>10</td><td>10</td><td>4.31</td><td></td></tr><tr><td>100</td><td>100</td><td>2.84</td><td></td></tr><tr><td>250</td><td>250</td><td>2.73</td><td></td></tr><tr><td>500</td><td>500</td><td>2.68</td><td></td></tr><tr><td>1000</td><td>1000</td><td>2.67</td><td></td></tr><tr><td>10000</td><td>10000</td><td>2.66</td><td></td></tr><tr><td>8</td><td>10</td><td>7.54</td><td>7.54</td></tr><tr><td>8</td><td>100</td><td>2.90</td><td>2.91</td></tr><tr><td>8</td><td>250</td><td>2.74</td><td>2.76</td></tr><tr><td>8</td><td>500</td><td>2.69</td><td>2.72</td></tr><tr><td>8</td><td>1000</td><td>2.67</td><td>2.72</td></tr><tr><td>8</td><td>10000</td><td>2.65</td><td></td></tr><tr><td>8</td><td>8</td><td>2.65</td><td></td></tr></table>
+
+![](images/d9a6535ca7cea624ad43cc6e8acc7d7420e607f45e1ba25f3ff2279b3f38b92c.jpg)  
+Figure 5: Test set likelihoods during training, with/without Fourier features, and with/without learning the noise schedule to minimize variance.
+
+ANS [Townsend et al., 2018], an implementation of bits-back coding based on asymmetric numeral systems [Duda, 2009]. Details of our implementation are given in Appendix N. We achieve state-ofthe-art net codelengths, proving our model can be used as the basis of a lossless compression algorithm. However, for large $T _ { e v a l }$ a gap remains with the theoretically optimal codelength corresponding to the negative VLB, and compression becomes computationally expensive due to the large number of neural network forward passes required. Closing this gap with more efficient implementations of bits-back coding suitable for very deep models is an interesting avenue for future work.
+
+# 7 Conclusion
+
+We presented state-of-the-art results on modeling the density of natural images using a new class of diffusion models that incorporates a learnable diffusion specification, Fourier features for fine-scale modeling, as well as other architectural innovations. In addition, we obtained new theoretical insight into likelihood-based generative modeling with diffusion models, showing a surprising invariance of the VLB to the forward time diffusion process in continuous time, as well as an equivalence between various diffusion processes from the literature previously thought to be different.
+
+# Acknowledgments
+
+We thank Yang Song, Kevin Murphy and Mohammad Norouzi for feedback on drafts of this paper.
+
+# References
+
+Nanxin Chen, Yu Zhang, Heiga Zen, Ron J. Weiss, Mohammad Norouzi, and William Chan. Wavegrad: Estimating gradients for waveform generation. arXiv preprint arXiv:2009.00713, 2020.
+
+Rewon Child. Very deep VAEs generalize autoregressive models and can outperform them on images. arXiv preprint arXiv:2011.10650, 2020.
+
+Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. arXiv preprint arXiv:1904.10509, 2019.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
+
+Jarek Duda. Asymmetric numeral systems. arXiv preprint arXiv:0902.0271, 2009.
+
+Ruiqi Gao, Yang Song, Ben Poole, Ying Nian Wu, and Diederik P Kingma. Learning energy-based models by diffusion recovery likelihood. arXiv preprint arXiv:2012.08125, 2020.
+
+Geoffrey E Hinton and Drew Van Camp. Keeping the neural networks simple by minimizing the description length of the weights. In Proceedings of the Sixth Annual Conference on Computational Learning Theory, pages 5–13, 1993.
+
+Jonathan Ho, Xi Chen, Aravind Srinivas, Yan Duan, and Pieter Abbeel. Flow $^ { + + }$ : Improving flowbased generative models with variational dequantization and architecture design. In International Conference on Machine Learning, pages 2722–2730. PMLR, 2019a.
+
+Jonathan Ho, Evan Lohn, and Pieter Abbeel. Compression with flows via local bits-back coding. In Advances in Neural Information Processing Systems, pages 3874–3883, 2019b.
+
+Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. arXiv preprint arXiv:2006.11239, 2020.
+
+Chin-Wei Huang, Jae Hyun Lim, and Aaron Courville. A variational perspective on diffusion-based generative models and score matching. arXiv preprint arXiv:2106.02808, 2021.
+
+Heewoo Jun, Rewon Child, Mark Chen, John Schulman, Aditya Ramesh, Alec Radford, and Ilya Sutskever. Distribution augmentation for generative modeling. In International Conference on Machine Learning, pages 5006–5019. PMLR, 2020.
+
+Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. arXiv preprint arXiv:1710.10196, 2017.
+
+Diederik P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. arXiv preprint arXiv:1807.03039, 2018.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational Bayes. International Conference on Learning Representations, 2013.
+
+Diederik P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improved variational inference with inverse autoregressive flow. In Advances in Neural Information Processing Systems, pages 4743–4751, 2016.
+
+Friso Kingma, Pieter Abbeel, and Jonathan Ho. Bit-swap: Recursive bits-back coding for lossless compression with hierarchical latent variables. In International Conference on Machine Learning, pages 3408–3417. PMLR, 2019.
+
+Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. Diffwave: A versatile diffusion model for audio synthesis. arXiv preprint arXiv:2009.09761, 2020.
+
+Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
+
+Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017.
+
+David JC MacKay. Information theory, inference and learning algorithms. Cambridge university press, 2003.
+
+Jacob Menick and Nal Kalchbrenner. Generating high fidelity images with subscale pixel networks and multidimensional upscaling. arXiv preprint arXiv:1812.01608, 2018.
+
+Alex Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. arXiv preprint arXiv:2102.09672, 2021.
+
+Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016.
+
+Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Image transformer. In International Conference on Machine Learning, pages 4055–4064. PMLR, 2018.
+
+Danilo J Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning, pages 1278–1286, 2014.
+
+Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computerassisted intervention, pages 234–241. Springer, 2015.
+
+Aurko Roy, Mohammad Saffar, Ashish Vaswani, and David Grangier. Efficient content-based sparse attention with routing transformers. Transactions of the Association for Computational Linguistics, 9:53–68, 2021.
+
+Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelcnn $^ { + + }$ : Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017.
+
+Samarth Sinha and Adji B Dieng. Consistency regularization for variational auto-encoders. arXiv preprint arXiv:2105.14859, 2021.
+
+Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, pages 2256–2265, 2015.
+
+Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020.
+
+Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In Advances in Neural Information Processing Systems, pages 11895–11907, 2019.
+
+Yang Song and Stefano Ermon. Improved techniques for training score-based generative models. In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.
+
+Yang Song, Conor Durkan, Iain Murray, and Stefano Ermon. Maximum likelihood training of score-based diffusion models. arXiv e-prints, pages arXiv–2101, 2021a.
+
+Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-Based Generative Modeling Through Stochastic Differential Equations. In International Conference on Learning Representations, 2021b.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. arXiv preprint arXiv:1409.3215, 2014.
+
+Matthew Tancik, Pratul P Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. arXiv preprint arXiv:2006.10739, 2020.
+
+James Townsend, Thomas Bird, and David Barber. Practical lossless compression with latent variables using bits back coding. In International Conference on Learning Representations, 2018.   
+James Townsend, Thomas Bird, Julius Kunze, and David Barber. HiLLoC: lossless image compression with hierarchical latent variable models. In International Conference on Learning Representations, 2020.   
+Arash Vahdat and Jan Kautz. Nvae: A deep hierarchical variational autoencoder. arXiv preprint arXiv:2007.03898, 2020.   
+Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. arXiv preprint arXiv:2106.05931, 2021.   
+Aaron Van Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In International Conference on Machine Learning, pages 1747–1756, 2016.   
+Pascal Vincent. A connection between score matching and denoising autoencoders. Neural computation, 23(7):1661–1674, 2011.   
+Fisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, and Jianxiong Xiao. Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015.

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+        "text": "Variational Diffusion Models ",
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+        "text": "Diederik P. Kingma⇤, Tim Salimans⇤, Ben Poole, Jonathan Ho Google Research ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Diffusion-based generative models have demonstrated a capacity for perceptually impressive synthesis, but can they also be great likelihood-based models? We answer this in the affirmative, and introduce a family of diffusion-based generative models that obtain state-of-the-art likelihoods on standard image density estimation benchmarks. Unlike other diffusion-based models, our method allows for efficient optimization of the noise schedule jointly with the rest of the model. We show that the variational lower bound (VLB) simplifies to a remarkably short expression in terms of the signal-to-noise ratio of the diffused data, thereby improving our theoretical understanding of this model class. Using this insight, we prove an equivalence between several models proposed in the literature. In addition, we show that the continuous-time VLB is invariant to the noise schedule, except for the signal-to-noise ratio at its endpoints. This enables us to learn a noise schedule that minimizes the variance of the resulting VLB estimator, leading to faster optimization. Combining these advances with architectural improvements, we obtain state-of-the-art likelihoods on image density estimation benchmarks, outperforming autoregressive models that have dominated these benchmarks for many years, with often significantly faster optimization. In addition, we show how to use the model as part of a bits-back compression scheme, and demonstrate lossless compression rates close to the theoretical optimum. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Likelihood-based generative modeling is a central task in machine learning that is the basis for a wide range of applications ranging from speech synthesis [Oord et al., 2016], to translation [Sutskever et al., 2014], to compression [MacKay, 2003], to many others. Autoregressive models have long been the dominant model class on this task due to their tractable likelihood and expressivity, as shown in Figure 1. Diffusion models have recently shown impressive results in image [Ho et al., 2020, Song et al., 2021b, Nichol and Dhariwal, 2021] and audio generation [Kong et al., 2020, Chen et al., 2020] in terms of perceptual quality, but have yet to match autoregressive models on density estimation benchmarks. In this paper we make several technical contributions that allow diffusion models to challenge the dominance of autoregressive models in this domain. Our main contributions are as follows: ",
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+        "text": "• We introduce a flexible family of diffusion-based generative models that achieve new stateof-the-art log-likelihoods on standard image density estimation benchmarks (CIFAR-10 and ImageNet). This is enabled by incorporating Fourier features into the diffusion model and using a learnable specification of the diffusion process, among other modeling innovations. • We improve our theoretical understanding of density modeling using diffusion models by analyzing their variational lower bound (VLB), deriving a remarkably simple expression in terms of the signal-to-noise ratio of the diffusion process. This result delivers new insight into the model class: for the continuous-time (infinite-depth) setting we prove a novel invariance of the generative model and its VLB to the specification of the diffusion process, and we show that various diffusion models from the literature are equivalent up to a trivial time-dependent rescaling of the data. ",
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+        "image_caption": [
+            "Figure 1: Autoregressive generative models were long dominant in standard image density estimation benchmarks. In contrast, we propose a family of diffusion-based generative models, Variational Diffusion Models (VDMs), that outperforms contemporary autoregressive models in these benchmarks. See Table 1 for more results and comparisons. "
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our work builds on diffusion probabilistic models (DPMs) [Sohl-Dickstein et al., 2015], or diffusion models in short. DPMs can be viewed as a type of variational autoencoder (VAE) [Kingma and Welling, 2013, Rezende et al., 2014], whose structure and loss function allows for efficient training of arbitrarily deep models. Interest in diffusion models has recently reignited due to their impressive image generation results [Ho et al., 2020, Song and Ermon, 2020]. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Ho et al. [2020] introduced a number of model innovations to the original DPM, with impressive results on image generation quality benchmarks. They showed that the VLB objective, for a diffusion model with discrete time and diffusion variances shared across input dimensions, is equivalent to multiscale denoising score matching, up to particular weightings per noise scale. Further improvements were proposed by Nichol and Dhariwal [2021], resulting in better log-likelihood scores. Gao et al. [2020] show how diffusion can also be used to efficiently optimize energy-based models (EBMs) towards a close approximation of the log-likelihood objective, resulting in high-fidelity samples even after long MCMC chains. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Song and Ermon [2019] first proposed learning generative models through a multi-scale denoising score matching objective, with improved methods in Song and Ermon [2020]. This was later extended to continuous-time diffusion with novel sampling algorithms based on reversing the diffusion process [Song et al., 2021b]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Concurrent to our work, Song et al. [2021a], Huang et al. [2021], and Vahdat et al. [2021] also derived variational lower bounds to the data likelihood under a continuous-time diffusion model. Where we consider the infinitely deep limit of a standard VAE, Song et al. [2021a] and Vahdat et al. [2021] present different derivations based on stochastic differential equations. Huang et al. [2021] considers both perspectives and discusses the similarities between the two approaches. An advantage of our analysis compared to these other works is that we present an intuitive expression of the VLB in terms of the signal-to-noise ratio of the diffused data, leading to much simplified expressions of the discrete-time and continuous-time loss, allowing for simple and numerically stable implementation. This also leads to new results on the invariance of the generative model and its VLB to the specification of the diffusion process. We empirically compare to these works, as well as others, in Table 1. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Previous approaches to diffusion probabilistic models fixed the diffusion process; in contrast optimize the diffusion process parameters jointly with the rest of the model. This turns the model into a type of VAE [Kingma and Welling, 2013, Rezende et al., 2014]. This is enabled by directly parameterizing the mean and variance of the marginal $q ( \\mathbf { z } _ { t } | \\mathbf { z } _ { 0 } )$ , where previous approaches instead parameterized the individual diffusion steps $q ( \\mathbf { z } _ { t + \\epsilon } | \\mathbf { z } _ { t } )$ . In addition, our denoising models include several architecture changes, the most important of which is the use of Fourier features, which enable us to reach much better likelihoods than previous diffusion probabilistic models. ",
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+    {
+        "type": "text",
+        "text": "3 Model ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We will focus on the most basic case of generative modeling, where we have a dataset of observations of $\\mathbf { x }$ , and the task is to estimate the marginal distribution $p ( \\mathbf { x } )$ . As with most generative models, the described methods can be extended to the case of multiple observed variables, and/or the task of estimating conditional densities $p ( \\mathbf { x } | \\mathbf { y } )$ . The proposed latent-variable model consists of a diffusion process (Section 3.1) that we invert to obtain a hierarchical generative model (Section 3.3). As we will show, the model choices below result in a surprisingly simple variational lower bound (VLB) of the marginal likelihood, which we use for optimization of the parameters. ",
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+    {
+        "type": "text",
+        "text": "3.1 Forward time diffusion process ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our starting point is a Gaussian diffusion process that begins with the data $\\mathbf { x }$ , and defines a sequence of increasingly noisy versions of $\\mathbf { x }$ which we call the latent variables $\\mathbf { z } _ { t }$ , where $t$ runs from $t = 0$ (least noisy) to $t = 1$ (most noisy). The distribution of latent variable $\\mathbf { z } _ { t }$ conditioned on $\\mathbf { x }$ , for any $t \\in [ 0 , 1 ]$ is given by: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/696251951d1b050885ea68e72f32cd8acdf73a5162ace8b265d442a807bd612b.jpg",
+        "text": "$$\nq ( \\mathbf { z } _ { t } | \\mathbf { x } ) = \\mathcal { N } \\left( \\alpha _ { t } \\mathbf { x } , \\sigma _ { t } ^ { 2 } \\mathbf { I } \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\alpha _ { t }$ and $\\sigma _ { t } ^ { 2 }$ are strictly positive scalar-valued functions of $t$ . Furthermore, let us define the signal-to-noise ratio (SNR): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/680633e3b584ce891a6c9c851dade4263cdb07f443090e5c7f7b15a857c3dc88.jpg",
+        "text": "$$\n\\mathrm { S N R } ( t ) = \\alpha _ { t } ^ { 2 } / \\sigma _ { t } ^ { 2 } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We assume that the $\\mathrm { S N R } ( t )$ is strictly monotonically decreasing in time, i.e. that $\\mathrm { S N R } ( t ) < \\mathrm { S N R } ( s )$ for any $t > s$ . This formalizes the notion that the $\\mathbf { z } _ { t }$ is increasingly noisy as we go forward in time. We also assume that both $\\alpha _ { t }$ and $\\sigma _ { t } ^ { 2 }$ are smooth, such that their derivatives with respect to time $t$ are finite. This diffusion process specification includes the variance-preserving diffusion process as used by [Sohl-Dickstein et al., 2015, Ho et al., 2020] as a special case, where $\\alpha _ { t } = \\sqrt { 1 - \\sigma _ { t } ^ { 2 } }$ . Another special case is the variance-exploding diffusion process as used by [Song and Ermon, 2019, Song et al., 2021b], where $\\alpha _ { t } ^ { 2 } = 1$ . In experiments, we use the variance-preserving version. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The distributions $q ( \\mathbf { z } _ { t } | \\mathbf { z } _ { s } )$ for any $t > s$ are also Gaussian, and given in Appendix A. The joint distribution of latent variables $\\left( { \\bf z } _ { s } , { \\bf z } _ { t } , { \\bf z } _ { u } \\right)$ at any subsequent timesteps $0 \\leq s < t < u \\leq 1$ is Markov: $q ( { \\bf z } _ { u } | { \\bf z } _ { t } , { \\bf z } _ { s } ) = q ( { \\bf z } _ { u } | { \\bf z } _ { t } )$ . Given the distributions above, it is relatively straightforward to verify through Bayes rule that ${ \\dot { q } } ( \\mathbf { z } _ { s } | \\mathbf { z } _ { t } , \\mathbf { x } )$ , for any $0 \\leq s < t \\leq 1$ , is also Gaussian. This distribution is also given in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "3.2 Noise schedule ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In previous work, the noise schedule has a fixed form (see Appendix $_ \\mathrm { H }$ , Fig. 4a). In contrast, we learn this schedule through the parameterization ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/aa16f89f0a03c8700f277338b083396386ca76db374e77b4763ff04d61abd92f.jpg",
+        "text": "$$\n\\sigma _ { t } ^ { 2 } = \\mathrm { s i g m o i d } ( \\gamma _ { \\eta } ( t ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where $\\gamma _ { \\eta } ( t )$ is a monotonic neural network with parameters $\\eta$ , as detailed in Appendix H. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Motivated by the equivalence discussed in Section 5.1, we use $\\alpha _ { t } = \\sqrt { 1 - \\sigma _ { t } ^ { 2 } }$ in our experiments for both the discrete-time and continuous-time models, i.e. variance-preserving diffusion processes. It is straightforward to verify that $\\alpha _ { t } ^ { 2 }$ and $\\mathrm { S N R } ( t )$ , as a function of $\\gamma _ { \\eta } ( t )$ , then simplify to: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/a728470a3ed0293c40e301a481f73ad5a71918f6a9867c50562fa4c69aea4b39.jpg",
+        "text": "$$\n\\begin{array} { r } { \\alpha _ { t } ^ { 2 } = \\mathrm { s i g m o i d } ( - \\gamma _ { \\pm } ( t ) ) } \\\\ { \\mathrm { S N R } ( t ) = \\exp ( - \\gamma _ { \\pm } ( t ) ) } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "3.3 Reverse time generative model ",
+        "text_level": 1,
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We define our generative model by inverting the diffusion process of Section 3.1, yielding a hierarchical generative model that samples a sequence of latents $\\mathbf { z } _ { t }$ , with time running backward from $t = 1$ to $t = 0$ . We consider both the case where this sequence consists of a finite number of steps $T$ , as well as a continuous time model corresponding to $T \\to \\infty$ . We start by presenting the discrete-time case. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Given finite $T$ , we discretize time uniformly into $T$ timesteps (segments) of width $\\tau = 1 / T$ . Defining $s ( i ) = ( i - 1 ) / T$ and $t ( i ) = i / T$ , our hierarchical generative model for data $\\mathbf { x }$ is then given by: ",
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+        "type": "equation",
+        "img_path": "images/0265bf5f34e7e1c0f2fe8d2d2a5add5d94fb2c9edeef708fe9abf09a035341fa.jpg",
+        "text": "$$\np ( \\mathbf { x } ) = \\int _ { \\mathbf { z } } p ( \\mathbf { z } _ { 1 } ) p ( \\mathbf { x } | \\mathbf { z } _ { 0 } ) \\prod _ { i = 1 } ^ { T } p ( \\mathbf { z } _ { s ( i ) } | \\mathbf { z } _ { t ( i ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "With the variance preserving diffusion specification and sufficiently small SNR(1), we have that $q ( \\mathbf { z } _ { 1 } | \\mathbf { x } ) \\approx \\mathcal { N } ( \\mathbf { z } _ { 1 } ; \\bar { 0 , } \\mathbf { I } )$ . We therefore model the marginal distribution of $\\mathbf { z } _ { 1 }$ as a spherical Gaussian: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/5517076bc7ee33cdfea8b89966752b516b53d2915691e6aad0cecfa565443ac0.jpg",
+        "text": "$$\np ( \\mathbf { z } _ { 1 } ) = \\mathcal { N } ( \\mathbf { z } _ { 1 } ; 0 , \\mathbf { I } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We wish to choose a model $p ( \\mathbf { x } | \\mathbf { z } _ { 0 } )$ that is close to the unknown $q ( \\mathbf { x } | \\mathbf { z } _ { 0 } )$ . Let $x _ { i }$ and $z _ { 0 , i }$ be the $i$ -th elements of $\\mathbf { x } , \\mathbf { z } _ { 0 }$ , respectively. We then use a factorized distribution of the form: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2d40f4f66687471c8582e5e116f2a59ffdfbf4ed0be5c225984fa23e299abb97.jpg",
+        "text": "$$\np ( \\mathbf { x } | \\mathbf { z } _ { 0 } ) = \\prod _ { i } p ( x _ { i } | z _ { 0 , i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we choose $p ( x _ { i } | z _ { 0 , i } ) \\propto q ( z _ { 0 , i } | x _ { i } )$ , which is normalized by summing over all possible discrete values of $x _ { i }$ (256 in the case of 8-bit image data). With sufficiently large $\\mathrm { S N R } ( 0 )$ , this becomes a very close approximation to the true $q ( \\mathbf { x } | \\mathbf { z } _ { 0 } )$ , as the influence of the unknown data distribution $q ( \\mathbf { x } )$ is overwhelmed by the likelihood $q ( \\mathbf { z } _ { 0 } | \\mathbf { x } )$ . Finally, we choose the conditional model distributions as ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/54a8771e083272423ce790556b4c80c33330b06b7f65565fdc20bc7558e234d0.jpg",
+        "text": "$$\np ( \\mathbf { z } _ { s } | \\mathbf { z } _ { t } ) = q ( \\mathbf { z } _ { s } | \\mathbf { z } _ { t } , \\mathbf { x } = \\hat { \\mathbf { x } } _ { \\theta } \\big ( \\mathbf { z } _ { t } ; t \\big ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "i.e. the same as $q ( \\mathbf { z } _ { s } | \\mathbf { z } _ { t } , \\mathbf { x } )$ , but with the original data $\\mathbf { x }$ replaced by the output of a denoising model $\\hat { \\mathbf { x } } _ { \\pmb { \\theta } } ( \\mathbf { z } _ { t } ; t )$ that predicts $\\mathbf { x }$ from its noisy version $\\mathbf { z } _ { t }$ . Note that in practice we parameterize the denoising model as a function of a noise prediction model (Section 3.4), bridging the gap with previous work on diffusion models [Ho et al., 2020]. The means and variances of $p ( \\mathbf { z } _ { s } | \\mathbf { z } _ { t } )$ simplify to a remarkable degree; see Appendix A. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3.4 Noise prediction model and Fourier features ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We parameterize the denoising model in terms of a noise prediction model $\\hat { \\epsilon } _ { \\theta } ( { \\bf z } _ { t } ; t )$ ",
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+    {
+        "type": "equation",
+        "img_path": "images/a70b11166060330670096df79c80bbd54f3d1553c61151632a52d95ae2b4f1c4.jpg",
+        "text": "$$\n\\hat { \\mathbf { x } } _ { \\theta } ( \\mathbf { z } _ { t } ; t ) = ( \\mathbf { z } _ { t } - \\sigma _ { t } \\hat { \\epsilon } _ { \\theta } ( \\mathbf { z } _ { t } ; t ) ) / \\alpha _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $\\hat { \\epsilon } _ { \\theta } ( { \\bf z } _ { t } ; t )$ is parameterized as a neural network. The noise prediction models we use in experiments closely follow Ho et al. [2020], except that they process the data solely at the original resolution. The exact parameterization of the noise prediction model and noise schedule is discussed in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "Prior work on diffusion models has mainly focused on the perceptual quality of generated samples, which emphasizes coarse scale patterns and global consistency of generated images. Here, we optimize for likelihood, which is sensitive to fine scale details and exact values of individual pixels. To capture the fine scale details of the data, we propose adding a set of Fourier features to the input of our noise prediction model. Let $\\mathbf { x }$ be the original data, scaled to the range $[ - 1 , 1 ]$ , and let $\\mathbf { z }$ be the resulting latent variable, with similar magnitudes. We then append channels $\\sin ( 2 ^ { n } \\pi \\mathbf { z } )$ and $\\cos ( 2 ^ { n } \\pi \\mathbf { z } )$ , where $n$ runs over a range of integers $\\{ n _ { m i n } , . . . , n _ { m a x } \\}$ . These features are high frequency periodic functions that amplify small changes in the input data $\\mathbf { z } _ { t }$ ; see Appendix C for further details. Including these features in the input of our denoising model leads to large improvements in likelihood as demonstrated in Section 6 and Figure 5, especially when combined with a learnable SNR function. We did not observe such improvements when incorporating Fourier features into autoregressive models. ",
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+    {
+        "type": "text",
+        "text": "3.5 Variational lower bound ",
+        "text_level": 1,
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We optimize the parameters towards the variational lower bound (VLB) of the marginal likelihood, which is given by ",
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+        "type": "equation",
+        "img_path": "images/3765581cdd488094a61ced242a3bffdb33abeb0a0b60696a3d9d1d2bbab6578c.jpg",
+        "text": "$$\n\\begin{array} { r } { - \\log p ( \\mathbf { x } ) \\leq - \\mathrm { V L B } ( \\mathbf { x } ) = \\underbrace { D _ { K L } \\big ( q ( \\mathbf { z } _ { 1 } | \\mathbf { x } ) | | p ( \\mathbf { z } _ { 1 } ) \\big ) } _ { \\mathrm { P r i o r ~ l o s s } } + \\underbrace { { \\mathbb { E } _ { q ( \\mathbf { z } _ { 0 } | \\mathbf { x } ) } } \\left[ - \\log p ( \\mathbf { x } | \\mathbf { z } _ { 0 } ) \\right] } _ { \\mathrm { R e c o n s t u r c i t o n ~ l o s s } } + \\underbrace { \\mathcal { L } _ { T } ( \\mathbf { x } ) . } _ { \\mathrm { D i f f u s i o n ~ l o s s } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            147,
+            771,
+            184
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The prior loss and reconstruction loss can be (stochastically and differentiably) estimated using standard techniques; see [Kingma and Welling, 2013]. The diffusion loss, $\\mathcal { L } _ { T } ( \\mathbf { x } )$ , is more complicated, and depends on the hyperparameter $T$ that determines the depth of the generative model. ",
+        "bbox": [
+            174,
+            185,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Discrete-time model ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            246,
+            374,
+            263
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the case of finite $T$ , using $s ( i ) = ( i - 1 ) / T , t ( i ) = i / T$ , the diffusion loss is: ",
+        "bbox": [
+            173,
+            275,
+            697,
+            292
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b9abd6ab5f1fd76c7d9991f88bf36b7214821ea693bcff5d9df85bcf71949bbd.jpg",
+        "text": "$$\n\\mathcal { L } _ { T } ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { T } \\mathbb { E } _ { q ( \\mathbf { z } _ { t ( i ) } \\mid \\mathbf { x } ) } D _ { K L } \\big [ q ( \\mathbf { z } _ { s ( i ) } \\vert \\mathbf { z } _ { t ( i ) } , \\mathbf { x } ) \\vert \\big \\vert p ( \\mathbf { z } _ { s ( i ) } \\vert \\mathbf { z } _ { t ( i ) } ) \\big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            296,
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+            338
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In appendix $\\mathrm { E }$ we show that this expression simplifies considerably, yielding: ",
+        "bbox": [
+            173,
+            339,
+            679,
+            354
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f41c392645ea41e5d885ca9c8659c08d5d710e010fdeb32f5618608aafb6cdf2.jpg",
+        "text": "$$\n\\mathcal { L } _ { T } ( \\mathbf { x } ) = \\frac { T } { 2 } \\mathbb { E } _ { \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) , i \\sim U \\{ 1 , T \\} } \\left[ \\left( \\mathrm { S N R } ( s ) - \\mathrm { S N R } ( t ) \\right) | | \\mathbf { x } - \\hat { \\mathbf { x } } _ { \\theta } ( \\mathbf { z } _ { t } ; t ) | | _ { 2 } ^ { 2 } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            356,
+            741,
+            387
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $U \\{ 1 , T \\}$ is the uniform distribution on the integers $\\{ 1 , \\ldots , T \\}$ , and ${ \\bf z } _ { t } = \\alpha _ { t } { \\bf x } + \\sigma _ { t } \\epsilon$ . This is the general discrete-time loss for any choice of forward diffusion parameters $\\left( \\sigma _ { t } , \\alpha _ { t } \\right)$ . When plugging in the specifications of $\\sigma _ { t } , \\alpha _ { t }$ and $\\hat { \\mathbf { x } } _ { \\theta } ( \\mathbf { z } _ { t } ; t )$ that we use in experiments, given in Sections 3.2 and 3.4, the loss simplifies to: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9636fc4ce3941760c3b57db9af871727dc892c041dca94be2d149ea6be2b725.jpg",
+        "text": "$$\n\\mathcal { L } _ { T } ( \\mathbf { x } ) = \\frac { T } { 2 } \\mathbb { E } _ { \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) , i \\sim U \\lbrace 1 , T \\rbrace } \\left[ ( \\exp ( \\gamma _ { \\eta } ( t ) - \\gamma _ { \\eta } ( s ) ) - 1 ) \\| \\epsilon - \\hat { \\epsilon } _ { \\theta } ( \\mathbf { z } _ { t } ; t ) \\| _ { 2 } ^ { 2 } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            446,
+            754,
+            477
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { z } _ { t } = \\mathrm { s i g m o i d } ( - \\gamma _ { \\eta } ( t ) ) \\mathbf { x } + \\mathrm { s i g m o i d } ( \\gamma _ { \\eta } ( t ) ) \\pmb { \\epsilon }$ . In the discrete-time case, we simply jointly optimize $\\eta$ and $\\pmb \\theta$ by maximizing the VLB through a Monte Carlo estimator of Equation 14. ",
+        "bbox": [
+            171,
+            479,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that $\\exp ( . ) - 1$ has a numerically stable primitive $e x p m 1 ( . )$ in common numerical computing packages; see figure 6. Equation 14 allows for numerically stable implementation in 32-bit or lower-precision floating point, in contrast with previous implementations of discrete-time diffusion models (e.g. [Ho et al., 2020]), which had to resort to 64-bit floating point. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 More steps leads to a lower loss ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            585,
+            429,
+            601
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A natural question to ask is what the number of timesteps $T$ should be, and whether more timesteps is always better in terms of the VLB. In Appendix F we analyze the difference between the diffusion loss with $T$ timesteps, $\\mathcal { L } _ { T } ( \\mathbf { x } )$ , and the diffusion loss with double the timesteps, $\\mathcal { L } _ { 2 T } ( \\mathbf { x } )$ , while keeping the SNR function fixed. We then find that if our trained denoising model $\\hat { \\mathbf { x } } _ { \\pmb { \\theta } }$ is sufficiently good, we have that $\\mathcal { L } _ { 2 T } ( \\mathbf { x } ) < \\mathcal { L } _ { T } ( \\mathbf { x } )$ , i.e. that our VLB will be better for a larger number of timesteps. Intuitively, the discrete time diffusion loss is an upper Riemann sum approximation of an integral of a strictly decreasing function, meaning that a finer approximation yields a lower diffusion loss. This result is illustrated in Figure 2. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 Continuous-time model: $T \\to \\infty$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            741,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since taking more time steps leads to a better VLB, we now take $T \\to \\infty$ , effectively treating time $t$ as continuous rather than discrete. The model for $p ( \\mathbf { z } _ { t } )$ can in this case be described as a continuous time diffusion process [Song et al., 2021b] governed by a stochastic differential equation; see Appendix D. In Appendix E we show that in this limit the diffusion loss $\\mathcal { L } _ { T } ( \\mathbf { x } )$ simplifies further. Letting $\\mathrm { S N R } ^ { \\prime } ( t ) = d \\hat { \\mathrm { S N R } } ( t ) / d t$ , we have, with ${ \\bf z } _ { t } = \\alpha _ { t } { \\bf x } + \\sigma _ { t } \\epsilon$ : ",
+        "bbox": [
+            176,
+            770,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f21ba467fbcff601fac8f1632127982d0e70527bf3f8c0f014bf47211dabf533.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } _ { \\infty } ( \\mathbf { x } ) = - \\frac { 1 } { 2 } \\mathbb { E } _ { \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) } \\int _ { 0 } ^ { 1 } \\mathrm { S N R } ^ { \\prime } ( t ) \\left\\| \\mathbf { x } - \\hat { \\mathbf { x } } _ { \\theta } ( \\mathbf { z } _ { t } ; t ) \\right\\| _ { 2 } ^ { 2 } d t , } \\\\ { \\displaystyle \\quad \\quad = - \\frac { 1 } { 2 } \\mathbb { E } _ { \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) , t \\sim \\mathcal { U } ( 0 , 1 ) } \\left[ \\mathrm { S N R } ^ { \\prime } ( t ) \\left\\| \\mathbf { x } - \\hat { \\mathbf { x } } _ { \\theta } ( \\mathbf { z } _ { t } ; t ) \\right\\| _ { 2 } ^ { 2 } \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/a1bce51bea03c5d5f7bc9253564d5e084a5720fd426ee069732cd96a8af54609.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the diffusion loss with few segments $T$ (left), more segments $T$ (middle), and infinite segments $T$ (continuous time, right). The continuous-time loss (Equation 18) is an integral of mean squared error (MSE) over SNR, here visualized as a black curve. The black curve is strictly decreasing when the model is sufficiently well trained, so the discrete-time loss (Equation 13) is an upper bound (an upper Riemann sum approximation) of this integral that becomes better when segments are added. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            89,
+            785,
+            172
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This is the general continuous-time loss for any choice of forward diffusion parameters $\\left( \\sigma _ { t } , \\alpha _ { t } \\right)$ . When plugging in the specifications of $\\sigma _ { t } , \\alpha _ { t }$ and $\\hat { \\mathbf { x } } _ { \\pmb { \\theta } } ( \\mathbf { z } _ { t } ; t )$ that we use in experiments, given in Sections 3.2 and 3.4, the loss simplifies to: ",
+        "bbox": [
+            173,
+            295,
+            826,
+            337
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcb401c73561a782be762526e575c4e192508160ffd7f7554d0e65ba2347bca4.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\infty } ( \\mathbf { x } ) = \\frac { 1 } { 2 } \\mathbb { E } _ { \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) , t \\sim \\mathcal { U } ( 0 , 1 ) } \\left[ \\gamma _ { \\eta } ^ { \\prime } ( t ) \\left. \\epsilon - \\hat { \\epsilon } _ { \\theta } ( \\mathbf { z } _ { t } ; t ) \\right. _ { 2 } ^ { 2 } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            340,
+            687,
+            371
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma _ { \\pmb { \\eta } } ^ { \\prime } ( t ) = d \\gamma _ { \\pmb { \\eta } } ( t ) / d t$ . We use the Monte Carlo estimator of this loss for evaluation and optimization. ",
+        "bbox": [
+            174,
+            373,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 Equivalence of diffusion models in continuous time ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            417,
+            566,
+            433
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The signal-to-noise function $\\mathrm { S N R } ( t )$ is invertible due to the monotonicity assumption in Section 3.1. Due to this invertibility, we can perform a change of variables, and make everything a function of $v \\equiv \\operatorname { S N R } ( t )$ instead of $t$ , such that $t = \\mathrm { S N R } ^ { - \\mathrm { \\bar { 1 } } } ( v )$ . Let $\\alpha _ { v }$ and $\\sigma _ { v }$ be the functions $\\alpha _ { t }$ and $\\sigma _ { t }$ evaluated at $t = \\mathrm { S N R } ^ { - 1 } ( v )$ , and correspondingly let $\\mathbf { z } _ { v } = \\alpha _ { v } \\mathbf { x } + \\sigma _ { v } \\mathbf { \\epsilon }$ . Similarly, we rewrite our noise prediction model as $\\tilde { \\mathbf { x } } _ { \\pmb { \\theta } } ( \\mathbf { z } , v ) \\equiv \\hat { \\mathbf { x } } _ { \\pmb { \\theta } } ( \\mathbf { z } , \\mathrm { S N R } ^ { - 1 } ( v ) )$ . With this change of variables, our continuous-time loss in Equation 15 can equivalently be written as: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1ea0d33cdda1e38c8790ba5d691ec6795ab8e6699a09232de3250c9dc95eb69.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\infty } ( \\mathbf { x } ) = \\frac { 1 } { 2 } \\mathbb { E } _ { \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) } \\int _ { \\mathrm { S N R } _ { \\operatorname* { m i n } } } ^ { \\mathrm { S N R } _ { \\operatorname* { m a x } } } \\left\\| \\mathbf { x } - \\tilde { \\mathbf { x } } _ { \\pmb { \\theta } } ( \\mathbf { z } _ { v } , v ) \\right\\| _ { 2 } ^ { 2 } d v ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            535,
+            674,
+            573
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where instead of integrating w.r.t. time $t$ we now integrate w.r.t. the signal-to-noise ratio $v$ , and where $\\mathrm { S N R } _ { \\operatorname* { m i n } } = \\mathrm { S N R } ( 1 )$ and $\\mathrm { S N R } _ { \\operatorname* { m a x } } = \\mathrm { S N R } ( 0 )$ . ",
+        "bbox": [
+            174,
+            575,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "What this equation shows us is that the only effect the functions $\\alpha ( t )$ and $\\sigma ( t )$ have on the diffusion loss is through the values $\\mathrm { S N R } ( t ) = \\alpha _ { t } ^ { 2 } / \\sigma _ { t } ^ { 2 }$ at endpoints $t = 0$ and $t = 1$ . Given these values $\\operatorname { S N R } _ { \\operatorname* { m a x } }$ and $\\mathrm { { S N R } _ { \\operatorname* { m i n } } }$ , the diffusion loss is invariant to the shape of function $\\mathrm { S N R } ( t )$ between $t = 0$ and $t = 1$ . The VLB is thus only impacted by the function $\\mathrm { S N R } ( t )$ through its endpoints $\\mathrm { { S N R } _ { \\operatorname* { m i n } } }$ and $\\operatorname { S N R } _ { \\operatorname* { m a x } }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we find that the distribution $p ( \\mathbf { x } )$ defined by our generative model is also invariant to the specification of the diffusion process. Specifically, let $p ^ { A } ( \\mathbf { \\bar { x } } )$ denote the distribution defined by the combination of a diffusion specification and denoising function $\\{ \\alpha _ { v } ^ { A } , \\sigma _ { v } ^ { A } , \\tilde { \\mathbf { x } } _ { \\theta } ^ { A } \\}$ , and similarly let $p ^ { B } ( \\mathbf { x } )$ {be the distribution defined through a different specification $\\{ \\alpha _ { v } ^ { B } , \\sigma _ { v } ^ { B } , \\tilde { \\mathbf { x } } _ { \\theta } ^ { B } \\}$ , where both specifications have equal $\\mathrm { { S N R } _ { \\operatorname* { m i n } } }$ , $\\operatorname { S N R } _ { \\operatorname* { m a x } }$ ; as shown in Appendix G, we then have that $p ^ { A } ( \\mathbf { x } ) =$ $p ^ { \\mathbf { \\hat { \\cal B } } } ( \\mathbf { x } )$ if $\\tilde { \\mathbf { x } } _ { \\pmb { \\theta } } ^ { B } ( \\mathbf { z } , v ) \\equiv \\hat { \\tilde { \\mathbf { x } } } _ { \\pmb { \\theta } } ^ { A } ( ( \\alpha _ { v } ^ { A } / \\alpha _ { v } ^ { B } ) \\mathbf { z } , v )$ . The distribution on all latents $\\mathbf { z } _ { v }$ is then also the same under both specifications, up to a trivial rescaling. This means that any two diffusion models satisfying the mild constraints set in 3.1 (which includes e.g. the variance exploding and variance preserving specifications considered by Song et al. [2021b]), can thus be seen as equivalent in continuous time. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 Weighted diffusion loss ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This equivalence between diffusion specifications continues to hold even if, instead of the VLB, these models optimize a weighted diffusion loss of the form: ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/190a95e67e62441adafd4290585d1bbcab984b1628fe3c35779d2f6ba8a57564.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\infty } ( \\mathbf { x } , w ) = \\frac { 1 } { 2 } \\mathbb { E } _ { \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) } \\int _ { \\mathrm { S N R } _ { \\operatorname* { m i n } } } ^ { \\mathrm { S N R } _ { \\operatorname* { m a x } } } \\ d w ( v ) \\left. \\mathbf { x } - \\tilde { \\mathbf { x } } _ { \\theta } ( \\mathbf { z } _ { v } , v ) \\right. _ { 2 } ^ { 2 } d v ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/a9a8b115479135194e6d7dc7e3697749211aeb8bb6bf44e50de44c9b7293d1b7.jpg",
+        "table_caption": [
+            "Table 1: Summary of our findings for density modeling tasks, in terms of bits per dimension (BPD) on the test set. Model types are autoregressive (AR), normalizing flows (Flow), variational autoencoders (VAE), or diffusion models (Diff). Our results were obtained using the continuous-time formulation of our model. CIFAR-10 data augmentations are: (A) extensive, (B) small translations, or (C) horizontal flips. The numbers for VDM are variational bounds, and can likely be improved by estimating the marginal likelihood through importance sampling, or through evaluation of the corresponding continuous normalizing flow as done by Song et al. [2021a]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model (Bits per dim on test set)</td><td>Type</td><td>CIFAR10 no data aug.</td><td>CIFAR10 data aug.</td><td>ImageNet 32x32</td><td>ImageNet 64x64</td></tr><tr><td>Previous work</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ResNet VAE with IAF [Kingma et al., 2016]</td><td>VAE</td><td>3.11</td><td></td><td></td><td></td></tr><tr><td>Very Deep VAE [Child,2020]</td><td>VAE</td><td>2.87</td><td></td><td>3.80</td><td>3.52</td></tr><tr><td>NVAE [Vahdat and Kautz,2020]</td><td>VAE</td><td>2.91</td><td></td><td>3.92</td><td></td></tr><tr><td>Glow [Kingma and Dhariwal, 2018]</td><td>Flow</td><td></td><td>3.35(B)</td><td>4.09</td><td>3.81</td></tr><tr><td>Flow++ [Ho et al.,2019a]</td><td>Flow</td><td>3.08</td><td></td><td>3.86</td><td>3.69</td></tr><tr><td>PixelCNN [Van Oord et al., 2016]</td><td>AR</td><td>3.03</td><td></td><td>3.83</td><td>3.57</td></tr><tr><td>PixelCNN++ [Salimans et al.,2017]</td><td>AR</td><td>2.92</td><td></td><td></td><td></td></tr><tr><td>Image Transformer [Parmar et al., 2018]</td><td>AR</td><td>2.90</td><td></td><td>3.77</td><td></td></tr><tr><td>SPN [Menick and Kalchbrenner, 2018]</td><td>AR</td><td></td><td></td><td></td><td>3.52</td></tr><tr><td>Sparse Transformer [Child et al.,2019]</td><td>AR</td><td>2.80</td><td></td><td></td><td>3.44</td></tr><tr><td>Routing Transformer [Roy et al., 2021]</td><td>AR</td><td></td><td></td><td></td><td>3.43</td></tr><tr><td>Sparse Transformer + DistAug [Jun et al.,2020]</td><td>AR</td><td></td><td>2.53(4)</td><td></td><td></td></tr><tr><td>DDPM [Ho et al., 2020]</td><td>Diff</td><td></td><td>3.69(C)</td><td></td><td></td></tr><tr><td>EBM-DRL [Gao et al., 2020]</td><td>Diff</td><td></td><td>3.18(C)</td><td></td><td></td></tr><tr><td>Score SDE [Song et al., 2021b]</td><td>Diff</td><td>2.99</td><td></td><td></td><td></td></tr><tr><td>Improved DDPM [Nichol and Dhariwal, 2021]</td><td>Diff</td><td>2.94</td><td></td><td></td><td>3.54</td></tr><tr><td>Concurrent work</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CR-NVAE [Sinha and Dieng,2021]</td><td>VAE</td><td></td><td>2.51(4)</td><td></td><td></td></tr><tr><td>LSGM [Vahdat et al., 2021]</td><td>Diff</td><td>2.87</td><td></td><td></td><td></td></tr><tr><td>ScoreFlow [Song et al.,2O21a] (variational bound)</td><td>Diff</td><td></td><td>2.90(C)</td><td>3.86</td><td></td></tr><tr><td>ScoreFlow [Song et al.,202la] (cont. norm. flow)</td><td>Diff</td><td>2.83</td><td>2.80(C)</td><td>3.76</td><td></td></tr><tr><td>Our work</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>VDM (variational bound)</td><td>Diff</td><td>2.65</td><td>2.49(A)</td><td>3.72</td><td>3.40</td></tr></table>",
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "which e.g. captures all the different objectives discussed by Song et al. [2021b], see Appendix K. Here, $w ( v )$ is a weighting function that generally puts increased emphasis on the noisier data compared to the VLB, and which thereby can sometimes improve perceptual generation quality as measured by certain metrics like FID and Inception Score. For the models presented in this paper, we further use $w ( v ) = 1$ as corresponding to the (unweighted) VLB. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 Variance minimization ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lowering the variance of the Monte Carlo estimator of the continuous-time loss generally improves the efficiency of optimization. We found that using a low-discrepancy sampler for $t$ , as explained in Appendix I.1, leads to a significant reduction in variance. In addition, due to the invariance shown in Section 5.1 for the continous-time case, we can optimize the schedule between its endpoints w.r.t. to minimize the variance of our estimator of loss, as detailed in Appendix I. The endpoints of the noise schedule are simply optimized w.r.t. the VLB. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We demonstrate our proposed class of diffusion models, which we call Variational Diffusion Models (VDMs), on the CIFAR-10 [Krizhevsky et al., 2009] dataset, and the downsampled ImageNet [Van Oord et al., 2016, Deng et al., 2009] dataset, where we focus on maximizing likelihood. For our result with data augmentation we used random flips, 90-degree rotations, and color channel swapping. More details of our model specifications are in Appendix B. ",
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+    {
+        "type": "image",
+        "img_path": "images/54d231c9ca10990f784c07d9d3cd51bc63cfe05ce3e48f5ad4ff0710f43df11c.jpg",
+        "image_caption": [
+            "Figure 3: Non cherry-picked unconditional samples from our Imagenet $6 4 \\mathrm { x } 6 4$ model, trained in continuous time and generated using $T = 1 0 0 0$ . The model’s hyper-parameters and parameters are optimized w.r.t. the likelihood bound, so the model is not optimized for synthesis quality. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "6.1 Likelihood and samples ",
+        "text_level": 1,
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Table 1 shows our results on modeling the CIFAR-10 dataset, and the downsampled ImageNet dataset. We establish a new state-of-the-art in terms of test set likelihood on all considered benchmarks, by a significant margin. Our model for CIFAR-10 without data augmentation surpasses the previous best result of 2.80 about $1 0 \\mathrm { x }$ faster than it takes the Sparse Transformer to reach this, in wall clock time on equivalent hardware. Our CIFAR-10 model, whose hyper-parameters were tuned for likelihood, results in a FID (perceptual quality) score of 7.41. This would have been state-of-the-art until recently, but is worse than recent diffusion models that specifically target FID scores [Nichol and Dhariwal, 2021, Song et al., 2021b, Ho et al., 2020]. By instead using a weighted diffusion loss, with the weighting function $w ( \\mathbf { S N R } )$ used by Ho et al. [2020] and described in Appendix K, our FID score improves to 4.0. We did not pursue further tuning of the model to improve FID instead of likelihood. A random sample of generated images from our model is provided in Figure 3. We provide additional samples from this model, as well as our other models for the other datasets, in Appendix M. ",
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+    {
+        "type": "text",
+        "text": "6.2 Ablations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Next, we investigate the relative importance of our contributions. In Table 2 we compare our discretetime and continuous-time specifications of the diffusion model: When evaluating our model with a small number of steps, our discretely trained models perform better by learning the diffusion schedule to optimize the VLB. However, as argued theoretically in Section 4.1, we find experimentally that more steps $T$ indeed gives better likelihood. When $T$ grows large, our continuously trained model performs best, helped by training its diffusion schedule to minimize variance instead. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Minimizing the variance also helps the continuous time model to train faster, as shown in Figure 5. This effect is further examined in Table 4b, where we find dramatic variance reductions compared to our baselines in continuous time. Figure 4a shows how this effect is achieved: Compared to the other schedules, our learned schedule spends much more time in the high $\\mathrm { S N R } ( t )$ / low $\\bar { \\sigma } _ { t } ^ { 2 }$ range. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In Figure 5 we further show training curves for our model including and excluding the Fourier features proposed in Appendix C: with Fourier features enabled our model achieves much better likelihood. For comparison we also implemented Fourier features in a PixelCNN++ model [Salimans et al., 2017], where we do not see a benefit. In addition, we find that learning the SNR is necessary to get the most out of including Fourier features: if we fix the SNR schedule to that used by Ho et al. [2020], the maximum log-SNR is fixed to approximately 8 (see figure 7), and test set negative likelihood stays above 4 bits per dim. When learning the SNR endpoints, our maximum log-SNR ends up at 13.3, which, combined with the inclusion of Fourier features, leads to the SOTA test set likelihoods reported in Table 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6.3 Lossless compression ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For a fixed number of evaluation timesteps $T _ { e v a l }$ , our diffusion model in discrete time is a hierarchical latent variable model that can be turned into a lossless compression algorithm using bits-back coding [Hinton and Van Camp, 1993]. As a proof of concept of practical lossless compression, Table 2 reports net codelengths on the CIFAR10 test set for various settings of $T _ { e v a l }$ using BB",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/aaa5a5353d4ef15cbf77343aa7eddc81de46e3564641ccc5decb6dc68551eb38.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Variance of VLB estimate "
+        ],
+        "table_body": "<table><tr><td>SNR(t) schedule</td><td>Var(BPD)</td></tr><tr><td>Learned (ours)</td><td>0.53</td></tr><tr><td>log SNR-linear</td><td>6.35</td></tr><tr><td>β-Linear [1]</td><td>31.6</td></tr><tr><td>α-Cosine [2]</td><td>31.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c24191f17672e99720cf2abbe47c7f69d11815f1effee8dfcf8e9806285e04b4.jpg",
+        "image_caption": [
+            "Figure 4: Our learned continuous-time variance-minimizing noise schedule $\\mathrm { S N R } ( t )$ for CIFAR-10, compared to its log-linear initialization and to schedules from the literature: [1] The $\\beta$ -Linear schedule from Ho et al. [2020], [2] The $\\alpha$ -Cosine schedule from Nichol and Dhariwal [2021]. All schedules were scaled and shifted on the log scale such that the resulting $\\mathrm { { S N R } _ { \\operatorname* { m i n } } }$ , $\\operatorname { S N R } _ { \\operatorname* { m a x } }$ were the equal to our learned endpoints, resulting in the same VLB estimate of 2.66. We report the variance of our VLB estimate per data point, computed on the test set, and conditional on the data: This does not include the noise due to sampling minibatches of data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/496993b59ec80b16fa97e4fc4f5017457597ef46565d0a3ef842f08e4c9af8ba.jpg",
+        "table_caption": [
+            "Table 2: Discrete versus continuous-time training and evaluation with CIFAR-10, in terms of bits per dimension (BPD). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Ttrain</td><td>Teual</td><td>BPD</td><td>Bits-Back Net BPD</td></tr><tr><td>10</td><td>10</td><td>4.31</td><td></td></tr><tr><td>100</td><td>100</td><td>2.84</td><td></td></tr><tr><td>250</td><td>250</td><td>2.73</td><td></td></tr><tr><td>500</td><td>500</td><td>2.68</td><td></td></tr><tr><td>1000</td><td>1000</td><td>2.67</td><td></td></tr><tr><td>10000</td><td>10000</td><td>2.66</td><td></td></tr><tr><td>8</td><td>10</td><td>7.54</td><td>7.54</td></tr><tr><td>8</td><td>100</td><td>2.90</td><td>2.91</td></tr><tr><td>8</td><td>250</td><td>2.74</td><td>2.76</td></tr><tr><td>8</td><td>500</td><td>2.69</td><td>2.72</td></tr><tr><td>8</td><td>1000</td><td>2.67</td><td>2.72</td></tr><tr><td>8</td><td>10000</td><td>2.65</td><td></td></tr><tr><td>8</td><td>8</td><td>2.65</td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9a6535ca7cea624ad43cc6e8acc7d7420e607f45e1ba25f3ff2279b3f38b92c.jpg",
+        "image_caption": [
+            "Figure 5: Test set likelihoods during training, with/without Fourier features, and with/without learning the noise schedule to minimize variance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ANS [Townsend et al., 2018], an implementation of bits-back coding based on asymmetric numeral systems [Duda, 2009]. Details of our implementation are given in Appendix N. We achieve state-ofthe-art net codelengths, proving our model can be used as the basis of a lossless compression algorithm. However, for large $T _ { e v a l }$ a gap remains with the theoretically optimal codelength corresponding to the negative VLB, and compression becomes computationally expensive due to the large number of neural network forward passes required. Closing this gap with more efficient implementations of bits-back coding suitable for very deep models is an interesting avenue for future work. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We presented state-of-the-art results on modeling the density of natural images using a new class of diffusion models that incorporates a learnable diffusion specification, Fourier features for fine-scale modeling, as well as other architectural innovations. In addition, we obtained new theoretical insight into likelihood-based generative modeling with diffusion models, showing a surprising invariance of the VLB to the forward time diffusion process in continuous time, as well as an equivalence between various diffusion processes from the literature previously thought to be different. ",
+        "bbox": [
+            174,
+            760,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            864,
+            328,
+            881
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Yang Song, Kevin Murphy and Mohammad Norouzi for feedback on drafts of this paper. ",
+        "bbox": [
+            173,
+            897,
+            818,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            267,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nanxin Chen, Yu Zhang, Heiga Zen, Ron J. Weiss, Mohammad Norouzi, and William Chan. Wavegrad: Estimating gradients for waveform generation. arXiv preprint arXiv:2009.00713, 2020. ",
+        "bbox": [
+            171,
+            113,
+            826,
+            142
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rewon Child. Very deep VAEs generalize autoregressive models and can outperform them on images. arXiv preprint arXiv:2011.10650, 2020. ",
+        "bbox": [
+            173,
+            151,
+            825,
+            180
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. arXiv preprint arXiv:1904.10509, 2019. ",
+        "bbox": [
+            173,
+            188,
+            823,
+            217
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009. ",
+        "bbox": [
+            173,
+            226,
+            823,
+            268
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jarek Duda. Asymmetric numeral systems. arXiv preprint arXiv:0902.0271, 2009. ",
+        "bbox": [
+            171,
+            276,
+            717,
+            294
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ruiqi Gao, Yang Song, Ben Poole, Ying Nian Wu, and Diederik P Kingma. Learning energy-based models by diffusion recovery likelihood. arXiv preprint arXiv:2012.08125, 2020. ",
+        "bbox": [
+            169,
+            300,
+            823,
+            330
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey E Hinton and Drew Van Camp. Keeping the neural networks simple by minimizing the description length of the weights. In Proceedings of the Sixth Annual Conference on Computational Learning Theory, pages 5–13, 1993. ",
+        "bbox": [
+            174,
+            338,
+            823,
+            382
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Ho, Xi Chen, Aravind Srinivas, Yan Duan, and Pieter Abbeel. Flow $^ { + + }$ : Improving flowbased generative models with variational dequantization and architecture design. In International Conference on Machine Learning, pages 2722–2730. PMLR, 2019a. ",
+        "bbox": [
+            173,
+            390,
+            825,
+            434
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Ho, Evan Lohn, and Pieter Abbeel. Compression with flows via local bits-back coding. In Advances in Neural Information Processing Systems, pages 3874–3883, 2019b. ",
+        "bbox": [
+            169,
+            440,
+            823,
+            470
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. arXiv preprint arXiv:2006.11239, 2020. ",
+        "bbox": [
+            171,
+            478,
+            823,
+            507
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chin-Wei Huang, Jae Hyun Lim, and Aaron Courville. A variational perspective on diffusion-based generative models and score matching. arXiv preprint arXiv:2106.02808, 2021. ",
+        "bbox": [
+            171,
+            516,
+            823,
+            546
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Heewoo Jun, Rewon Child, Mark Chen, John Schulman, Aditya Ramesh, Alec Radford, and Ilya Sutskever. Distribution augmentation for generative modeling. In International Conference on Machine Learning, pages 5006–5019. PMLR, 2020. ",
+        "bbox": [
+            178,
+            554,
+            823,
+            598
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. arXiv preprint arXiv:1710.10196, 2017. ",
+        "bbox": [
+            169,
+            604,
+            823,
+            635
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. arXiv preprint arXiv:1807.03039, 2018. ",
+        "bbox": [
+            171,
+            642,
+            823,
+            672
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Max Welling. Auto-encoding variational Bayes. International Conference on Learning Representations, 2013. ",
+        "bbox": [
+            169,
+            680,
+            825,
+            710
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improved variational inference with inverse autoregressive flow. In Advances in Neural Information Processing Systems, pages 4743–4751, 2016. ",
+        "bbox": [
+            174,
+            718,
+            825,
+            762
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Friso Kingma, Pieter Abbeel, and Jonathan Ho. Bit-swap: Recursive bits-back coding for lossless compression with hierarchical latent variables. In International Conference on Machine Learning, pages 3408–3417. PMLR, 2019. ",
+        "bbox": [
+            174,
+            770,
+            825,
+            813
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. Diffwave: A versatile diffusion model for audio synthesis. arXiv preprint arXiv:2009.09761, 2020. ",
+        "bbox": [
+            173,
+            820,
+            821,
+            851
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009. ",
+        "bbox": [
+            171,
+            858,
+            823,
+            875
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017. ",
+        "bbox": [
+            171,
+            882,
+            826,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "David JC MacKay. Information theory, inference and learning algorithms. Cambridge university press, 2003. ",
+        "bbox": [
+            171,
+            92,
+            823,
+            121
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jacob Menick and Nal Kalchbrenner. Generating high fidelity images with subscale pixel networks and multidimensional upscaling. arXiv preprint arXiv:1812.01608, 2018. ",
+        "bbox": [
+            171,
+            128,
+            823,
+            159
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Alex Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. arXiv preprint arXiv:2102.09672, 2021. ",
+        "bbox": [
+            171,
+            165,
+            825,
+            195
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016. ",
+        "bbox": [
+            174,
+            203,
+            823,
+            247
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Image transformer. In International Conference on Machine Learning, pages 4055–4064. PMLR, 2018. ",
+        "bbox": [
+            176,
+            255,
+            825,
+            297
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Danilo J Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning, pages 1278–1286, 2014. ",
+        "bbox": [
+            173,
+            306,
+            823,
+            349
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computerassisted intervention, pages 234–241. Springer, 2015. ",
+        "bbox": [
+            174,
+            358,
+            825,
+            401
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aurko Roy, Mohammad Saffar, Ashish Vaswani, and David Grangier. Efficient content-based sparse attention with routing transformers. Transactions of the Association for Computational Linguistics, 9:53–68, 2021. ",
+        "bbox": [
+            173,
+            409,
+            825,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelcnn $^ { + + }$ : Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017. ",
+        "bbox": [
+            171,
+            460,
+            825,
+            503
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Samarth Sinha and Adji B Dieng. Consistency regularization for variational auto-encoders. arXiv preprint arXiv:2105.14859, 2021. ",
+        "bbox": [
+            173,
+            512,
+            823,
+            541
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, pages 2256–2265, 2015. ",
+        "bbox": [
+            173,
+            549,
+            823,
+            593
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020. ",
+        "bbox": [
+            169,
+            601,
+            825,
+            631
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In Advances in Neural Information Processing Systems, pages 11895–11907, 2019. ",
+        "bbox": [
+            169,
+            638,
+            825,
+            669
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yang Song and Stefano Ermon. Improved techniques for training score-based generative models. In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020. ",
+        "bbox": [
+            173,
+            676,
+            826,
+            733
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yang Song, Conor Durkan, Iain Murray, and Stefano Ermon. Maximum likelihood training of score-based diffusion models. arXiv e-prints, pages arXiv–2101, 2021a. ",
+        "bbox": [
+            169,
+            742,
+            825,
+            772
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-Based Generative Modeling Through Stochastic Differential Equations. In International Conference on Learning Representations, 2021b. ",
+        "bbox": [
+            176,
+            779,
+            823,
+            823
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. arXiv preprint arXiv:1409.3215, 2014. ",
+        "bbox": [
+            169,
+            832,
+            825,
+            861
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Matthew Tancik, Pratul P Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. arXiv preprint arXiv:2006.10739, 2020. ",
+        "bbox": [
+            174,
+            868,
+            826,
+            912
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "James Townsend, Thomas Bird, and David Barber. Practical lossless compression with latent variables using bits back coding. In International Conference on Learning Representations, 2018.   \nJames Townsend, Thomas Bird, Julius Kunze, and David Barber. HiLLoC: lossless image compression with hierarchical latent variable models. In International Conference on Learning Representations, 2020.   \nArash Vahdat and Jan Kautz. Nvae: A deep hierarchical variational autoencoder. arXiv preprint arXiv:2007.03898, 2020.   \nArash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. arXiv preprint arXiv:2106.05931, 2021.   \nAaron Van Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In International Conference on Machine Learning, pages 1747–1756, 2016.   \nPascal Vincent. A connection between score matching and denoising autoencoders. Neural computation, 23(7):1661–1674, 2011.   \nFisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, and Jianxiong Xiao. Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015. ",
+        "bbox": [
+            168,
+            90,
+            828,
+            375
+        ],
+        "page_idx": 11
+    }
+]

parse/train/BJfvAoC9YQ/BJfvAoC9YQ.md

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+# FEATURE TRANSFORMERS: A UNIFIED REPRESENTATION LEARNING FRAMEWORK FOR LIFELONG LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Despite the recent advances in representation learning, lifelong learning continues to be one of the most challenging and unconquered problems. Catastrophic forgetting and data privacy constitute two of the important challenges for a successful lifelong learner. Further, existing techniques are designed to handle only specific manifestations of lifelong learning, whereas a practical lifelong learner is expected to switch and adapt seamlessly to different scenarios. In this paper, we present a single, unified mathematical framework for handling the myriad variants of lifelong learning, while alleviating these two challenges. We utilize an external memory to store only the features representing past data and learn richer and newer representations incrementally through transformation neural networks - feature transformers. We define, simulate and demonstrate exemplary performance on a realistic lifelong experimental setting using the MNIST rotations dataset, paving the way for practical lifelong learners. To illustrate the applicability of our method in data sensitive domains like healthcare, we study the pneumothorax classification problem from X-ray images, achieving near gold standard performance. We also benchmark our approach with a number of state-of-the art methods on MNIST rotations and iCIFAR100 datasets demonstrating superior performance.
+
+# 1 INTRODUCTION
+
+Deep learning algorithms have achieved tremendous success on various challenging tasks like object detection, language translation, medical image segmentation, etc. Lifelong learning - the ability to adapt, benefit and sustain performance post deployment with more data and feedback, is an important goal for artificial intelligence (AI) and extremely crucial for sustained utility of these algorithms in domains like healthcare (learning from rare cases), self driving (learning new object detectors), etc. While there is no single standard definition of lifelong learning, most of the research in this field can be classified into one of the following sub-categories:
+
+1. Incremental learning - encountering new data, with no change in distribution, for same task   
+2. Domain adaptation - data from modified target distributions but for the same task   
+3. New-task learning - data from tasks that were not presented before
+
+Majority of the successful techniques study these variants in isolation. However, a realistic lifelong learning scenario would involve a mixture of these manifestations over time. Another major impediment to a successful lifelong learner is data privacy. In domains like healthcare, it is near impossible to have data access beyond scope (both time and geography) which leads to catastophic forgetting (McCloskey & Cohen, 1989)- the inability of machine learning models to retain past knowledge while learning from new data. In this paper, we provide a unified framework - feature transformers - for practical lifelong learning to handle data privacy and catastrophic forgetting.
+
+In summary, the major contributions of our work are as follows:
+
+• Define a realistic and a challenging lifelong learning scenario and formulate a unique and generic mathematical framework - feature transformer, to work in this setting successfully. • Ensure data privacy by storing only features from previous episodes while successfully combating catastrophic forgetting.
+
+• Principled way of utilizing additional neural-compute to tackle complex incremental tasks. • Demonstrate state-of-the-art results on common continuous learning benchmarks for new task learning like MNIST rotations, iCIFAR dataset and a challenging healthcare problem. Demonstrate exemplary performance even under severe constraints on memory and compute, thus operating under the assumptions of practical lifelong learning.
+
+Following review of related work in section 2, we describe the mathematical formulation of feature transformers and their practical realization in section 3. Section 4 contains experiments and results, followed by discussion in section 5.
+
+# 2 RELATED WORK
+
+A conventional deep learning classifier can be viewed as an automatic feature extractor followed by a classifier, trained jointly. For lifelong learning, the methods in literature can be broadly classified into: 1) learning with fixed feature representations and 2) incrementally evolving representations.
+
+Fixed Represenations: In this class of approaches, a representation for the data is learnt from the initial task and remains frozen for ensuing tasks, while only the classification layers are modified. The simplest baseline approaches include fine-tuning (Oquab et al., 2014) and Mensink et al. (2012) where the idea of using fixed representations was extended further by using the nearest class mean classifier. In spite of their simplicity, these algorithms do not perform well in practice due to the limiting constraint of a fixed representation throughout the incremental learning phase.
+
+Incrementally evolving representations: The last few years have witnessed renewed interest in developing methods that allow changing data representation with the addition of new tasks. Naive methods of retraining with only new data suffer from catastrophic forgetting (McCloskey & Cohen, 1989). To overcome this problem, most methods have attempted different manifestations of rehearsal (Robins, 1995) - replaying data from the previous tasks. However, access to all the previous data is usually not feasible due to size, compute and privacy concerns, researchers have attempted to reproduce past through proxy information, known as pseudorehearsal. Pseudorehearsal techniques can be further classified depending on how the previous information is stored /used:
+
+• Knowledge distillation based approaches: An example method for lifelong learning using knowledge distillation (Hinton et al., 2015) is learning without forgetting (LwF) (Li & Hoiem, 2017) - where distillation loss was added to match output of the updated network to that of the old network on the old task output variables. LwF and its extensions does not scale to a large number of new tasks, suffers from catastrophic forgetting and importantly do not address incremental learning or domain adapatation settings.
+
+• Rehearsal using exemplar sets: Rebuffi et al. (2017) propose iCaRL, an incremental learning approach by storing an exemplar set of data from the previous tasks and augmenting it with the new data. In more recent works, Javed & Paracha (2018) and Castro et al. (2018) have argued that a decoupled nearest mean classifier from Rebuffi et al. (2017) is not essential and have proposed joint learning of feature extraction and classification. Lopez-Paz et al. (2017) propose Gradient Episodic Memory, a technique which stores data from previous classes and constrains the gradient update while learning new tasks.
+
+• Rehearsal using generative models: Triki Rannen et al. (2017) propose to reproduce past knowledge by using task-specific under-complete autoencoders. When a new task is presented, optimization is constrained by preventing the reconstructions of these autoencoders from changing, thereby ensuring that the features necessary for the previous tasks are not destroyed. Shin et al. (2017); He et al. (2018); Venkatesan et al. (2017) employ generative adversarial networks for recreating the history from previous tasks.
+
+• Network regularization strategies: Methods belonging to this class aim to identify and selectively modify parts of the neural network which are critical to remember past knowledge or by explicitly penalizing loss of performance on old tasks. Kirkpatrick et al. (2017); Liu et al. (2018) use the Fischer information matrix to identify the most crucial weights responsible for prediction of a given task and lower the learning rate for these tasks. Lee et al. (2017) propose a dynamically expanding network to increase the capacity for new tasks if the previous architecture is insufficient to represent the data.
+
+Our framework lies at the intersection of pseudorehearsal methods and progressive neural networks, with scope for judiciously utilizing extra capacity, while resisting catastrophic forgetting.
+
+# 3 LIFE-LONG LEARNING VIA FEATURE TRANSFORMATIONS
+
+Before we present the feature transformer method, we introduce the terminologies and notations. We consider training a deep neural network which classifies an input to one of the classes $c \in [ C ] ^ { \triangleq }$ $\{ 1 , 2 , \cdots , C \}$ . We refer to the operation of classifying an input to a particular class as a task. To this end, the classifier is trained with training dataset $( X , Y )$ , drawn from a joint distribution $( \mathcal { X } , \mathcal { Y } )$ .
+
+We view the deep neural network, defined by the parameters $( \theta , \kappa )$ , as the composition of a feature extractor $\Phi _ { \theta } : X \to F$ , and a classifier $\Psi _ { \kappa }$
+
+$$
+\Psi _ { \kappa } \circ \Phi _ { \theta } : X  [ C ] ,
+$$
+
+where $\boldsymbol { X }$ is the space of input data, and $\pmb { F }$ is a space of low-dimensional feature vectors.
+
+We concisely denote the training of the neural network by $\mathrm { T R A I N } ( \pmb \theta , \pmb \kappa ; ~ D )$ , which minimizes a loss function on training data $D = ( X , Y )$ and produces the network parameters $( \theta , \kappa )$ . Let us also define the set of all computed features on input ${ \mathcal { F } } = \Phi _ { \theta } ( X )$ . 1
+
+When the loss function only penalizes misclassification, the network is expected to learn only the class separation boundaries in the feature space. However, as we demonstrate experimentally, good separation of class specific features enables stable learning of representations, which directly has a bearing on the performance of life-long learning. Therefore, in all our training procedures, we also use a feature loss which promotes feature separation
+
+$$
+\operatorname { m o d e l } \operatorname { l o s s } = \operatorname { c l a s s i f i c a t i o n } \operatorname { l o s s } _ { ( \theta , \kappa ) } + \lambda \cdot \operatorname { f e a t u r e - l o s s } _ { ( \theta ) } .
+$$
+
+In the lifelong learning context, we denote the time varying aspect of the network, training data and the classes, by using the time symbol $t \in \mathbb { N } \triangleq \{ 0 , 1 , 2 , \cdots \}$ in superscript on these objects. Realistically, at any time $t > 0$ , the classifier encounters any of the following canonical situations:
+
+1. The number of classes remain the same for the classifier as at $t - 1$ . The model encounters new training data for a subset of classes, without change in their distribution: $\forall t > 0$ , $C ^ { ( t ) } = C ^ { ( t - 1 ) }$ , ${ \mathcal { T } _ { 1 } ^ { ( t ) } \subseteq [ C ^ { ( t ) } ] }$ and $\in \mathcal { T } _ { 1 } ^ { ( t ) } , \ ( X _ { \tau } ^ { ( t ) } , \bar { Y } _ { \tau } ^ { ( t ) } ) \sim ( \mathcal { X } _ { \tau } ^ { ( t ) } , \mathcal { Y } _ { \tau } ^ { ( t ) } ) =$ $( \mathcal { X } _ { \tau } ^ { ( t - 1 ) } , \mathcal { Y } _ { \tau } ^ { ( t - 1 ) } )$ .   
+2. The number of classes remain the same for the classifier as at $t - 1$ . The model encounters new training data from modified input distribution(s), for a subset of classes: $\forall t > 0$ , $C ^ { ( t ) } = C ^ { ( t - 1 ) }$ , $\mathcal { T } _ { 2 } ^ { ( t ) } \subseteq [ C ^ { ( t ) } ]$ and $\forall \tau \in T _ { 2 } ^ { ( t ) }$ , $( X _ { \tau } ^ { ( t ) } , Y _ { \tau } ^ { ( t ) } ) \sim ( \mathcal { X } _ { \tau } ^ { ( t ) } , \mathcal { Y } _ { \tau } ^ { ( t ) } ) \neq$ $( \mathcal { X } _ { \tau } ^ { ( t - 1 ) } , \mathcal { Y } _ { \tau } ^ { ( t - 1 ) } )$ .   
+3. The model encounters new class(es) and corresponding new data: $\forall t \ > \ 0$ , $C ^ { ( t ) } >$ $C ^ { ( t - 1 ) }$ , $\mathcal { T } _ { 3 } ^ { ( t ) } = [ C ^ { ( t ) } ] \backslash [ C ^ { ( t - 1 ) } ]$ and $\forall \tau \in T _ { 3 } ^ { ( t ) }$ , $( \bar { X } _ { \tau } ^ { ( t ) } , \bar { Y _ { \tau } ^ { ( t ) } } ) \sim ( \mathcal { X } _ { \tau } ^ { ( t ) } , \mathcal { y } _ { \tau } ^ { ( t ) } )$ .
+
+In the most generic scenario, a combination of all the three situations can occur at any time index $t$ , with training data available for the classes in the set $\mathcal { T } ^ { ( t ) } \triangleq \mathcal { T } _ { 1 } ^ { ( t ) } \cup \mathcal { T } _ { 2 } ^ { ( t ) } \cup \mathcal { T } _ { 3 } ^ { ( t ) }$ . However, it is important to note that at every index $t$ , the classifier is trained to classify all the $C ^ { ( t ) }$ classes.
+
+# 3.1 FEATURE TRANSFORMATION BY AUGMENTING NETWORK CAPACITY
+
+At any time $t - 1$ , the classifier is optimized to classify all the classes $[ C ^ { ( t - 1 ) } ]$ and also the set of features $\mathcal { F } ^ { ( t - 1 ) }$ are well separated according to classes. At $t$ , when new training data $D ^ { ( t ) } =$ $\cup _ { \tau \in \mathcal { T } ^ { ( t ) } } ( X _ { \tau } ^ { ( t ) } , Y _ { \tau } ^ { ( t ) } )$ is encountered, the features extracted using the previous feature extractor
+
+$$
+\partial \mathcal { F } ^ { ( t ) } = \cup _ { \tau \in \mathcal { T } ^ { ( t ) } } \left( \Phi _ { \theta ^ { ( t - 1 ) } } \big ( X _ { \tau } ^ { ( t ) } \big ) \right) ,
+$$
+
+are not guaranteed to be optimized for classifying the new data and new classes. In order to achieve good performance on new data and classes, we propose to change the feature representation at time $t$ , just before the classification stage. We achieve this by defining a feature transformer
+
+$$
+\Phi _ { \Delta \theta ^ { ( t ) } } : { \cal F } ^ { ( t - 1 ) }  { \cal F } ^ { ( t ) } ,
+$$
+
+parameterized by $\Delta \pmb { \theta } ^ { ( t ) }$ , which maps any feature extracted by $\Phi _ { \pmb { \theta } ( t - 1 ) }$ to a new representation. The new feature extractor is now given by $\Phi _ { \theta ^ { ( t ) } } \triangleq \Phi _ { \Delta \theta ^ { ( t ) } } \circ \Phi _ { \theta ^ { ( t - 1 ) } }$ , where $\pmb { \theta } ^ { ( t ) } \triangleq \pmb { \theta } ^ { ( t - 1 ) } \cup \Delta \pmb { \theta } ^ { ( t ) }$ . Practically, this is realized by augmenting the capacity of the feature extractor at each time $t$ by using one or more fully connected layers2. It is however possible that $\Phi _ { \Delta \theta ^ { ( t ) } }$ could be simply an identity transform and feature transformers learnt in previous episodes could be adapted for new data. This helps in controlling the growth of network capacity over time and this aspect of our work is described in section 4.5.
+
+The feature transformer is trained, along with a new classifier layer, using the composite loss function of the form in equation 2, by invoking $\mathrm { T R A I N } \big ( \Delta \pmb { \theta } ^ { ( t ) } , \pmb { \kappa } ^ { ( t ) } ; \mathbf { \Delta } D ^ { ( t ) } \big )$ , with $D ^ { ( t ) } \stackrel { \textstyle = } { \textstyle = } ( \partial \mathcal { F } ^ { ( t ) } , Y ^ { ( t ) } ) ^ { 3 }$ . This ensures that the classifier performs well on the new data. However, strikingly, training a feature transformer at $t$ does not involve changing the feature extractor $\Phi _ { \pmb { \theta } ( t - 1 ) }$ at all, and this helps us in alleviating catastrophic forgetting by efficiently making use of already computed features $\mathcal { F } ^ { ( t - 1 ) }$ through a memory module.
+
+# 3.2 REMEMBERING HISTORY VIA MEMORY
+
+The set of all extracted features $\mathcal { F } ^ { ( t - 1 ) }$ serves as a good abstraction of the model, for all the tasks and data encountered till $t - 1$ . Therefore, if $\mathcal { F } ^ { ( t - 1 ) }$ is available to the model when it encounters new tasks and data, then the feature transformer at $t$ can take advantage of this knowledge to retain the classification performance on previous tasks and data as well. To this end, we assume the availability of an un-ending memory module $\mathcal { M }$ , equipped with READ(), WRITE() and ERASE() procedures, that can store $\mathcal { F } ^ { ( t - 1 ) }$ and can retrieve the same at $t$ . In situations where memory is scarce, only a relevant subset of $\mathcal { F } ^ { ( t - 1 ) }$ can be stored and retrieved.
+
+We train the feature transformer at any $t > 0$ by invoking $\mathrm { T R A I N } \big ( \Delta \pmb { \theta } ^ { ( t ) } , \pmb { \kappa } ^ { ( t ) } ; ~ \pmb { D } ^ { ( t ) } \big )$ , with the combined set of features
+
+$$
+D ^ { ( t ) } = ( \partial \mathcal { F } ^ { ( t ) } \cup \mathcal { F } ^ { ( t - 1 ) } , \cup _ { t ^ { \prime } \in [ 1 , 2 , \cdots , t ] } Y ^ { ( t ^ { \prime } ) } ) , \forall t > 0 .
+$$
+
+We then obtain the new set of features
+
+$$
+\mathcal { F } ^ { ( t ) } = \Phi _ { \Delta \theta ^ { ( t ) } } ( \partial \mathcal { F } ^ { ( t ) } ) \cup \Phi _ { \Delta \theta ^ { ( t ) } } ( \mathcal { F } ^ { ( t - 1 ) } ) ,
+$$
+
+and replace $\mathcal { F } ^ { ( t - 1 ) }$ in memory by (a subset of) $\mathcal { F } ^ { ( t ) }$
+
+With the assumption of infinite memory and capacity augmentation at every episode, our feature transformers framework is presented in algorithm 1.
+
+# 3.3 FEATURE TRANSFORMERS IN ACTION
+
+Before we present experimental results to demonstrate the ability of feature transformers for lifelong learning of new tasks, we first show how feature transformers operate when simply new episodes of data are presented. As described in section 3, we use a composite loss comprising of classification loss and feature loss to train the classifier. To promote feature clustering/separation, we propose to use center-loss as described in Wen et al. (2016).
+
+Dropping the time index $t$ for the brevity, with the convention that the ground truth class for each $x$ is encoded using one-hot vector $\bar { y } = [ \dot { y _ { 1 } } , y _ { 2 } , \cdot \cdot \cdot , y _ { C } ] ^ { T }$ , and $( \Psi _ { \kappa } \circ \Phi _ { \theta } ( x ) ) _ { c }$ is the $c ^ { t h }$ component of the classifier output, we use the following loss function for all the classifier training procedures:
+
+$$
+\log \left( \pmb { \theta } , \kappa \right) = - \sum _ { ( x , \bar { y } ) \in D } \sum _ { c \in [ C ] } y _ { c } \cdot \log \left( \left( \Psi _ { \kappa } \circ \Phi _ { \pmb { \theta } } ( x ) \right) _ { c } \right) + \lambda \cdot \sum _ { ( x , \bar { y } ) \in D } \sum _ { c \in [ C ] } \| \Phi _ { \pmb { \theta } } ( x ) - \mu _ { c } \| _ { 2 } ,
+$$
+
+where $D = \cup _ { \tau \in \mathcal { T } } ( X _ { \tau } , Y _ { \tau } )$ is the given training data set, and $\pmb { \mu } _ { c }$ is the centroid of all features corresponding to input data labelled as $c$ .
+
+Figure 1 provides snapshot of feature transformation algorithm when a new episode of data is encountered. We consider X-Ray lung images (from Wang et al. (2017a)) consisting of two classes: (i) normal and (ii) pneumothorax. At a time index $( t - 1 )$ , the classifier model is trained on 6000
+
+Input Task set $\mathcal { T } ^ { ( t ) }$ , and training data $\cup _ { \tau \in \mathcal { T } ^ { ( t ) } } ( X _ { \tau } ^ { ( t ) } , Y _ { \tau } ^ { ( t ) } ) , \forall t \geq 0$ Output $( \pmb \theta ^ { ( t ) } , \pmb \kappa ^ { ( t ) } ) , \forall t$
+
+$t \gets 0$ , ERASE $( \mathcal { M } )$ $D ^ { ( 0 ) } \gets \cup _ { \tau \in \mathcal { T } ^ { ( 0 ) } } ( X _ { \tau } ^ { ( 0 ) } , Y _ { \tau } ^ { ( 0 ) } )$ $\mathrm { T R A I N } \big ( { \pmb \theta } ^ { ( 0 ) } , { \pmb \kappa } ^ { ( 0 ) } ; D ^ { ( 0 ) } \big )$ $\mathcal { F } ^ { ( 0 ) }  \cup _ { \tau \in \mathcal { T } ^ { ( 0 ) } } ( \Phi _ { \theta ^ { ( 0 ) } } ( X _ { \tau } ^ { ( 0 ) } ) )$ WRITE $( \mathcal { M } , ( \mathcal { F } ^ { ( 0 ) } , Y ^ { ( 0 ) } ) )$ )
+
+$/ { } ^ { * }$ Set initial time, erase memory $^ { * } /$ $/ { } ^ { * }$ Obtain initial tasks and training data $^ { * } /$ $/ { } ^ { * }$ Train initial network $^ { * } /$ /\* Compute features $^ { * } /$ /\* Write features to memory $^ { * } /$
+
+# while T RUE do
+
+$t \gets t + 1$ , obtain $\mathscr { T } ^ { ( t ) } , \cup _ { \tau \in \mathcal { T } ^ { ( t ) } } ( X _ { \tau } ^ { ( t ) } , Y _ { \tau } ^ { ( t ) } )$ $/ { } ^ { * }$ Obtain current tasks and training data $^ { * } /$ Compute $\partial \mathcal { F } ^ { ( t ) }$ using equation 3 $/ *$ Compute old model features on new data $^ { * } /$ $( \mathcal { F } ^ { ( t - 1 ) } , Y ^ { ( t - 1 ) } ) \gets \mathtt { R E A D } ( \mathcal { M } )$ $/ { * }$ Read previously computed features from memory $^ { * } /$ Form $D ^ { ( t ) }$ using equation 5 /\* Form composite training data $^ { * } /$ $\mathrm { T R A I N } \big ( \Delta \pmb { \theta } ^ { ( t ) } , \bar { \pmb { \kappa } } ^ { ( t ) } ; D ^ { ( t ) } \big )$ /\* Train feature transformer $^ { * } /$ $\Phi _ { \theta ^ { ( t ) } } \gets \Phi _ { \Delta \theta ^ { ( t ) } } \circ \Phi _ { \theta ^ { ( t - 1 ) } }$ $/ *$ Obtain new feature extractor $^ { * } /$ Compute $\mathcal { F } ^ { ( t ) }$ using equation 6 $/ *$ Compute new features $^ { * } /$ $\mathrm { E R A S E } ( \mathcal { M } ) , \mathrm { W R I T E } ( \mathcal { M } , ( \mathcal { F } ^ { ( t ) } , \cup _ { t ^ { \prime } \in [ 1 , 2 , \cdots , t ] } Y ^ { ( t ^ { \prime } ) } ) )$ /\* Erase & write new features\*/
+
+# end
+
+![](images/0b52e95803e81c650f264a23584ba62d283704183ee71b3017f2193fb9330a41.jpg)  
+Algorithm 1: The life-long learning framework   
+Figure 1: Visual depiction of feature transformation process on new episodes.
+
+images with the loss in equation 7. As shown by the t-SNE plot (A), the feature extractor $\Phi _ { \pmb { \theta } ( t - 1 ) }$ produces features which are well-separated, and these features get stored in memory $\mathcal { M }$ . However, at time $t$ , when a set of 2000 new images is encountered, $\Phi _ { \pmb { \theta } ^ { ( t - 1 ) } }$ produces features that are scattered (t-SNE plot (B)). To improve the separation on the new data, the feature transformer is trained using the (well-separated) features in $\mathcal { M }$ as well as poorly separated features (from new data), with the loss function promoting good separation in new representation. This ensures that all the 8000 images seen until time $t$ is well separated (t-SNE plots (C) and (D)). This is repeated for all time indices $t$ . Thus, the feature transformer, along with appropriate loss function, continuously changes the representation to ensure good classification performance.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we benchmark our algorithm’s performance in various scenarios on relevant datasets.
+
+• Realistic Lifelong scenario along with traditional multi-task(MT) on MNIST rotations • Incremental learning on Pneumothorax identification (data-sensitive domain) • New-task learning on iCIFAR100
+
+# 4.1 PRACTICAL LIFELONG LEARNER - MNIST ROTATIONS DATASET
+
+To simulate a realistic lifelong learning scenario as described in Section 2, we use MNIST rotations dataset - each task contains digits rotated by a fixed angle between 0 and 180 degrees. Firstly, we randomly permute the rotation angles (to test domain adaptation). For every rotation angle, we randomly permute the class labels (0-10) and divide them into two subsets of 5 classes each (learning new tasks). We divide the number of training samples into two different subsets (incremental learning). Table 1 details the episodes of a lifelong learning experiment from the described procedure.
+
+Table 1: Episodes descriptions in one realistic life-long learning experiment scenario   
+
+<table><tr><td rowspan=1 colspan=1>Episode No</td><td rowspan=1 colspan=1>Rotation Angle</td><td rowspan=1 colspan=3>Class Labels</td><td rowspan=1 colspan=1>Samplesseenso far/Total Number of samples available</td><td rowspan=1 colspan=1>Description</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2(5)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0,2,3,9,7]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>12.5k/50k</td><td rowspan=1 colspan=1>Start</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>25)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[6,4,5,1,8]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>25k/50k</td><td rowspan=1 colspan=1>New-task Learning</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0,2,3,9,7]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>37.5.5k/50k</td><td rowspan=1 colspan=1>IncrementalLearning</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>∠(5)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[6,4,5,1,8]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>50k/50k</td><td rowspan=1 colspan=1>Incremental Learning</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>Z(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[2,8,6,4,7]</td><td></td><td rowspan=1 colspan=1>62.5k/100k</td><td rowspan=1 colspan=1>Domain Adaptation</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>Z(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[5,0,1,3,9]</td><td></td><td rowspan=1 colspan=1>75k/100k</td><td rowspan=1 colspan=1>Domain Adaptation</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>2(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[2,8,6,4,7]</td><td></td><td rowspan=1 colspan=1>87.5k/100k</td><td rowspan=1 colspan=1>Domain Adaptation+ Incremental Learning</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>2(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[5,0,1,3,9]</td><td></td><td rowspan=1 colspan=1>100k/100k</td><td rowspan=1 colspan=1>Domain Adaptation+ Incremental Learning</td></tr><tr><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=3>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td></tr><tr><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=3>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td></tr></table>
+
+# 4.1.1 EXPERIMENT DETAILS AND RESULTS
+
+We used a basic CNN architecture composed of 3 conv-layers of 32 filters and kernel-size - (3x3) followed by 2 dense layers “fc1” and “fc2” of feature length 256 and a softmax. Our feature transformer networks at every episode aims to transform “fc1” features using 2 additional dense layers of feature length 256. Feature transformers from previous episode serve as initialization for current episode and these models were optimized for the cumulative loss (equation 7), with $\lambda = 0 . 2$ . All the feature transformers were trained for only 3 epochs, with batch size of 32.
+
+We compare our results to the two obvious life-long learners - naive and cumulative training, which serve as the lower and upper bounds of performance respectively in the absence of other competitive algorithms for this setting. While the naive learner finetunes the entire network on the latest episode data, cumulative learner accumulates data from all the episodes seen so far and retrains the model. Fig. 2a highlights the remarkable performance of the proposed approach. We show the performance evolution over the first 25 episodes in Fig. 2a while, performance comparisons over 80 episodes across multiple experiment runs are averaged and shown in Table. 2b. We demonstrate that just by storing features in memory and learning feature transformers at every episode, we achieve almost similar performance to the gold-standard result. Further, it also underlines the applicability of our approach as single framework to combat different requirements of lifelong learning.
+
+Finally, we also compare our results (figure 3) in the conventional MT setting described in LopezPaz et al. (2017), whose open source implementation we used for our experiments. For details on the terminology and compared methods, the readers are encouraged to refer to Lopez-Paz et al. (2017). Fig. 3a demonstrates clear superiority of our approach over multiple methods compared. Additionally, backward transfer (BWT) - quantitative metric that models the deterioration of performance on older tasks while learning new tasks, is also negligible for our method, which shows resistance to catastrophic forgetting. Fig. 3b highlights this further as shown by accuracy on first task after learning subsequent tasks.
+
+# .2 PNEUMOTHORAX IDENTIFICATION FROM X-RAY IMAGES
+
+We simulate a practical manifestation of lifelong learning where a model trained to detect pneumothorax is deployed in a hospital with data arriving incrementally. We utilize a subset of ChestXRay (Wang et al., 2017b) dataset, which consists of chest $\mathrm { X }$ -rays labeled with corresponding diagnoses. We simulated incremental learning by providing the 8k training images in incremental batches of $2 \mathrm { k }$ and measured the performance on held-out validation set of $2 \mathrm { k }$ images. Fig. 4a establishes the baselines for the experiment. As in previous experiment, naive and cumulative training define the performance bounds. To clearly highlight the value of our feature transfom, we also add another strong baseline - naive Learner with center loss, which learns on the recent batch but with an augmented loss function (equation 7). In spite of a gain of $5 \%$ due to center loss, there is still a loss of $4 \%$ performance in the incremental learning set-up in the fourth batch of 2k.
+
+Experimental Details and Results: We used a pre-trained VGG-network (Simonyan & Zisserman, 2014) as the base network and explored the use of features from different layers of the VGG-network namely - post the two pooling layers and fully connected layers. Feature transformer network essentially had one additional dense layer per step and was optimized for (equation 7).
+
+![](images/4162168d34baeb3bcd95a8c308df9930fbcde7ccadbffd0465c37f7ccda9786a.jpg)  
+Figure 2: (a) Performance evolution over first 25 episodes and (b) Average performance across multiple runs at the end of lifelong experiments
+
+![](images/3511fb12edf5b80024a9081dfa0fad6b2357699e2a5a5e4a522714ddc71373ed.jpg)  
+Figure 3: Comparison of proposed approach in conventional multi-task setting
+
+![](images/7b8f4de1e5d4cdd04aac0b1d1733bb9cdee426a0236cea34c72a01f569a3bb20.jpg)  
+Figure 4: (a) Performance Comparison on validation dataset and (b) Comparison of feature transformers from different base layers
+
+Fig. 4a captures the performance of feature transformer with the base features being extracted from first pooling layer - block3 pool. After fourth batch of data, feature transformer result almost matches the performance of cumulative training. This performance is achieved despite not having access to the full images but only the stored features. Table. 4b also presents the performance of feature transformer depending upon the base features used. It can be noted that performance is lowest for the layer that is closer to the classification layer - fc 2. This is intuitively satisfying because, the further layers in a deep neural network will be more finely tuned towards the specific task and deprives the feature transform of any general features.
+
+![](images/ed66f02a2b13bb94fb23cfd8e9f354637f53b5d0740ac508396f4608ac541ef6.jpg)  
+Figure 5: Comparison with state-of-the-art methods - iCIFAR100 dataset
+
+# 4.3 ICIFAR100 DATASET
+
+We present the 100 classes from CIFAR100 dataset in a sequence of 20 tasks comprising of 5 classes each. Similar to Lomonaco & Maltoni (2017), we use the definitions for MT and single-incremental task (SIT). In an MT setting, evaluation is performed only on the new tasks exposed to the learner in the current episode. We start with a VGG type architecture, pretrained on iCIFAR10 dataset as suggested by Lopez-Paz et al. (2017) and Lomonaco & Maltoni (2017). Base features are extracted from flatten layer (before the fully connected layers) and our feature transformers included two dense layers with feature length $= 2 5 6$ . Similar to our earlier experiments, the feature transformer module from the previous episode initializes the current episode transformer and is optimized for the cumulative loss, with NoEpochs $= 3 0$ and batch size $= 3 2$ .
+
+# 4.3.1 MULTI-TASK SETTING
+
+Fig 5a demonstrates the superiority of our approach by a significant margin of ${ > } 1 0 \%$ . Further, we have negligible Backward Transfer.
+
+# 4.3.2 SINGLE INCREMENTAL TASK SETTING
+
+Fig 5b captures the performance over 20 batches of data for our method along with cumulative and naive learner. Unsurprisingly, naive learner performs very poorly, while feature transformer displays exemplary performance numbers of $4 0 \%$ validation acccuracy after encountering all 20 episodes, while compared to cumulative learner at $5 0 \%$ . Our method significantly outperforms AR1 (Lomonaco & Maltoni, 2017), while iCaRL (Rebuffi et al., 2017) achieves best-in-class performance close to gold-standard cumulative learner. This is not surprising because iCaRL is an explicit rehearsal technique, where exemplar images from previous episodes are stored and replayed while learning new tasks, while we only store low dimensional features.
+
+# 4.4 EFFECT OF LIMITED MEMORY
+
+One of the drawbacks of the proposed approach is the assumption of infinite memory and need to store features computed on all samples observed so far. To understand the extent of this limitation, we performed ablation experiments limiting the amount of history replayed as well as computing the storage requirements involved.
+
+Storing all features is not necessary We studied the effect of size of memory by limiting the number of samples stored to a smaller percentage. We observed that performance dropped from $9 7 \%$ to $9 4 \%$ when we reduced the memory size to only $2 5 \%$ the original size on MNIST rotations (Table. 2). We performed similar experiments on Pneumothorax classification problem and achieved similar trends as shown Table. 3, clearly demonstrating the resistance of the proposed method.
+
+Storing all features is not prohibitive Additionally, calculations of size-on-disk suggests that storing features of the entire history is not prohibitive. A typical natural/medical image is $2 5 6 ^ { * } 2 5 6 ^ { * } 3$ integers or more, whereas our representation is only 4096 floats (16kb). Even the largest available medical image repository of $1 0 0 \mathrm { k }$ X-ray images takes 1.6GB which is not huge. These are con
+
+Table 2: MNIST rotations   
+
+<table><tr><td rowspan=1 colspan=1>%ofSamples storedfrom history</td><td rowspan=1 colspan=1>FeatureTransformerVal Acc</td><td rowspan=1 colspan=1>CumulativeLearnerVal Acc</td></tr><tr><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>94%</td><td rowspan=1 colspan=1>95%</td></tr><tr><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>94.8%</td><td rowspan=1 colspan=1>98%</td></tr><tr><td rowspan=1 colspan=1>75</td><td rowspan=1 colspan=1>96%</td><td rowspan=1 colspan=1>98.5%</td></tr><tr><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>97 %</td><td rowspan=1 colspan=1>99 %</td></tr></table>
+
+Table 3: Pneumothorax dataset   
+
+<table><tr><td rowspan=1 colspan=1>%ofSamples storedfrom history</td><td rowspan=1 colspan=1>FeatureTransformerVal Acc</td></tr><tr><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>80.95%</td></tr><tr><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>82.95%</td></tr><tr><td rowspan=1 colspan=1>75</td><td rowspan=1 colspan=1>85.85 %</td></tr><tr><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>86.94 %</td></tr></table>
+
+Table: Performance comparisons with limited memory budget servative estimates. A standard medical image can be of much larger size $1 0 2 4 ^ { * } 1 0 2 4 )$ and in 3-D (minimum 10 slices). Any exemplar-based method (iCARL) will have severe storage limitations than our method. Additionally, storing 50 low-dimensional features occupies same memory as storing one exemplar image. This directly leads to storing more history compactly while addressing catastrophic forgetting and privacy.
+
+# 4.5 CONTROLLING THE GROWTH OF NETWORK CAPACITY
+
+In the description of the feature transformer framework in section 3.1 and section 3.2, we provided a generic treatment of the method, where, at the end of the each episode, the features are transformed up to date and then stored in memory for the next episode. With this scheme, it becomes imperative to always augment the capacity of the network in order to learn new representations, resulting in ever-growing network capacity. However, this problem can be easily alleviated by partitioning the entire network into a base feature extractor and feature transformer layers. The base feature extractor remains always fixed, and it is only the output of base feature extractor that is always stored in memory. With this scheme, the feature transformer layers need not grow in capacity and only a few already existing layers can be adapted for the new data. When the capacity of the feature transformer layers is not sufficient, then it can be augmented by adding one or more extra layers. In either case, the stored base features are sufficient to train.
+
+# Effect of varying additional capacity
+
+We varied the size of feature transformers and observed the difference in performance. Table 4 shows that by halving the additional capacity does not change the performance on MNIST rotations dataset at all. In addition, we froze the capacity of feature transformers after 5th episode and adapted them till end of 80 episodes. It is striking that performance is still high. Similarly, for Pneumothorax classification, Table 5 shows the performance comparisons with varying capacity of 2 , 1 and zero fc layers post third episode. These experiments (along with Sec 4.4) clearly demonstrate that power of the proposed approach comes from learning separable representations continually and not necessarily from storing all features or additional capacity.
+
+Table 4: MNIST rotations   
+
+<table><tr><td rowspan=1 colspan=1>Incremental capacityadded per episode</td><td rowspan=1 colspan=1>Feature TransformerVal Acc</td></tr><tr><td rowspan=1 colspan=1>2 dense layers</td><td rowspan=1 colspan=1>96.4%</td></tr><tr><td rowspan=1 colspan=1>1 dense layers</td><td rowspan=1 colspan=1>96.5%</td></tr><tr><td rowspan=1 colspan=1>No additonal capacity(after 5th episode)</td><td rowspan=1 colspan=1>96.2%</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>Incremental capacityadded per episode</td><td rowspan=1 colspan=1>FeatureTransformerVal Acc</td></tr><tr><td rowspan=1 colspan=1>2 dense layers</td><td rowspan=1 colspan=1>86.94%</td></tr><tr><td rowspan=1 colspan=1>1 dense layers</td><td rowspan=1 colspan=1>86.43%</td></tr><tr><td rowspan=1 colspan=1>No additonal capacity(after 3rd episode)</td><td rowspan=1 colspan=1>86.41%</td></tr></table>
+
+Table: Performance comparisons with limited incremental compute
+
+# 5 DISCUSSION
+
+In the final section, we discuss various points concerning our proposed approach.
+
+# Bayesian interpretation of Feature transformers
+
+Our feature transformers framework - learning a new representation with every new episode of data, can be interpreted as a maximum-a-posteriori representation learning, with the previous representation acting as prior. The implementation described in this paper using the combination loss (equation 7) is one instantiation of a general incremental representation learning possible in our framework. We have cast the MAP estimate problem to a tractable optimization problem constrained by a center-loss. In future, we plan to explore other manifestations of our approach with different loss functions that can ensure better separability.
+
+# Information Loss, Incremental Capacity, Data Privacy
+
+As shown in Table. 4b, feature transformer becomes less effective if the base features do not contain enough relevant information. This also means that additional capacity that every feature transformer adds may not help or in-fact be counter-productive. If the base features are extracted from layers close to the input image, there will be problem of traceability which violates the data privacy requirement we want to accomplish. We feel this a potential trade-off between performance and data privacy which we will investigate in future.
+
+# Model compaction for cascade of feature transformers
+
+Another approach to control the growth of network capacity is model compaction, which will be our future work. At any point in time, the entire set of feature transformer layers can be replaced by a smaller and simpler network (possibly using distillation techniques) and then again allowed to grow subsequently. This cycle of grow-and-purge can be used to effectively manage the overall capacity of the network.
+
+# REFERENCES
+
+F. M. Castro, M. J. Mar´ın-Jimenez, N. Guil, C. Schmid, and K. Alahari. End-to-End Incremental ´ Learning. ArXiv e-prints, July 2018.
+
+Chen He, Ruiping Wang, Shiguang Shan, and Xilin Chen. Exemplar-supported generative reproduction for class incremental learning. In $2 9 ^ { t h }$ British Machine Vision Conference (BMVC 2018), 3–6 Sep 2018.
+
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
+
+Khurram Javed and Muhammad Talha Paracha. Incremental classifier & representation learning. Learning, 3:4, 2018.
+
+James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the national academy of sciences, pp. 201611835, 2017.
+
+Jeongtae Lee, Jaehong Yoon, Eunho Yang, and Sung Ju Hwang. Lifelong learning with dynamically expandable networks. CoRR, abs/1708.01547, 2017.
+
+Zhizhong Li and Derek Hoiem. Learning without forgetting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017.
+
+Xialei Liu, Marc Masana, Luis Herranz, Joost Van de Weijer, Antonio M Lopez, and Andrew D Bagdanov. Rotate your networks: Better weight consolidation and less catastrophic forgetting. arXiv preprint arXiv:1802.02950, 2018.
+
+Vincenzo Lomonaco and Davide Maltoni. Core50: a new dataset and benchmark for continuous object recognition. In Sergey Levine, Vincent Vanhoucke, and Ken Goldberg (eds.), Proceedings of the 1st Annual Conference on Robot Learning, volume 78 of Proceedings of Machine Learning Research, pp. 17–26. PMLR, 13–15 Nov 2017. URL http://proceedings.mlr.press/ v78/lomonaco17a.html.
+
+David Lopez-Paz et al. Gradient episodic memory for continual learning. In Advances in Neural Information Processing Systems, pp. 6467–6476, 2017.
+
+Michael McCloskey and Neal J Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pp. 109–165. Elsevier, 1989.
+
+Thomas Mensink, Jakob Verbeek, Florent Perronnin, and Gabriela Csurka. Metric learning for large scale image classification: Generalizing to new classes at near-zero cost. In Computer Vision– ECCV 2012, pp. 488–501. Springer, 2012.
+
+Maxime Oquab, Leon Bottou, Ivan Laptev, and Josef Sivic. Learning and transferring mid-level image representations using convolutional neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1717–1724, 2014.
+
+Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, and Christoph H. Lampert. iCaRL: Incremental classifier and representation learning. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5533–5542, 2017.
+
+Anthony Robins. Catastrophic forgetting, rehearsal and pseudorehearsal. Connection Science, 7: 123–146, 1995.
+
+Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual learning with deep generative replay. In Advances in Neural Information Processing Systems, pp. 2990–2999, 2017.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
+
+Amal Triki Rannen, Rahaf Aljundi, Mathew B Blaschko, and Tinne Tuytelaars. Encoder based lifelong learning. IEEE International Conference of Computer Vision, 2017.
+
+Ragav Venkatesan, Hemanth Venkateswara, Sethuraman Panchanathan, and Baoxin Li. A strategy for an uncompromising incremental learner. arXiv preprint arXiv:1705.00744, 2017.
+
+X. Wang, Y. Peng, L. Lu, Z. Lu, M. Bagheri, and R. M. Summers. Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3462–3471, July 2017a. doi: 10.1109/CVPR.2017.369.
+
+X. Wang, Y. Peng, L. Lu, Z. Lu, M. Bagheri, and R. M. Summers. Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3462–3471, July 2017b. doi: 10.1109/CVPR.2017.369.
+
+Yandong Wen, Kaipeng Zhang, Zhifeng Li, and Yu Qiao. A discriminative feature learning approach for deep face recognition. In ECCV, 2016.

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+        "type": "text",
+        "text": "FEATURE TRANSFORMERS: A UNIFIED REPRESENTATION LEARNING FRAMEWORK FOR LIFELONG LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            261,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Despite the recent advances in representation learning, lifelong learning continues to be one of the most challenging and unconquered problems. Catastrophic forgetting and data privacy constitute two of the important challenges for a successful lifelong learner. Further, existing techniques are designed to handle only specific manifestations of lifelong learning, whereas a practical lifelong learner is expected to switch and adapt seamlessly to different scenarios. In this paper, we present a single, unified mathematical framework for handling the myriad variants of lifelong learning, while alleviating these two challenges. We utilize an external memory to store only the features representing past data and learn richer and newer representations incrementally through transformation neural networks - feature transformers. We define, simulate and demonstrate exemplary performance on a realistic lifelong experimental setting using the MNIST rotations dataset, paving the way for practical lifelong learners. To illustrate the applicability of our method in data sensitive domains like healthcare, we study the pneumothorax classification problem from X-ray images, achieving near gold standard performance. We also benchmark our approach with a number of state-of-the art methods on MNIST rotations and iCIFAR100 datasets demonstrating superior performance. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            549,
+            336,
+            564
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning algorithms have achieved tremendous success on various challenging tasks like object detection, language translation, medical image segmentation, etc. Lifelong learning - the ability to adapt, benefit and sustain performance post deployment with more data and feedback, is an important goal for artificial intelligence (AI) and extremely crucial for sustained utility of these algorithms in domains like healthcare (learning from rare cases), self driving (learning new object detectors), etc. While there is no single standard definition of lifelong learning, most of the research in this field can be classified into one of the following sub-categories: ",
+        "bbox": [
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+            571,
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+            670
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. Incremental learning - encountering new data, with no change in distribution, for same task   \n2. Domain adaptation - data from modified target distributions but for the same task   \n3. New-task learning - data from tasks that were not presented before ",
+        "bbox": [
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+            731
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Majority of the successful techniques study these variants in isolation. However, a realistic lifelong learning scenario would involve a mixture of these manifestations over time. Another major impediment to a successful lifelong learner is data privacy. In domains like healthcare, it is near impossible to have data access beyond scope (both time and geography) which leads to catastophic forgetting (McCloskey & Cohen, 1989)- the inability of machine learning models to retain past knowledge while learning from new data. In this paper, we provide a unified framework - feature transformers - for practical lifelong learning to handle data privacy and catastrophic forgetting. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "In summary, the major contributions of our work are as follows: ",
+        "bbox": [
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+            843,
+            591,
+            857
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Define a realistic and a challenging lifelong learning scenario and formulate a unique and generic mathematical framework - feature transformer, to work in this setting successfully. • Ensure data privacy by storing only features from previous episodes while successfully combating catastrophic forgetting. ",
+        "bbox": [
+            217,
+            864,
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+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Principled way of utilizing additional neural-compute to tackle complex incremental tasks. • Demonstrate state-of-the-art results on common continuous learning benchmarks for new task learning like MNIST rotations, iCIFAR dataset and a challenging healthcare problem. Demonstrate exemplary performance even under severe constraints on memory and compute, thus operating under the assumptions of practical lifelong learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following review of related work in section 2, we describe the mathematical formulation of feature transformers and their practical realization in section 3. Section 4 contains experiments and results, followed by discussion in section 5. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            244,
+            341,
+            260
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A conventional deep learning classifier can be viewed as an automatic feature extractor followed by a classifier, trained jointly. For lifelong learning, the methods in literature can be broadly classified into: 1) learning with fixed feature representations and 2) incrementally evolving representations. ",
+        "bbox": [
+            176,
+            267,
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+            310
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fixed Represenations: In this class of approaches, a representation for the data is learnt from the initial task and remains frozen for ensuing tasks, while only the classification layers are modified. The simplest baseline approaches include fine-tuning (Oquab et al., 2014) and Mensink et al. (2012) where the idea of using fixed representations was extended further by using the nearest class mean classifier. In spite of their simplicity, these algorithms do not perform well in practice due to the limiting constraint of a fixed representation throughout the incremental learning phase. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Incrementally evolving representations: The last few years have witnessed renewed interest in developing methods that allow changing data representation with the addition of new tasks. Naive methods of retraining with only new data suffer from catastrophic forgetting (McCloskey & Cohen, 1989). To overcome this problem, most methods have attempted different manifestations of rehearsal (Robins, 1995) - replaying data from the previous tasks. However, access to all the previous data is usually not feasible due to size, compute and privacy concerns, researchers have attempted to reproduce past through proxy information, known as pseudorehearsal. Pseudorehearsal techniques can be further classified depending on how the previous information is stored /used: ",
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+    {
+        "type": "text",
+        "text": "• Knowledge distillation based approaches: An example method for lifelong learning using knowledge distillation (Hinton et al., 2015) is learning without forgetting (LwF) (Li & Hoiem, 2017) - where distillation loss was added to match output of the updated network to that of the old network on the old task output variables. LwF and its extensions does not scale to a large number of new tasks, suffers from catastrophic forgetting and importantly do not address incremental learning or domain adapatation settings. ",
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+    {
+        "type": "text",
+        "text": "• Rehearsal using exemplar sets: Rebuffi et al. (2017) propose iCaRL, an incremental learning approach by storing an exemplar set of data from the previous tasks and augmenting it with the new data. In more recent works, Javed & Paracha (2018) and Castro et al. (2018) have argued that a decoupled nearest mean classifier from Rebuffi et al. (2017) is not essential and have proposed joint learning of feature extraction and classification. Lopez-Paz et al. (2017) propose Gradient Episodic Memory, a technique which stores data from previous classes and constrains the gradient update while learning new tasks. ",
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+    {
+        "type": "text",
+        "text": "• Rehearsal using generative models: Triki Rannen et al. (2017) propose to reproduce past knowledge by using task-specific under-complete autoencoders. When a new task is presented, optimization is constrained by preventing the reconstructions of these autoencoders from changing, thereby ensuring that the features necessary for the previous tasks are not destroyed. Shin et al. (2017); He et al. (2018); Venkatesan et al. (2017) employ generative adversarial networks for recreating the history from previous tasks. ",
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+    {
+        "type": "text",
+        "text": "• Network regularization strategies: Methods belonging to this class aim to identify and selectively modify parts of the neural network which are critical to remember past knowledge or by explicitly penalizing loss of performance on old tasks. Kirkpatrick et al. (2017); Liu et al. (2018) use the Fischer information matrix to identify the most crucial weights responsible for prediction of a given task and lower the learning rate for these tasks. Lee et al. (2017) propose a dynamically expanding network to increase the capacity for new tasks if the previous architecture is insufficient to represent the data. ",
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+        "type": "text",
+        "text": "Our framework lies at the intersection of pseudorehearsal methods and progressive neural networks, with scope for judiciously utilizing extra capacity, while resisting catastrophic forgetting. ",
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+    {
+        "type": "text",
+        "text": "3 LIFE-LONG LEARNING VIA FEATURE TRANSFORMATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Before we present the feature transformer method, we introduce the terminologies and notations. We consider training a deep neural network which classifies an input to one of the classes $c \\in [ C ] ^ { \\triangleq }$ $\\{ 1 , 2 , \\cdots , C \\}$ . We refer to the operation of classifying an input to a particular class as a task. To this end, the classifier is trained with training dataset $( X , Y )$ , drawn from a joint distribution $( \\mathcal { X } , \\mathcal { Y } )$ . ",
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+        "text": "We view the deep neural network, defined by the parameters $( \\theta , \\kappa )$ , as the composition of a feature extractor $\\Phi _ { \\theta } : X \\to F$ , and a classifier $\\Psi _ { \\kappa }$ ",
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+    {
+        "type": "equation",
+        "img_path": "images/321b528817989e337a214ec3c0b0e523d0d70606b9d56892dfc05a961e725f8b.jpg",
+        "text": "$$\n\\Psi _ { \\kappa } \\circ \\Phi _ { \\theta } : X  [ C ] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\boldsymbol { X }$ is the space of input data, and $\\pmb { F }$ is a space of low-dimensional feature vectors. ",
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+        "type": "text",
+        "text": "We concisely denote the training of the neural network by $\\mathrm { T R A I N } ( \\pmb \\theta , \\pmb \\kappa ; ~ D )$ , which minimizes a loss function on training data $D = ( X , Y )$ and produces the network parameters $( \\theta , \\kappa )$ . Let us also define the set of all computed features on input ${ \\mathcal { F } } = \\Phi _ { \\theta } ( X )$ . 1 ",
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+        "type": "text",
+        "text": "When the loss function only penalizes misclassification, the network is expected to learn only the class separation boundaries in the feature space. However, as we demonstrate experimentally, good separation of class specific features enables stable learning of representations, which directly has a bearing on the performance of life-long learning. Therefore, in all our training procedures, we also use a feature loss which promotes feature separation ",
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+    {
+        "type": "equation",
+        "img_path": "images/1afed3c469f6a2aa9ffe16c64f90472cd893234e18c870f3a25386ef2e62510e.jpg",
+        "text": "$$\n\\operatorname { m o d e l } \\operatorname { l o s s } = \\operatorname { c l a s s i f i c a t i o n } \\operatorname { l o s s } _ { ( \\theta , \\kappa ) } + \\lambda \\cdot \\operatorname { f e a t u r e - l o s s } _ { ( \\theta ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the lifelong learning context, we denote the time varying aspect of the network, training data and the classes, by using the time symbol $t \\in \\mathbb { N } \\triangleq \\{ 0 , 1 , 2 , \\cdots \\}$ in superscript on these objects. Realistically, at any time $t > 0$ , the classifier encounters any of the following canonical situations: ",
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+    {
+        "type": "text",
+        "text": "1. The number of classes remain the same for the classifier as at $t - 1$ . The model encounters new training data for a subset of classes, without change in their distribution: $\\forall t > 0$ , $C ^ { ( t ) } = C ^ { ( t - 1 ) }$ , ${ \\mathcal { T } _ { 1 } ^ { ( t ) } \\subseteq [ C ^ { ( t ) } ] }$ and $\\in \\mathcal { T } _ { 1 } ^ { ( t ) } , \\ ( X _ { \\tau } ^ { ( t ) } , \\bar { Y } _ { \\tau } ^ { ( t ) } ) \\sim ( \\mathcal { X } _ { \\tau } ^ { ( t ) } , \\mathcal { Y } _ { \\tau } ^ { ( t ) } ) =$ $( \\mathcal { X } _ { \\tau } ^ { ( t - 1 ) } , \\mathcal { Y } _ { \\tau } ^ { ( t - 1 ) } )$ .   \n2. The number of classes remain the same for the classifier as at $t - 1$ . The model encounters new training data from modified input distribution(s), for a subset of classes: $\\forall t > 0$ , $C ^ { ( t ) } = C ^ { ( t - 1 ) }$ , $\\mathcal { T } _ { 2 } ^ { ( t ) } \\subseteq [ C ^ { ( t ) } ]$ and $\\forall \\tau \\in T _ { 2 } ^ { ( t ) }$ , $( X _ { \\tau } ^ { ( t ) } , Y _ { \\tau } ^ { ( t ) } ) \\sim ( \\mathcal { X } _ { \\tau } ^ { ( t ) } , \\mathcal { Y } _ { \\tau } ^ { ( t ) } ) \\neq$ $( \\mathcal { X } _ { \\tau } ^ { ( t - 1 ) } , \\mathcal { Y } _ { \\tau } ^ { ( t - 1 ) } )$ .   \n3. The model encounters new class(es) and corresponding new data: $\\forall t \\ > \\ 0$ , $C ^ { ( t ) } >$ $C ^ { ( t - 1 ) }$ , $\\mathcal { T } _ { 3 } ^ { ( t ) } = [ C ^ { ( t ) } ] \\backslash [ C ^ { ( t - 1 ) } ]$ and $\\forall \\tau \\in T _ { 3 } ^ { ( t ) }$ , $( \\bar { X } _ { \\tau } ^ { ( t ) } , \\bar { Y _ { \\tau } ^ { ( t ) } } ) \\sim ( \\mathcal { X } _ { \\tau } ^ { ( t ) } , \\mathcal { y } _ { \\tau } ^ { ( t ) } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the most generic scenario, a combination of all the three situations can occur at any time index $t$ , with training data available for the classes in the set $\\mathcal { T } ^ { ( t ) } \\triangleq \\mathcal { T } _ { 1 } ^ { ( t ) } \\cup \\mathcal { T } _ { 2 } ^ { ( t ) } \\cup \\mathcal { T } _ { 3 } ^ { ( t ) }$ . However, it is important to note that at every index $t$ , the classifier is trained to classify all the $C ^ { ( t ) }$ classes. ",
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+    {
+        "type": "text",
+        "text": "3.1 FEATURE TRANSFORMATION BY AUGMENTING NETWORK CAPACITY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "At any time $t - 1$ , the classifier is optimized to classify all the classes $[ C ^ { ( t - 1 ) } ]$ and also the set of features $\\mathcal { F } ^ { ( t - 1 ) }$ are well separated according to classes. At $t$ , when new training data $D ^ { ( t ) } =$ $\\cup _ { \\tau \\in \\mathcal { T } ^ { ( t ) } } ( X _ { \\tau } ^ { ( t ) } , Y _ { \\tau } ^ { ( t ) } )$ is encountered, the features extracted using the previous feature extractor ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/afe65a48a7d026c78cb31a576326d313ae335537c6aebb4687667f91bccfc817.jpg",
+        "text": "$$\n\\partial \\mathcal { F } ^ { ( t ) } = \\cup _ { \\tau \\in \\mathcal { T } ^ { ( t ) } } \\left( \\Phi _ { \\theta ^ { ( t - 1 ) } } \\big ( X _ { \\tau } ^ { ( t ) } \\big ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "are not guaranteed to be optimized for classifying the new data and new classes. In order to achieve good performance on new data and classes, we propose to change the feature representation at time $t$ , just before the classification stage. We achieve this by defining a feature transformer ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c75a868cfd64add0c76d53934bc538cd6dda5d19358d14cf5f092bb79f782181.jpg",
+        "text": "$$\n\\Phi _ { \\Delta \\theta ^ { ( t ) } } : { \\cal F } ^ { ( t - 1 ) }  { \\cal F } ^ { ( t ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "parameterized by $\\Delta \\pmb { \\theta } ^ { ( t ) }$ , which maps any feature extracted by $\\Phi _ { \\pmb { \\theta } ( t - 1 ) }$ to a new representation. The new feature extractor is now given by $\\Phi _ { \\theta ^ { ( t ) } } \\triangleq \\Phi _ { \\Delta \\theta ^ { ( t ) } } \\circ \\Phi _ { \\theta ^ { ( t - 1 ) } }$ , where $\\pmb { \\theta } ^ { ( t ) } \\triangleq \\pmb { \\theta } ^ { ( t - 1 ) } \\cup \\Delta \\pmb { \\theta } ^ { ( t ) }$ . Practically, this is realized by augmenting the capacity of the feature extractor at each time $t$ by using one or more fully connected layers2. It is however possible that $\\Phi _ { \\Delta \\theta ^ { ( t ) } }$ could be simply an identity transform and feature transformers learnt in previous episodes could be adapted for new data. This helps in controlling the growth of network capacity over time and this aspect of our work is described in section 4.5. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The feature transformer is trained, along with a new classifier layer, using the composite loss function of the form in equation 2, by invoking $\\mathrm { T R A I N } \\big ( \\Delta \\pmb { \\theta } ^ { ( t ) } , \\pmb { \\kappa } ^ { ( t ) } ; \\mathbf { \\Delta } D ^ { ( t ) } \\big )$ , with $D ^ { ( t ) } \\stackrel { \\textstyle = } { \\textstyle = } ( \\partial \\mathcal { F } ^ { ( t ) } , Y ^ { ( t ) } ) ^ { 3 }$ . This ensures that the classifier performs well on the new data. However, strikingly, training a feature transformer at $t$ does not involve changing the feature extractor $\\Phi _ { \\pmb { \\theta } ( t - 1 ) }$ at all, and this helps us in alleviating catastrophic forgetting by efficiently making use of already computed features $\\mathcal { F } ^ { ( t - 1 ) }$ through a memory module. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 REMEMBERING HISTORY VIA MEMORY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The set of all extracted features $\\mathcal { F } ^ { ( t - 1 ) }$ serves as a good abstraction of the model, for all the tasks and data encountered till $t - 1$ . Therefore, if $\\mathcal { F } ^ { ( t - 1 ) }$ is available to the model when it encounters new tasks and data, then the feature transformer at $t$ can take advantage of this knowledge to retain the classification performance on previous tasks and data as well. To this end, we assume the availability of an un-ending memory module $\\mathcal { M }$ , equipped with READ(), WRITE() and ERASE() procedures, that can store $\\mathcal { F } ^ { ( t - 1 ) }$ and can retrieve the same at $t$ . In situations where memory is scarce, only a relevant subset of $\\mathcal { F } ^ { ( t - 1 ) }$ can be stored and retrieved. ",
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+    {
+        "type": "text",
+        "text": "We train the feature transformer at any $t > 0$ by invoking $\\mathrm { T R A I N } \\big ( \\Delta \\pmb { \\theta } ^ { ( t ) } , \\pmb { \\kappa } ^ { ( t ) } ; ~ \\pmb { D } ^ { ( t ) } \\big )$ , with the combined set of features ",
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+    {
+        "type": "equation",
+        "img_path": "images/e88b4ee7bab6235c1fe3378164e12dbc1ef6f548f082c88428b8ed9969f4a626.jpg",
+        "text": "$$\nD ^ { ( t ) } = ( \\partial \\mathcal { F } ^ { ( t ) } \\cup \\mathcal { F } ^ { ( t - 1 ) } , \\cup _ { t ^ { \\prime } \\in [ 1 , 2 , \\cdots , t ] } Y ^ { ( t ^ { \\prime } ) } ) , \\forall t > 0 .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We then obtain the new set of features ",
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+    {
+        "type": "equation",
+        "img_path": "images/735a3683e1dc37162cbb0ec6c0c1801b5c67abba7ae0bd0a350ba32955730489.jpg",
+        "text": "$$\n\\mathcal { F } ^ { ( t ) } = \\Phi _ { \\Delta \\theta ^ { ( t ) } } ( \\partial \\mathcal { F } ^ { ( t ) } ) \\cup \\Phi _ { \\Delta \\theta ^ { ( t ) } } ( \\mathcal { F } ^ { ( t - 1 ) } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "and replace $\\mathcal { F } ^ { ( t - 1 ) }$ in memory by (a subset of) $\\mathcal { F } ^ { ( t ) }$ ",
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+    {
+        "type": "text",
+        "text": "With the assumption of infinite memory and capacity augmentation at every episode, our feature transformers framework is presented in algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "3.3 FEATURE TRANSFORMERS IN ACTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Before we present experimental results to demonstrate the ability of feature transformers for lifelong learning of new tasks, we first show how feature transformers operate when simply new episodes of data are presented. As described in section 3, we use a composite loss comprising of classification loss and feature loss to train the classifier. To promote feature clustering/separation, we propose to use center-loss as described in Wen et al. (2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Dropping the time index $t$ for the brevity, with the convention that the ground truth class for each $x$ is encoded using one-hot vector $\\bar { y } = [ \\dot { y _ { 1 } } , y _ { 2 } , \\cdot \\cdot \\cdot , y _ { C } ] ^ { T }$ , and $( \\Psi _ { \\kappa } \\circ \\Phi _ { \\theta } ( x ) ) _ { c }$ is the $c ^ { t h }$ component of the classifier output, we use the following loss function for all the classifier training procedures: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d52835a8a94c71dc3d23cf4c178d91f1de02f1faeda123f2125db3fafc2bf9c0.jpg",
+        "text": "$$\n\\log \\left( \\pmb { \\theta } , \\kappa \\right) = - \\sum _ { ( x , \\bar { y } ) \\in D } \\sum _ { c \\in [ C ] } y _ { c } \\cdot \\log \\left( \\left( \\Psi _ { \\kappa } \\circ \\Phi _ { \\pmb { \\theta } } ( x ) \\right) _ { c } \\right) + \\lambda \\cdot \\sum _ { ( x , \\bar { y } ) \\in D } \\sum _ { c \\in [ C ] } \\| \\Phi _ { \\pmb { \\theta } } ( x ) - \\mu _ { c } \\| _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $D = \\cup _ { \\tau \\in \\mathcal { T } } ( X _ { \\tau } , Y _ { \\tau } )$ is the given training data set, and $\\pmb { \\mu } _ { c }$ is the centroid of all features corresponding to input data labelled as $c$ . ",
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+    {
+        "type": "text",
+        "text": "Figure 1 provides snapshot of feature transformation algorithm when a new episode of data is encountered. We consider X-Ray lung images (from Wang et al. (2017a)) consisting of two classes: (i) normal and (ii) pneumothorax. At a time index $( t - 1 )$ , the classifier model is trained on 6000 ",
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+    {
+        "type": "text",
+        "text": "Input Task set $\\mathcal { T } ^ { ( t ) }$ , and training data $\\cup _ { \\tau \\in \\mathcal { T } ^ { ( t ) } } ( X _ { \\tau } ^ { ( t ) } , Y _ { \\tau } ^ { ( t ) } ) , \\forall t \\geq 0$ Output $( \\pmb \\theta ^ { ( t ) } , \\pmb \\kappa ^ { ( t ) } ) , \\forall t$ ",
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+    },
+    {
+        "type": "text",
+        "text": "$t \\gets 0$ , ERASE $( \\mathcal { M } )$ $D ^ { ( 0 ) } \\gets \\cup _ { \\tau \\in \\mathcal { T } ^ { ( 0 ) } } ( X _ { \\tau } ^ { ( 0 ) } , Y _ { \\tau } ^ { ( 0 ) } )$ $\\mathrm { T R A I N } \\big ( { \\pmb \\theta } ^ { ( 0 ) } , { \\pmb \\kappa } ^ { ( 0 ) } ; D ^ { ( 0 ) } \\big )$ $\\mathcal { F } ^ { ( 0 ) }  \\cup _ { \\tau \\in \\mathcal { T } ^ { ( 0 ) } } ( \\Phi _ { \\theta ^ { ( 0 ) } } ( X _ { \\tau } ^ { ( 0 ) } ) )$ WRITE $( \\mathcal { M } , ( \\mathcal { F } ^ { ( 0 ) } , Y ^ { ( 0 ) } ) )$ ) ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$/ { } ^ { * }$ Set initial time, erase memory $^ { * } /$ $/ { } ^ { * }$ Obtain initial tasks and training data $^ { * } /$ $/ { } ^ { * }$ Train initial network $^ { * } /$ /\\* Compute features $^ { * } /$ /\\* Write features to memory $^ { * } /$ ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "while T RUE do ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "$t \\gets t + 1$ , obtain $\\mathscr { T } ^ { ( t ) } , \\cup _ { \\tau \\in \\mathcal { T } ^ { ( t ) } } ( X _ { \\tau } ^ { ( t ) } , Y _ { \\tau } ^ { ( t ) } )$ $/ { } ^ { * }$ Obtain current tasks and training data $^ { * } /$ Compute $\\partial \\mathcal { F } ^ { ( t ) }$ using equation 3 $/ *$ Compute old model features on new data $^ { * } /$ $( \\mathcal { F } ^ { ( t - 1 ) } , Y ^ { ( t - 1 ) } ) \\gets \\mathtt { R E A D } ( \\mathcal { M } )$ $/ { * }$ Read previously computed features from memory $^ { * } /$ Form $D ^ { ( t ) }$ using equation 5 /\\* Form composite training data $^ { * } /$ $\\mathrm { T R A I N } \\big ( \\Delta \\pmb { \\theta } ^ { ( t ) } , \\bar { \\pmb { \\kappa } } ^ { ( t ) } ; D ^ { ( t ) } \\big )$ /\\* Train feature transformer $^ { * } /$ $\\Phi _ { \\theta ^ { ( t ) } } \\gets \\Phi _ { \\Delta \\theta ^ { ( t ) } } \\circ \\Phi _ { \\theta ^ { ( t - 1 ) } }$ $/ *$ Obtain new feature extractor $^ { * } /$ Compute $\\mathcal { F } ^ { ( t ) }$ using equation 6 $/ *$ Compute new features $^ { * } /$ $\\mathrm { E R A S E } ( \\mathcal { M } ) , \\mathrm { W R I T E } ( \\mathcal { M } , ( \\mathcal { F } ^ { ( t ) } , \\cup _ { t ^ { \\prime } \\in [ 1 , 2 , \\cdots , t ] } Y ^ { ( t ^ { \\prime } ) } ) )$ /\\* Erase & write new features\\*/ ",
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+    },
+    {
+        "type": "text",
+        "text": "end ",
+        "text_level": 1,
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+    },
+    {
+        "type": "image",
+        "img_path": "images/0b52e95803e81c650f264a23584ba62d283704183ee71b3017f2193fb9330a41.jpg",
+        "image_caption": [
+            "Algorithm 1: The life-long learning framework ",
+            "Figure 1: Visual depiction of feature transformation process on new episodes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "images with the loss in equation 7. As shown by the t-SNE plot (A), the feature extractor $\\Phi _ { \\pmb { \\theta } ( t - 1 ) }$ produces features which are well-separated, and these features get stored in memory $\\mathcal { M }$ . However, at time $t$ , when a set of 2000 new images is encountered, $\\Phi _ { \\pmb { \\theta } ^ { ( t - 1 ) } }$ produces features that are scattered (t-SNE plot (B)). To improve the separation on the new data, the feature transformer is trained using the (well-separated) features in $\\mathcal { M }$ as well as poorly separated features (from new data), with the loss function promoting good separation in new representation. This ensures that all the 8000 images seen until time $t$ is well separated (t-SNE plots (C) and (D)). This is repeated for all time indices $t$ . Thus, the feature transformer, along with appropriate loss function, continuously changes the representation to ensure good classification performance. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we benchmark our algorithm’s performance in various scenarios on relevant datasets. ",
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+        "text": "• Realistic Lifelong scenario along with traditional multi-task(MT) on MNIST rotations • Incremental learning on Pneumothorax identification (data-sensitive domain) • New-task learning on iCIFAR100 ",
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+    {
+        "type": "text",
+        "text": "4.1 PRACTICAL LIFELONG LEARNER - MNIST ROTATIONS DATASET ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To simulate a realistic lifelong learning scenario as described in Section 2, we use MNIST rotations dataset - each task contains digits rotated by a fixed angle between 0 and 180 degrees. Firstly, we randomly permute the rotation angles (to test domain adaptation). For every rotation angle, we randomly permute the class labels (0-10) and divide them into two subsets of 5 classes each (learning new tasks). We divide the number of training samples into two different subsets (incremental learning). Table 1 details the episodes of a lifelong learning experiment from the described procedure. ",
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+    {
+        "type": "table",
+        "img_path": "images/3d3fc2b7ae0a2069c6d36ecabf2ad8cf0df7754c23690485d1315b4768000a0d.jpg",
+        "table_caption": [
+            "Table 1: Episodes descriptions in one realistic life-long learning experiment scenario "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Episode No</td><td rowspan=1 colspan=1>Rotation Angle</td><td rowspan=1 colspan=3>Class Labels</td><td rowspan=1 colspan=1>Samplesseenso far/Total Number of samples available</td><td rowspan=1 colspan=1>Description</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2(5)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0,2,3,9,7]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>12.5k/50k</td><td rowspan=1 colspan=1>Start</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>25)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[6,4,5,1,8]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>25k/50k</td><td rowspan=1 colspan=1>New-task Learning</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[0,2,3,9,7]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>37.5.5k/50k</td><td rowspan=1 colspan=1>IncrementalLearning</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>∠(5)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[6,4,5,1,8]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>50k/50k</td><td rowspan=1 colspan=1>Incremental Learning</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>Z(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[2,8,6,4,7]</td><td></td><td rowspan=1 colspan=1>62.5k/100k</td><td rowspan=1 colspan=1>Domain Adaptation</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>Z(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[5,0,1,3,9]</td><td></td><td rowspan=1 colspan=1>75k/100k</td><td rowspan=1 colspan=1>Domain Adaptation</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>2(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[2,8,6,4,7]</td><td></td><td rowspan=1 colspan=1>87.5k/100k</td><td rowspan=1 colspan=1>Domain Adaptation+ Incremental Learning</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>2(270)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[5,0,1,3,9]</td><td></td><td rowspan=1 colspan=1>100k/100k</td><td rowspan=1 colspan=1>Domain Adaptation+ Incremental Learning</td></tr><tr><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=3>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td></tr><tr><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=3>：</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>：</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1.1 EXPERIMENT DETAILS AND RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We used a basic CNN architecture composed of 3 conv-layers of 32 filters and kernel-size - (3x3) followed by 2 dense layers “fc1” and “fc2” of feature length 256 and a softmax. Our feature transformer networks at every episode aims to transform “fc1” features using 2 additional dense layers of feature length 256. Feature transformers from previous episode serve as initialization for current episode and these models were optimized for the cumulative loss (equation 7), with $\\lambda = 0 . 2$ . All the feature transformers were trained for only 3 epochs, with batch size of 32. ",
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+    {
+        "type": "text",
+        "text": "We compare our results to the two obvious life-long learners - naive and cumulative training, which serve as the lower and upper bounds of performance respectively in the absence of other competitive algorithms for this setting. While the naive learner finetunes the entire network on the latest episode data, cumulative learner accumulates data from all the episodes seen so far and retrains the model. Fig. 2a highlights the remarkable performance of the proposed approach. We show the performance evolution over the first 25 episodes in Fig. 2a while, performance comparisons over 80 episodes across multiple experiment runs are averaged and shown in Table. 2b. We demonstrate that just by storing features in memory and learning feature transformers at every episode, we achieve almost similar performance to the gold-standard result. Further, it also underlines the applicability of our approach as single framework to combat different requirements of lifelong learning. ",
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+        "type": "text",
+        "text": "Finally, we also compare our results (figure 3) in the conventional MT setting described in LopezPaz et al. (2017), whose open source implementation we used for our experiments. For details on the terminology and compared methods, the readers are encouraged to refer to Lopez-Paz et al. (2017). Fig. 3a demonstrates clear superiority of our approach over multiple methods compared. Additionally, backward transfer (BWT) - quantitative metric that models the deterioration of performance on older tasks while learning new tasks, is also negligible for our method, which shows resistance to catastrophic forgetting. Fig. 3b highlights this further as shown by accuracy on first task after learning subsequent tasks. ",
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+        "type": "text",
+        "text": ".2 PNEUMOTHORAX IDENTIFICATION FROM X-RAY IMAGES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We simulate a practical manifestation of lifelong learning where a model trained to detect pneumothorax is deployed in a hospital with data arriving incrementally. We utilize a subset of ChestXRay (Wang et al., 2017b) dataset, which consists of chest $\\mathrm { X }$ -rays labeled with corresponding diagnoses. We simulated incremental learning by providing the 8k training images in incremental batches of $2 \\mathrm { k }$ and measured the performance on held-out validation set of $2 \\mathrm { k }$ images. Fig. 4a establishes the baselines for the experiment. As in previous experiment, naive and cumulative training define the performance bounds. To clearly highlight the value of our feature transfom, we also add another strong baseline - naive Learner with center loss, which learns on the recent batch but with an augmented loss function (equation 7). In spite of a gain of $5 \\%$ due to center loss, there is still a loss of $4 \\%$ performance in the incremental learning set-up in the fourth batch of 2k. ",
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+    {
+        "type": "text",
+        "text": "Experimental Details and Results: We used a pre-trained VGG-network (Simonyan & Zisserman, 2014) as the base network and explored the use of features from different layers of the VGG-network namely - post the two pooling layers and fully connected layers. Feature transformer network essentially had one additional dense layer per step and was optimized for (equation 7). ",
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+    {
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+        "img_path": "images/4162168d34baeb3bcd95a8c308df9930fbcde7ccadbffd0465c37f7ccda9786a.jpg",
+        "image_caption": [
+            "Figure 2: (a) Performance evolution over first 25 episodes and (b) Average performance across multiple runs at the end of lifelong experiments "
+        ],
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+        "image_caption": [
+            "Figure 3: Comparison of proposed approach in conventional multi-task setting "
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+    {
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+        "img_path": "images/7b8f4de1e5d4cdd04aac0b1d1733bb9cdee426a0236cea34c72a01f569a3bb20.jpg",
+        "image_caption": [
+            "Figure 4: (a) Performance Comparison on validation dataset and (b) Comparison of feature transformers from different base layers "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Fig. 4a captures the performance of feature transformer with the base features being extracted from first pooling layer - block3 pool. After fourth batch of data, feature transformer result almost matches the performance of cumulative training. This performance is achieved despite not having access to the full images but only the stored features. Table. 4b also presents the performance of feature transformer depending upon the base features used. It can be noted that performance is lowest for the layer that is closer to the classification layer - fc 2. This is intuitively satisfying because, the further layers in a deep neural network will be more finely tuned towards the specific task and deprives the feature transform of any general features. ",
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+    {
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+        "image_caption": [
+            "Figure 5: Comparison with state-of-the-art methods - iCIFAR100 dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 ICIFAR100 DATASET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We present the 100 classes from CIFAR100 dataset in a sequence of 20 tasks comprising of 5 classes each. Similar to Lomonaco & Maltoni (2017), we use the definitions for MT and single-incremental task (SIT). In an MT setting, evaluation is performed only on the new tasks exposed to the learner in the current episode. We start with a VGG type architecture, pretrained on iCIFAR10 dataset as suggested by Lopez-Paz et al. (2017) and Lomonaco & Maltoni (2017). Base features are extracted from flatten layer (before the fully connected layers) and our feature transformers included two dense layers with feature length $= 2 5 6$ . Similar to our earlier experiments, the feature transformer module from the previous episode initializes the current episode transformer and is optimized for the cumulative loss, with NoEpochs $= 3 0$ and batch size $= 3 2$ . ",
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+    {
+        "type": "text",
+        "text": "4.3.1 MULTI-TASK SETTING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Fig 5a demonstrates the superiority of our approach by a significant margin of ${ > } 1 0 \\%$ . Further, we have negligible Backward Transfer. ",
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+    {
+        "type": "text",
+        "text": "4.3.2 SINGLE INCREMENTAL TASK SETTING",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Fig 5b captures the performance over 20 batches of data for our method along with cumulative and naive learner. Unsurprisingly, naive learner performs very poorly, while feature transformer displays exemplary performance numbers of $4 0 \\%$ validation acccuracy after encountering all 20 episodes, while compared to cumulative learner at $5 0 \\%$ . Our method significantly outperforms AR1 (Lomonaco & Maltoni, 2017), while iCaRL (Rebuffi et al., 2017) achieves best-in-class performance close to gold-standard cumulative learner. This is not surprising because iCaRL is an explicit rehearsal technique, where exemplar images from previous episodes are stored and replayed while learning new tasks, while we only store low dimensional features. ",
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+    {
+        "type": "text",
+        "text": "4.4 EFFECT OF LIMITED MEMORY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "One of the drawbacks of the proposed approach is the assumption of infinite memory and need to store features computed on all samples observed so far. To understand the extent of this limitation, we performed ablation experiments limiting the amount of history replayed as well as computing the storage requirements involved. ",
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+    {
+        "type": "text",
+        "text": "Storing all features is not necessary We studied the effect of size of memory by limiting the number of samples stored to a smaller percentage. We observed that performance dropped from $9 7 \\%$ to $9 4 \\%$ when we reduced the memory size to only $2 5 \\%$ the original size on MNIST rotations (Table. 2). We performed similar experiments on Pneumothorax classification problem and achieved similar trends as shown Table. 3, clearly demonstrating the resistance of the proposed method. ",
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+    {
+        "type": "text",
+        "text": "Storing all features is not prohibitive Additionally, calculations of size-on-disk suggests that storing features of the entire history is not prohibitive. A typical natural/medical image is $2 5 6 ^ { * } 2 5 6 ^ { * } 3$ integers or more, whereas our representation is only 4096 floats (16kb). Even the largest available medical image repository of $1 0 0 \\mathrm { k }$ X-ray images takes 1.6GB which is not huge. These are con",
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+    {
+        "type": "table",
+        "img_path": "images/e9e4729c16bb88e9211864d8dedd4ae44730c20f3d20205788059608a6c598be.jpg",
+        "table_caption": [
+            "Table 2: MNIST rotations "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>%ofSamples storedfrom history</td><td rowspan=1 colspan=1>FeatureTransformerVal Acc</td><td rowspan=1 colspan=1>CumulativeLearnerVal Acc</td></tr><tr><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>94%</td><td rowspan=1 colspan=1>95%</td></tr><tr><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>94.8%</td><td rowspan=1 colspan=1>98%</td></tr><tr><td rowspan=1 colspan=1>75</td><td rowspan=1 colspan=1>96%</td><td rowspan=1 colspan=1>98.5%</td></tr><tr><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>97 %</td><td rowspan=1 colspan=1>99 %</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d5bc4e41a4d12ef0c3db2a84280ff1067dcc3d0a1cebc050b5cb0f94dc6bd1b.jpg",
+        "table_caption": [
+            "Table 3: Pneumothorax dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>%ofSamples storedfrom history</td><td rowspan=1 colspan=1>FeatureTransformerVal Acc</td></tr><tr><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>80.95%</td></tr><tr><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>82.95%</td></tr><tr><td rowspan=1 colspan=1>75</td><td rowspan=1 colspan=1>85.85 %</td></tr><tr><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>86.94 %</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table: Performance comparisons with limited memory budget servative estimates. A standard medical image can be of much larger size $1 0 2 4 ^ { * } 1 0 2 4 )$ and in 3-D (minimum 10 slices). Any exemplar-based method (iCARL) will have severe storage limitations than our method. Additionally, storing 50 low-dimensional features occupies same memory as storing one exemplar image. This directly leads to storing more history compactly while addressing catastrophic forgetting and privacy. ",
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+    {
+        "type": "text",
+        "text": "4.5 CONTROLLING THE GROWTH OF NETWORK CAPACITY ",
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+    {
+        "type": "text",
+        "text": "In the description of the feature transformer framework in section 3.1 and section 3.2, we provided a generic treatment of the method, where, at the end of the each episode, the features are transformed up to date and then stored in memory for the next episode. With this scheme, it becomes imperative to always augment the capacity of the network in order to learn new representations, resulting in ever-growing network capacity. However, this problem can be easily alleviated by partitioning the entire network into a base feature extractor and feature transformer layers. The base feature extractor remains always fixed, and it is only the output of base feature extractor that is always stored in memory. With this scheme, the feature transformer layers need not grow in capacity and only a few already existing layers can be adapted for the new data. When the capacity of the feature transformer layers is not sufficient, then it can be augmented by adding one or more extra layers. In either case, the stored base features are sufficient to train. ",
+        "bbox": [
+            173,
+            387,
+            825,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Effect of varying additional capacity ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            553,
+            419,
+            568
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We varied the size of feature transformers and observed the difference in performance. Table 4 shows that by halving the additional capacity does not change the performance on MNIST rotations dataset at all. In addition, we froze the capacity of feature transformers after 5th episode and adapted them till end of 80 episodes. It is striking that performance is still high. Similarly, for Pneumothorax classification, Table 5 shows the performance comparisons with varying capacity of 2 , 1 and zero fc layers post third episode. These experiments (along with Sec 4.4) clearly demonstrate that power of the proposed approach comes from learning separable representations continually and not necessarily from storing all features or additional capacity. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/2bc1fea6dad7e304097258309b2d6b33340b7655e85e0d14fc19c29cba7ff726.jpg",
+        "table_caption": [
+            "Table 4: MNIST rotations "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Incremental capacityadded per episode</td><td rowspan=1 colspan=1>Feature TransformerVal Acc</td></tr><tr><td rowspan=1 colspan=1>2 dense layers</td><td rowspan=1 colspan=1>96.4%</td></tr><tr><td rowspan=1 colspan=1>1 dense layers</td><td rowspan=1 colspan=1>96.5%</td></tr><tr><td rowspan=1 colspan=1>No additonal capacity(after 5th episode)</td><td rowspan=1 colspan=1>96.2%</td></tr></table>",
+        "bbox": [
+            174,
+            700,
+            491,
+            789
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/ae7aad5758c788f6134654dee55eefd87dbdc3c1c547a9c804b0359e7a35a7c9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Incremental capacityadded per episode</td><td rowspan=1 colspan=1>FeatureTransformerVal Acc</td></tr><tr><td rowspan=1 colspan=1>2 dense layers</td><td rowspan=1 colspan=1>86.94%</td></tr><tr><td rowspan=1 colspan=1>1 dense layers</td><td rowspan=1 colspan=1>86.43%</td></tr><tr><td rowspan=1 colspan=1>No additonal capacity(after 3rd episode)</td><td rowspan=1 colspan=1>86.41%</td></tr></table>",
+        "bbox": [
+            504,
+            700,
+            821,
+            789
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table: Performance comparisons with limited incremental compute ",
+        "bbox": [
+            276,
+            825,
+            720,
+            840
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            858,
+            312,
+            873
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In the final section, we discuss various points concerning our proposed approach. ",
+        "bbox": [
+            174,
+            882,
+            704,
+            897
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Bayesian interpretation of Feature transformers ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            909,
+            498,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our feature transformers framework - learning a new representation with every new episode of data, can be interpreted as a maximum-a-posteriori representation learning, with the previous representation acting as prior. The implementation described in this paper using the combination loss (equation 7) is one instantiation of a general incremental representation learning possible in our framework. We have cast the MAP estimate problem to a tractable optimization problem constrained by a center-loss. In future, we plan to explore other manifestations of our approach with different loss functions that can ensure better separability. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            200
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Information Loss, Incremental Capacity, Data Privacy ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            214,
+            540,
+            229
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table. 4b, feature transformer becomes less effective if the base features do not contain enough relevant information. This also means that additional capacity that every feature transformer adds may not help or in-fact be counter-productive. If the base features are extracted from layers close to the input image, there will be problem of traceability which violates the data privacy requirement we want to accomplish. We feel this a potential trade-off between performance and data privacy which we will investigate in future. ",
+        "bbox": [
+            174,
+            236,
+            825,
+            319
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Model compaction for cascade of feature transformers ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            333,
+            540,
+            347
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Another approach to control the growth of network capacity is model compaction, which will be our future work. At any point in time, the entire set of feature transformer layers can be replaced by a smaller and simpler network (possibly using distillation techniques) and then again allowed to grow subsequently. This cycle of grow-and-purge can be used to effectively manage the overall capacity of the network. ",
+        "bbox": [
+            174,
+            354,
+            825,
+            424
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            433,
+            287,
+            448
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "F. M. Castro, M. J. Mar´ın-Jimenez, N. Guil, C. Schmid, and K. Alahari. End-to-End Incremental ´ Learning. ArXiv e-prints, July 2018. ",
+        "bbox": [
+            176,
+            455,
+            823,
+            484
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chen He, Ruiping Wang, Shiguang Shan, and Xilin Chen. Exemplar-supported generative reproduction for class incremental learning. In $2 9 ^ { t h }$ British Machine Vision Conference (BMVC 2018), 3–6 Sep 2018. ",
+        "bbox": [
+            174,
+            493,
+            821,
+            536
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015. ",
+        "bbox": [
+            173,
+            545,
+            821,
+            574
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Khurram Javed and Muhammad Talha Paracha. Incremental classifier & representation learning. Learning, 3:4, 2018. ",
+        "bbox": [
+            174,
+            583,
+            821,
+            612
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the national academy of sciences, pp. 201611835, 2017. ",
+        "bbox": [
+            173,
+            621,
+            825,
+            678
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jeongtae Lee, Jaehong Yoon, Eunho Yang, and Sung Ju Hwang. Lifelong learning with dynamically expandable networks. CoRR, abs/1708.01547, 2017. ",
+        "bbox": [
+            173,
+            686,
+            823,
+            717
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhizhong Li and Derek Hoiem. Learning without forgetting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017. ",
+        "bbox": [
+            173,
+            724,
+            823,
+            755
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xialei Liu, Marc Masana, Luis Herranz, Joost Van de Weijer, Antonio M Lopez, and Andrew D Bagdanov. Rotate your networks: Better weight consolidation and less catastrophic forgetting. arXiv preprint arXiv:1802.02950, 2018. ",
+        "bbox": [
+            176,
+            763,
+            823,
+            806
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Vincenzo Lomonaco and Davide Maltoni. Core50: a new dataset and benchmark for continuous object recognition. In Sergey Levine, Vincent Vanhoucke, and Ken Goldberg (eds.), Proceedings of the 1st Annual Conference on Robot Learning, volume 78 of Proceedings of Machine Learning Research, pp. 17–26. PMLR, 13–15 Nov 2017. URL http://proceedings.mlr.press/ v78/lomonaco17a.html. ",
+        "bbox": [
+            174,
+            815,
+            825,
+            886
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "David Lopez-Paz et al. Gradient episodic memory for continual learning. In Advances in Neural Information Processing Systems, pp. 6467–6476, 2017. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Michael McCloskey and Neal J Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pp. 109–165. Elsevier, 1989. ",
+        "bbox": [
+            173,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thomas Mensink, Jakob Verbeek, Florent Perronnin, and Gabriela Csurka. Metric learning for large scale image classification: Generalizing to new classes at near-zero cost. In Computer Vision– ECCV 2012, pp. 488–501. Springer, 2012. ",
+        "bbox": [
+            178,
+            155,
+            823,
+            198
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Maxime Oquab, Leon Bottou, Ivan Laptev, and Josef Sivic. Learning and transferring mid-level image representations using convolutional neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1717–1724, 2014. ",
+        "bbox": [
+            174,
+            207,
+            823,
+            250
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, and Christoph H. Lampert. iCaRL: Incremental classifier and representation learning. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5533–5542, 2017. ",
+        "bbox": [
+            174,
+            257,
+            823,
+            301
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Anthony Robins. Catastrophic forgetting, rehearsal and pseudorehearsal. Connection Science, 7: 123–146, 1995. ",
+        "bbox": [
+            173,
+            309,
+            821,
+            338
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual learning with deep generative replay. In Advances in Neural Information Processing Systems, pp. 2990–2999, 2017. ",
+        "bbox": [
+            174,
+            347,
+            823,
+            376
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. ",
+        "bbox": [
+            173,
+            385,
+            823,
+            414
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Amal Triki Rannen, Rahaf Aljundi, Mathew B Blaschko, and Tinne Tuytelaars. Encoder based lifelong learning. IEEE International Conference of Computer Vision, 2017. ",
+        "bbox": [
+            173,
+            422,
+            821,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ragav Venkatesan, Hemanth Venkateswara, Sethuraman Panchanathan, and Baoxin Li. A strategy for an uncompromising incremental learner. arXiv preprint arXiv:1705.00744, 2017. ",
+        "bbox": [
+            173,
+            460,
+            823,
+            489
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "X. Wang, Y. Peng, L. Lu, Z. Lu, M. Bagheri, and R. M. Summers. Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3462–3471, July 2017a. doi: 10.1109/CVPR.2017.369. ",
+        "bbox": [
+            173,
+            498,
+            825,
+            555
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "X. Wang, Y. Peng, L. Lu, Z. Lu, M. Bagheri, and R. M. Summers. Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3462–3471, July 2017b. doi: 10.1109/CVPR.2017.369. ",
+        "bbox": [
+            174,
+            564,
+            825,
+            619
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yandong Wen, Kaipeng Zhang, Zhifeng Li, and Yu Qiao. A discriminative feature learning approach for deep face recognition. In ECCV, 2016. ",
+        "bbox": [
+            171,
+            628,
+            823,
+            659
+        ],
+        "page_idx": 10
+    }
+]

parse/train/H1Y8hhg0b/H1Y8hhg0b.md

@@ -0,0 +1,341 @@
+# LEARNING SPARSE NEURAL NETWORKSTHROUGH $L _ { 0 }$ REGULARIZATION
+
+Christos Louizos∗ University of Amsterdam TNO, Intelligent Imaging c.louizos@uva.nl
+
+Max Welling   
+University of Amsterdam CIFAR   
+m.welling@uva.nl
+
+Diederik P. Kingma OpenAI dpkingma@openai.com
+
+# ABSTRACT
+
+We propose a practical method for $L _ { 0 }$ norm regularization for neural networks: pruning the network during training by encouraging weights to become exactly zero. Such regularization is interesting since (1) it can greatly speed up training and inference, and (2) it can improve generalization. AIC and BIC, well-known model selection criteria, are special cases of $L _ { 0 }$ regularization. However, since the $L _ { 0 }$ norm of weights is non-differentiable, we cannot incorporate it directly as a regularization term in the objective function. We propose a solution through the inclusion of a collection of non-negative stochastic gates, which collectively determine which weights to set to zero. We show that, somewhat surprisingly, for certain distributions over the gates, the expected $L _ { 0 }$ regularized objective is differentiable with respect to the distribution parameters. We further propose the hard concrete distribution for the gates, which is obtained by “stretching” a binary concrete distribution and then transforming its samples with a hard-sigmoid. The parameters of the distribution over the gates can then be jointly optimized with the original network parameters. As a result our method allows for straightforward and efficient learning of model structures with stochastic gradient descent and allows for conditional computation in a principled way. We perform various experiments to demonstrate the effectiveness of the resulting approach and regularizer.
+
+# 1 INTRODUCTION
+
+Deep neural networks are flexible function approximators that have been very successful in a broad range of tasks. They can easily scale to millions of parameters while allowing for tractable optimization with mini-batch stochastic gradient descent (SGD), graphical processing units (GPUs) and parallel computation. Nevertheless they do have drawbacks. Firstly, it has been shown in recent works (Han et al., 2015; Ullrich et al., 2017; Molchanov et al., 2017) that they are greatly overparametrized as they can be pruned significantly without any loss in accuracy; this exhibits unnecessary computation and resources. Secondly, they can easily overfit and even memorize random patterns in the data (Zhang et al., 2016), if not properly regularized. This overfitting can lead to poor generalization in practice.
+
+A way to address both of these issues is by employing model compression and sparsification techniques. By sparsifying the model, we can avoid unnecessary computation and resources, since irrelevant degrees of freedom are pruned away and do not need to be computed. Furthermore, we reduce its complexity, thus penalizing memorization and alleviating overfitting.
+
+A conceptually attractive approach is the $L _ { 0 }$ norm regularization of (blocks of) parameters; this explicitly penalizes parameters for being different than zero with no further restrictions. However, the combinatorial nature of this problem makes for an intractable optimization for large models.
+
+In this paper we propose a general framework for surrogate $L _ { 0 }$ regularized objectives. It is realized by smoothing the expected $L _ { 0 }$ regularized objective with continuous distributions in a way that can maintain the exact zeros in the parameters while still allowing for efficient gradient based optimization. This is achieved by transforming continuous random variables (r.v.s) with a hard nonlinearity, the hard-sigmoid. We further propose and employ a novel distribution obtained by this procedure; the hard concrete. It is obtained by “stretching” a binary concrete random variable (Maddison et al., 2016; Jang et al., 2016) and then passing its samples through a hard-sigmoid. We demonstrate the effectiveness of this simple procedure in various experiments.
+
+![](images/1ce9b8a3efa0b28dff6f2b5f0d0b5baf383aeb8c996ef35676c1753856894556.jpg)  
+Figure 1: $L _ { p }$ norm penalties for a parameter $\theta$ according to different values of $p$ . It is easily observed that both weight decay and Lasso, $p = 2$ and $p = 1$ respectively, impose shrinkage for large values of $\theta$ . By gradually allowing $p < 1$ we observe that the shrinkage is reduced and at the limit of $p = 0$ we observe that the penalty is a constant for $\theta \neq 0$ .
+
+# 2 MINIMIZING THE $L _ { 0 }$ NORM OF PARAMETRIC MODELS
+
+One way to sparsify parametric models, such as deep neural networks, with the least assumptions about the parameters is the following; let $\mathcal { D }$ be a dataset consisting of $N$ i.i.d. input output pairs $\left\{ ( \mathbf { x } _ { 1 } , \mathbf { y } _ { 1 } ) , \dotsc , ( \mathbf { x } _ { N } , \mathbf { y } _ { N } ) \right\}$ and consider a regularized empirical risk minimization procedure with an $L _ { 0 }$ regularization on the parameters $\pmb \theta$ of a hypothesis (e.g. a neural network) $h ( \cdot ; \bar { \pmb \theta } ) ^ { 1 }$ :
+
+$$
+\mathcal { R } ( \pmb { \theta } ) = \frac { 1 } { N } \bigg ( \sum _ { i = 1 } ^ { N } \mathcal { L } \big ( h ( \mathbf { x } _ { i } ; \pmb { \theta } ) , \mathbf { y } _ { i } \big ) \bigg ) + \lambda \| \pmb { \theta } \| _ { 0 } , \qquad \| \pmb { \theta } \| _ { 0 } = \sum _ { j = 1 } ^ { | \theta | } \mathbb { I } [ \theta _ { j } \neq 0 ] ,
+$$
+
+where $| \theta |$ is the dimensionality of the parameters, $\lambda$ is a weighting factor for the regularization and $\mathcal { L } ( \cdot )$ corresponds to a loss function, e.g. cross-entropy loss for classification or mean-squared error for regression. The $L _ { 0 }$ norm penalizes the number of non-zero entries of the parameter vector and thus encourages sparsity in the final estimates $\pmb { \theta } ^ { * }$ . The Akaike Information Criterion (AIC) (Akaike, 1998) and the Bayesian Information Criterion (BIC) (Schwarz et al., 1978), well-known model selection criteria, correspond to specific choices of $\lambda$ . Notice that the $L _ { 0 }$ norm induces no shrinkage on the actual values of the parameters $\pmb { \theta }$ ; this is in contrast to e.g. $L _ { 1 }$ regularization and the Lasso (Tibshirani, 1996), where the sparsity is due to shrinking the actual values of $\pmb { \theta }$ . We provide a visualization of this effect in Figure 1.
+
+Unfortunately, optimization under this penalty is computationally intractable due to the nondifferentiability and combinatorial nature of $2 ^ { | \theta | }$ possible states of the parameter vector $\pmb \theta$ . How can we relax the discrete nature of the $L _ { 0 }$ penalty such that we allow for efficient continuous optimization of Eq. 1, while allowing for exact zeros in the parameters? This section will present the necessary details of our approach.
+
+# 2.1 A GENERAL RECIPE FOR EFFICIENTLY MINIMIZING $L _ { 0 }$ NORMS
+
+Consider the $L _ { 0 }$ norm under a simple re-parametrization of $\pmb \theta$ :
+
+$$
+\theta _ { j } = \tilde { \theta } _ { j } z _ { j } , \qquad z _ { j } \in \{ 0 , 1 \} , \qquad \tilde { \theta } _ { j } \neq 0 , \qquad \lVert \theta \rVert _ { 0 } = \sum _ { j = 1 } ^ { \lvert \theta \rvert } z _ { j } ,
+$$
+
+where the $z _ { j }$ correspond to binary “gates” that denote whether a parameter is present and the $L _ { 0 }$ norm corresponds to the amount of gates being “on”. By letting $q ( z _ { j } | \pi _ { j } ) = \mathbf { B } \mathrm { e r n } ( \pi _ { j } )$ be a Bernoulli distribution over each gate $z _ { j }$ we can reformulate the minimization of Eq. 1 as penalizing the number of parameters being used, on average, as follows:
+
+$$
+\begin{array} { r l r } {  { \mathcal { R } ( \tilde { \pmb { \theta } } , \boldsymbol { \pi } ) = \mathbb { E } _ { q ( \mathbf { z } | \boldsymbol { \pi } ) } [ \frac { 1 } { N } \bigg ( \displaystyle \sum _ { i = 1 } ^ { N } \mathcal { L } \big ( h ( \mathbf { x } _ { i } ; \tilde { \pmb { \theta } } \odot \mathbf { z } ) , \mathbf { y } _ { i } \big ) \bigg ) ] + \lambda \displaystyle \sum _ { j = 1 } ^ { | \theta | } \pi _ { j } , } } \\ & { } & { \tilde { \pmb { \theta } } ^ { * } , \boldsymbol { \pi } ^ { * } = \underset { \tilde { \pmb { \theta } } , \boldsymbol { \pi } } { \operatorname { a r g m i n } } \{ \mathcal { R } ( \tilde { \pmb { \theta } } , \boldsymbol { \pi } ) \} , } \end{array}
+$$
+
+where $\odot$ corresponds to the elementwise product. The objective described in Eq. 3 is in fact a special case of a variational bound over the parameters involving spike and slab (Mitchell & Beauchamp, 1988) priors and approximate posteriors; we refer interested readers to appendix A.
+
+Now the second term of the r.h.s. of Eq. 3 is straightforward to minimize however the first term is problematic for $\pi$ due to the discrete nature of $\mathbf { z }$ , which does not allow for efficient gradient based optimization. While in principle a gradient estimator such as the REINFORCE (Williams, 1992) could be employed, it suffers from high variance and control variates (Mnih & Gregor, 2014; Mnih & Rezende, 2016; Tucker et al., 2017), that require auxiliary models or multiple evaluations of the network, have to be employed. Two simpler alternatives would be to use either the straightthrough (Bengio et al., 2013) estimator as done at Srinivas et al. (2017) or the concrete distribution as e.g. at Gal et al. (2017). Unfortunately both of these approach have drawbacks; the first one provides biased gradients due to ignoring the Heaviside function in the likelihood during the gradient evaluation whereas the second one does not allow for the gates (and hence parameters) to be exactly zero during optimization, thus precluding the benefits of conditional computation (Bengio et al., 2013).
+
+Fortunately, there is a simple alternative way to smooth the objective such that we allow for efficient gradient based optimization of the expected $L _ { 0 }$ norm along with zeros in the parameters $\pmb \theta$ . Let s be a continuous random variable with a distribution $q ( \mathbf { s } )$ that has parameters $\phi$ . We can now let the gates $\mathbf { z }$ be given by a hard-sigmoid rectification of $\mathbf { s } ^ { 2 }$ , as follows:
+
+$$
+\begin{array} { l } { \mathbf { s } \sim q ( \mathbf { s } | \phi ) } \\ { \mathbf { z } = \operatorname* { m i n } ( \mathbf { 1 } , \operatorname* { m a x } ( \mathbf { 0 } , \mathbf { s } ) ) . } \end{array}
+$$
+
+This would then allow the gate to be exactly zero and, due to the underlying continuous random variable s, we can still compute the probability of the gate being non-zero (active). This is easily obtained by the cumulative distribution function (CDF) ${ \bf { \bar { Q } } } ( \cdot )$ of $\mathbf { s }$ :
+
+$$
+q ( \mathbf { z } \neq 0 | \phi ) = 1 - Q ( \mathbf { s } \leq 0 | \phi ) ,
+$$
+
+i.e. it is the probability of the s variable being positive. We can thus smooth the binary Bernoulli gates $\mathbf { z }$ appearing in Eq. 3 by employing continuous distributions in the aforementioned way:
+
+$$
+\mathcal { R } ( \tilde { \pmb { \theta } } , \phi ) = \mathbb { E } _ { q ( \{ \pmb { \theta } \} ) } \left[ \frac { 1 } { N } \bigg ( \sum _ { i = 1 } ^ { N } \mathcal { L } \big ( h ( \mathbf { x } _ { i } ; \tilde { \pmb { \theta } } \odot g ( \mathbf { s } ) ) , \mathbf { y } _ { i } \big ) \bigg ) \right] + \lambda \sum _ { j = 1 } ^ { | \theta | } \big ( 1 - Q ( s _ { j } \le 0 | \phi _ { j } ) \big ) ,
+$$
+
+Notice that this is a close surrogate to the original objective function in Eq. 3, as we similarly have a cost that explicitly penalizes the probability of a gate being different from zero. Now for continuous
+
+distributions $q ( \mathbf { s } )$ that allow for the reparameterization trick (Kingma & Welling, 2014; Rezende et al., 2014) we can express the objective in Eq. 7 as an expectation over a parameter free noise distribution $p ( \epsilon )$ and a deterministic and differentiable transformation $f ( \cdot )$ of the parameters $\phi$ and $\epsilon$
+
+$$
+\mathcal { R } ( \tilde { \pmb { \theta } } , \phi ) = \mathbb { E } _ { p ( \epsilon ) } \left[ \frac { 1 } { N } \bigg ( \sum _ { i = 1 } ^ { N } \mathcal { L } \big ( h ( \mathbf { x } _ { i } ; \tilde { \pmb { \theta } } \odot g ( f ( \phi , \epsilon ) ) ) , \mathbf { y } _ { i } \big ) \bigg ) \right] + \lambda \sum _ { j = 1 } ^ { | \theta | } \big ( 1 - Q ( s _ { j } \le 0 | \phi _ { j } ) \big ) ,
+$$
+
+which allows us to make the following Monte Carlo approximation to the (generally) intractable expectation over the noise distribution $p ( \epsilon )$ :
+
+$$
+\begin{array} { r l } & { \hat { \mathcal { R } } ( \tilde { \pmb { \theta } } , \phi ) = \displaystyle \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \left( \frac { 1 } { N } \bigg ( \sum _ { i = 1 } ^ { N } \mathcal { L } \big ( h ( \mathbf { x } _ { i } ; \tilde { \pmb { \theta } } \odot \mathbf { z } ^ { ( l ) } ) , \mathbf { y } _ { i } \big ) \bigg ) \right) + \lambda \sum _ { j = 1 } ^ { | \boldsymbol { \theta } | } \big ( 1 - Q ( s _ { j } \le 0 | \phi _ { j } ) \big ) } \\ & { \qquad = \mathcal { L } _ { E } ( \tilde { \pmb { \theta } } , \phi ) + \lambda \mathcal { L } _ { C } ( \phi ) , \quad \mathrm { w h e r e ~ } \ \mathbf { z } ^ { ( l ) } = g \big ( f ( \phi , \epsilon ^ { ( l ) } ) \big ) \ \mathrm { a n d } \ \epsilon ^ { ( l ) } \sim p ( \epsilon ) . } \end{array}
+$$
+
+$\mathcal { L } _ { E }$ corresponds to the error loss that measures how well the model is fitting the current dataset whereas $\mathcal { L } _ { C }$ refers to the complexity loss that measures the flexibility of the model. Crucially, the total cost in Eq. 9 is now differentiable w.r.t. $\phi$ , thus enabling for efficient stochastic gradient based optimization, while still allowing for exact zeros at the parameters. One price we pay is that now the gradient of the log-likelihood w.r.t. the parameters $\phi$ of $q ( \mathbf { s } )$ is sparse due to the rectifications; nevertheless this should not pose an issue considering the prevalence of rectified linear units in neural networks. Furthermore, due to the stochasticity at s the hard-sigmoid gate $\mathbf { z }$ is smoothed to a soft version on average, thus allowing for gradient based optimization to succeed, even when the mean of s is negative or larger than one. An example visualization can be seen in Figure 2b. It should be noted that a similar argument was also shown at Bengio et al. (2013), where with logistic noise a rectifier nonlinearity was smoothed to a softplus3 on average.
+
+# 2.2 THE HARD CONCRETE DISTRIBUTION
+
+The framework described in Section 2.1 gives us the freedom to choose an appropriate smoothing distribution $q ( \mathbf { s } )$ . A choice that seems to work well in practice is the following; assume that we have a binary concrete (Maddison et al., 2016; Jang et al., 2016) random variable $s$ distributed in the $( 0 , 1 )$ interval with probability density $q _ { s } ( s | \phi )$ and cumulative density $Q _ { s } ( s | \phi )$ . The parameters of the distribution are $\phi = ( \log \alpha , \beta )$ , where $\log \alpha$ is the location and $\beta$ is the temperature. We can “stretch” this distribution to the $( \gamma , \zeta )$ interval, with $\gamma < 0$ and $\zeta > 1$ , and then apply a hard-sigmoid on its random samples:
+
+$$
+\begin{array} { r l } & { u \sim \mathcal { U } ( 0 , 1 ) , \quad s = \mathrm { S i g m o i d } \big ( ( \log u - \log ( 1 - u ) + \log \alpha ) / \beta \big ) , \quad \bar { s } = s ( \zeta - \gamma ) + \gamma , } \\ & { \qquad z = \operatorname* { m i n } ( 1 , \operatorname* { m a x } ( 0 , \bar { s } ) ) . } \end{array}
+$$
+
+This would then induce a distribution where the probability mass of $q _ { \bar { s } } ( { \bar { s } } | \phi )$ on the negative values, $Q _ { \bar { s } } ( 0 | \phi )$ , is “folded” to a delta peak at zero, the probability mass on values larger than one, $1 -$ $Q _ { \bar { s } } ( 1 | \phi )$ , is “folded” to a delta peak at one and the original distribution $q _ { \bar { s } } ( { \bar { s } } | \phi )$ is truncated to the (0, 1) range. We provide more information and the density of the resulting distribution at the appendix.
+
+Notice that a similar behavior would have been obtained even if we passed samples from any other distribution over the real line through a hard-sigmoid. The only requirement of the approach is that we can evaluate the CDF of $\bar { s }$ at 0 and 1. The main reason for picking the binary concrete is its close ties with Bernoulli r.v.s. It was originally proposed at Maddison et al. (2016); Jang et al. (2016) as a smooth approximation to Bernoulli r.vs, a fact that allows for gradient based optimization of its parameters through the reparametrization trick. The temperature $\beta$ controls the degree of approximation, as with $\beta = 0$ we can recover the original Bernoulli r.v. (but lose the differentiable properties) whereas with $0 < \beta < 1$ we obtain a probability density that concentrates its mass near the endpoints (e.g. as shown in Figure 2a). As a result, the hard concrete also inherits the same theoretical properties w.r.t. the Bernoulli distribution. Furthermore, it can serve as a better approximation of the discrete nature, since it includes $\{ 0 , 1 \}$ in its support, while still allowing for (sub)gradient optimization of its parameters due to the continuous probability mass that connects those two values. We can also view this distribution as a “rounded" version of the original binary concrete, where values larger than $\frac { 1 - \gamma } { \zeta - \gamma }$ are rounded to one whereas values smaller than $\frac { - \gamma } { \zeta - \gamma }$ are rounded to zero. We provide an example visualization of the hard concrete distribution in Figure 2a.
+
+![](images/acf2c6fdf4de450085133af4706b8591a0adb311cdedf797b18bc8fa891d39d7.jpg)  
+Figure 2: (a) The binary concrete distribution with location $\log \alpha = 0$ and temperature $\beta = 0 . 5$ and the hard concrete equivalent distribution obtained by stretching the concrete distribution to $( \gamma = - 0 . 1 , \zeta = 1 . 1 )$ and then applying a hard-sigmoid. Under this specification the hard concrete distribution assigns, roughly, half of its mass to $\{ 0 , 1 \}$ and the rest to $( 0 , 1 )$ . (b) The expected value of the afforementioned concrete and hard concrete gate as a function of the location $\log \alpha$ , obtained by averaging 10000 samples. We also added the value of the gates obtained by removing the noise entirely. We can see that the noise smooths the hard-sigmoid to a sigmoid on average.
+
+The $L _ { 0 }$ complexity loss of the objective in Eq. 9 under the hard concrete r.v. is conveniently expressed as follows:
+
+$$
+\mathcal { L } _ { C } = \sum _ { j = 1 } ^ { | \theta | } \big ( 1 - Q _ { \bar { s } _ { j } } ( 0 | \phi ) \big ) = \sum _ { j = 1 } ^ { | \theta | } \mathrm { S i g m o i d } \big ( \log \alpha _ { j } - \beta \log \frac { - \gamma } { \zeta } \big ) .
+$$
+
+At test time we use the following estimator for the final parameters $\pmb { \theta } ^ { * }$ under a hard concrete gate:
+
+$$
+\hat { \bf z } = \mathrm { m i n } ( { \bf 1 } , \mathrm { m a x } ( { \bf 0 } , \mathrm { S i g m o i d } ( \mathrm { l o g } { \pmb \alpha } ) ( \zeta - \gamma ) + \gamma ) ) , \qquad { \pmb \theta } ^ { \ast } = \tilde { \pmb \theta } ^ { \ast } \odot \hat { \bf z } .
+$$
+
+# 2.3 COMBINING THE $L _ { 0 }$ NORM WITH OTHER NORMS
+
+While the $L _ { 0 }$ norm leads to sparse estimates without imposing any shrinkage on $\pmb \theta$ it might still be desirable to impose some form of prior assumptions on the values of $\pmb { \theta }$ with alternative norms, e.g. impose smoothness with the $L _ { 2 }$ norm (i.e. weight decay). In the following we will show how this combination is feasible for the $L _ { 2 }$ norm. The expected $L _ { 2 }$ norm under the Bernoulli gating mechanism can be conveniently expressed as:
+
+$$
+\mathbb { E } _ { q ( \mathbf { z } | \pi ) } \left[ \lVert \pmb { \theta } \rVert _ { 2 } ^ { 2 } \right] = \sum _ { j = 1 } ^ { | \theta | } \mathbb { E } _ { q ( z _ { j } | \pi _ { j } ) } \left[ z _ { j } ^ { 2 } \tilde { \theta } _ { j } ^ { 2 } \right] = \sum _ { j = 1 } ^ { | \theta | } \pi _ { j } \tilde { \theta } _ { j } ^ { 2 } ,
+$$
+
+where $\pi _ { j }$ corresponds to the success probability of the Bernoulli gate $z _ { j }$ . To maintain a similar expression with our smoothing mechanism, and avoid extra shrinkage for the gates $z _ { j }$ , we can take into account that the standard $L _ { 2 }$ norm penalty is proportional to the negative log density of a zero mean Gaussian prior with a standard deviation of $\sigma = 1$ . We will then assume that the $\sigma$ for each $\theta$ is governed by $z$ in a way that when $z = 0$ we have that $\sigma = 1$ and when $z > 0$ we have that $\sigma = z$ . As a result, we can obtain the following expression for the $L _ { 2 }$ penalty (where $\begin{array} { r } { \hat { \theta } = \frac { \theta } { \sigma } , } \end{array}$ ):
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } _ { q ( \mathbf { z } | \phi ) } \left[ \| \hat { \theta } \| _ { 2 } ^ { 2 } \right] = \sum _ { j = 1 } ^ { | \theta | } \left( Q _ { \bar { s } _ { j } } ( 0 | \phi _ { j } ) \frac { 0 } { 1 } + \left( 1 - Q _ { \bar { s } _ { j } } ( 0 | \phi _ { j } ) \right) \mathbb { E } _ { q ( z _ { j } | \phi _ { j } , \bar { s } _ { j } > 0 ) } \left[ \frac { \widetilde { \theta } _ { j } ^ { 2 } \chi _ { j } ^ { 2 } } { \widetilde { \chi } _ { j } ^ { 2 } } \right] \right) } \\ { \displaystyle \qquad = \sum _ { j = 1 } ^ { | \theta | } \left( 1 - Q _ { \bar { s } _ { j } } ( 0 | \phi _ { j } ) \right) \widetilde { \theta } _ { j } ^ { 2 } . } \end{array}
+$$
+
+# 2.4 GROUP SPARSITY UNDER AN $L _ { 0 }$ NORM
+
+For reasons of computational efficiency it is usually desirable to perform group sparsity instead of parameter sparsity, as this can allow for practical computation savings. For example, in neural networks speedups can be obtained by employing a dropout (Srivastava et al., 2014) like procedure with neuron sparsity in fully connected layers or feature map sparsity for convolutional layers (Wen et al., 2016; Louizos et al., 2017; Neklyudov et al., 2017). This is straightforward to do with hard concrete gates; simply share the gate between all of the members of the group. The expected $L _ { 0 }$ and, according to section 2.3, $L _ { 2 }$ penalties in this scenario can be rewritten as:
+
+$$
+\begin{array} { r l } & { \displaystyle \mathbb { E } _ { q ( \mathbf { z } | \boldsymbol { \phi } ) } \left[ \| \pmb { \theta } \| _ { 0 } \right] = \sum _ { g = 1 } ^ { | G | } | g | \left( 1 - Q ( s _ { g } \leq 0 | \phi _ { g } ) \right) } \\ & { \displaystyle \mathbb { E } _ { q ( \mathbf { z } | \boldsymbol { \phi } ) } \left[ \| \hat { \pmb { \theta } } \| _ { 2 } ^ { 2 } \right] = \sum _ { g = 1 } ^ { | G | } \left( \big ( 1 - Q ( s _ { g } \leq 0 | \phi _ { g } ) \big ) \sum _ { j = 1 } ^ { | g | } \tilde { \theta } _ { j } ^ { 2 } \right) . } \end{array}
+$$
+
+where $| G |$ corresponds to the number of groups and $| g |$ corresponds to the number of parameters of group $g$ . For all of our subsequent experiments we employed neuron sparsity, where we introduced a gate per input neuron for fully connected layers and a gate per output feature map for convolutional layers. Notice that in the interpretation we adopt the gate is shared across all locations of the feature map for convolutional layers, akin to spatial dropout (Tompson et al., 2015). This can lead to practical computation savings while training, a benefit which is not possible with the commonly used independent dropout masks per spatial location (e.g. as at Zagoruyko & Komodakis (2016)).
+
+# 3 RELATED WORK
+
+Compression and sparsification of neural networks has recently gained much traction in the deep learning community. The most common and straightforward technique is parameter / neuron pruning (LeCun et al., 1990) according to some criterion. Whereas weight pruning (Han et al., 2015; Ullrich et al., 2017; Molchanov et al., 2017) is in general inefficient for saving computation time, neuron pruning (Wen et al., 2016; Louizos et al., 2017; Neklyudov et al., 2017) can lead to computation savings. Unfortunately, all of the aforementioned methods require training the original dense network thus precluding the benefits we can obtain by having exact sparsity on the computation during training. This is in contrast to our approach where sparsification happens during training, thus theoretically allowing conditional computation to speed-up training (Bengio et al., 2013; 2015).
+
+Emulating binary r.v.s with rectifications of continuous r.v.s is not a new concept and has been previously done with Gaussian distributions in the context of generative modelling (Hinton & Ghahramani, 1997; Harva & Kabán, 2007; Salimans, 2016) and with logistic distributions at (Bengio et al., 2013) in the context of conditional computation. These distributions can similarly represent the value of exact zero, while still maintaining the tractability of continuous optimization. Nevertheless, they are sub-optimal when we require approximations to binary r.v.s (as is the case for the $L _ { 0 }$ penalty); we cannot represent the bimodal behavior of a Bernoulli r.v. due to the fact that the underlying distribution is unimodal. Another technique that allows for gradient based optimization of discrete r.v.s are the smoothing transformations proposed by Rolfe (2016). There the core idea is that if a model has binary latent variables, then we can smooth them with continuous noise in a way that allows for reparametrization gradients. There are two main differences with the hard concrete distribution we employ here; firstly, the double rectification of the hard concrete r.v.s allows us to represent the values of exact zero and one (instead of just zero) and, secondly, due to the underlying concrete distribution the random samples from the hard concrete will better emulate binary r.v.s.
+
+# 4 EXPERIMENTS
+
+We validate the effectiveness of our method on two tasks. The first corresponds to the toy classification task of MNIST using a simple multilayer perceptron (MLP) with two hidden layers of size 300 and 100 (LeCun et al., 1998), and a simple convolutional network, the LeNet-5-Caffe4. The second corresponds to the more modern task of CIFAR 10 and CIFAR 100 classification using Wide Residual Networks (Zagoruyko & Komodakis, 2016). For all of our experiments we set $\gamma = - 0 . 1$ , $\zeta = 1 . 1$ and, following the recommendations from Maddison et al. (2016), set $\beta = 2 / 3$ for the concrete distributions. We initialized the locations $\log \alpha$ by sampling from a normal distribution with a standard deviation of 0.01 and a mean that yields $\frac { \alpha } { \alpha + 1 }$ to be approximately equal to the original dropout rate employed at each of the networks. We used a single sample of the gate $\mathbf { z }$ for each minibatch of datapoints during the optimization, even though this can lead to larger variance in the gradients (Kingma et al., 2015). In this way we show that we can obtain the speedups in training with practical implementations, without actually hurting the overall performance of the network.
+
+# 4.1 MNIST CLASSIFICATION AND SPARSIFICATION
+
+For these experiments we did no further regularization besides the $L _ { 0 }$ norm and optimization was done with Adam (Kingma & Ba, 2014) using the default hyper-parameters and temporal averaging. We can see at Table 1 that our approach is competitive with other methods that tackle neural network compression. However, it is worth noting that all of these approaches prune the network post-training using thresholds while requiring training the full network. We can further see that our approach minimizes the amount of parameters more at layers where the gates affect a larger part of the cost; for the MLP this corresponds to the input layer whereas for the LeNet5 this corresponds to the first fully connected layer. In contrast, the methods with sparsity inducing priors (Louizos et al., 2017; Neklyudov et al., 2017) sparsify parameters irrespective of that extra cost (since they are only encouraged by the prior to move parameters to zero) and as a result they achieve similar sparsity on all of the layers. Nonetheless, it should be mentioned that we can in principle increase the sparsification on specific layers simply by specifying a separate $\lambda$ for each layer, e.g. by increasing the $\lambda$ for gates that affect less parameters. We provide such results at the $^ { 6 6 } \lambda$ sep.” rows.
+
+Table 1: Comparison of the learned architectures and performance of the baselines from Louizos et al. (2017) and the proposed $L _ { 0 }$ minimization under $L _ { 0 _ { h c } }$ . We show the amount of neurons left after pruning with the estimator in Eq. 13 along with the error in the test set after 200 epochs. $N$ denotes the number of training datapoints.   
+
+<table><tr><td>Network&amp; size</td><td>Method</td><td>Pruned architecture</td><td>Error (%)</td></tr><tr><td rowspan="4">MLP 784-300-100</td><td>Sparse VD (Molchanov et al., 2017)</td><td>512-114-72</td><td>1.8</td></tr><tr><td>BC-GNJ (Louizos et al., 2017)</td><td>278-98-13</td><td>1.8</td></tr><tr><td>BC-GHS S (Louizos et al., 2017)</td><td>311-86-14</td><td>1.8</td></tr><tr><td>Lohe,入=0.1/N</td><td>219-214-100</td><td>1.4</td></tr><tr><td rowspan="7">LeNet-5-Caffe 20-50-800-500</td><td>LOne, X sep.</td><td>266-88-33</td><td>1.8</td></tr><tr><td>Sparse VD (Molchanov et al., 2017) GL (Wen et al., 2016)</td><td>14-19-242-131 3-12-192-500</td><td>1.0</td></tr><tr><td>GD (Srinivas &amp; Babu,2016)</td><td>7-13-208-16</td><td>1.0 1.1</td></tr><tr><td>SBP (Neklyudov et al., 2017)</td><td>3-18-284-283</td><td>0.9</td></tr><tr><td>BC-GNJ (Louizos et al., 2017)</td><td>8-13-88-13</td><td>1.0</td></tr><tr><td>BC-GHS (Louizos et al., 2017)</td><td>5-10-76-16</td><td>1.0</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>L0he,λ=0.1/N LOhe, 入 sep.</td><td>20-25-45-462 9-18-65-25</td><td>0.9 1.0</td></tr></table>
+
+To get a better idea about the potential speedup we can obtain in training we plot in Figure 3 the expected, under the probability of the gate being active, floating point operations (FLOPs) as a function of the training iterations. We also included the theoretical speedup we can obtain by using dropout (Srivastava et al., 2014) networks. As we can observe, our $L _ { 0 }$ minimization procedure that is targeted towards neuron sparsity can potentially yield significant computational benefits compared to the original or dropout architectures, with minimal or no loss in performance. We further observe that there is a significant difference in the flop count for the LeNet model between the $\lambda = 0 . 1 / N$ and $\lambda$ sep. settings. This is because we employed larger values for $\lambda$ $( 1 0 / N$ and $0 . 5 / N )$ for the convolutional layers (which contribute the most to the computation) in the $\lambda$ sep. setting. As a result, this setting is more preferable when we are concerned with speedup, rather than network compression (which is affected only by the number of parameters).
+
+![](images/a2429c72848dca80dc4c12b8e376e6cbc831d91a74dd589a3bdba2cc13c1c6ac.jpg)  
+Figure 3: Expected number of floating point operations (FLOPs) during training for the original, dropout and $L _ { 0 }$ regularized networks. These were computed by assuming one flop for multiplication and one flop for addition.
+
+# 4.2 CIFAR CLASSIFICATION
+
+For WideResNets we apply $L _ { 0 }$ regularization on the weights of the hidden layer of the residual blocks, i.e. where dropout is usually employed. We also employed an $L _ { 2 }$ regularization term as described in Section 2.3 with the weight decay coefficient used in Zagoruyko & Komodakis (2016). For the layers with the hard concrete gates we divided the weight decay coefficient by 0.7 to ensure that a-priori we assume the same length-scale as the 0.3 dropout equivalent network. For optimization we employed the procedure described in Zagoruyko & Komodakis (2016) with a minibatch of 128 datapoints, which was split between two GPUs, and used a single sample for the gates for each GPU.
+
+Table 2: Results on the benchmark classification tasks of CIFAR 10 and CIFAR 100. All of the baseline results are taken from Zagoruyko & Komodakis (2016). For the $L _ { 0 }$ regularized WRN we report the median of the error on the test set after 200 epochs over 5 runs.   
+
+<table><tr><td>Network</td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td>original-ResNet-110 (He et al., 2016a)</td><td>6.43</td><td>25.16</td></tr><tr><td>pre-act-ResNet-110 (He et al., 2016b)</td><td>6.37</td><td>1</td></tr><tr><td>WRN-28-10 (Zagoruyko &amp; Komodakis, 2016)</td><td>4.00</td><td>21.18</td></tr><tr><td>WRN-28-10-dropout (Zagoruyko &amp; Komodakis,2016)</td><td>3.89</td><td>18.85</td></tr><tr><td>WRN-28-10-L0hc,入= 0.001/N</td><td>3.83</td><td>18.75</td></tr><tr><td>WRN-28-10-L0hc,λ = 0.002/N</td><td>3.93</td><td>19.04</td></tr></table>
+
+![](images/43e3baa6be118a350690ba6da0ab3f6b2e3caa76fc34dad8b75cdee09d5351f6.jpg)  
+Figure 4: (a, b) Expected number of FLOPs during training for the dropout and $L _ { 0 }$ regularized WRNs for CIFAR 10 (a) and CIFAR 100 (b). The original WRN is not shown as it has the same practical FLOPs as the dropout equivalent network. (c) Train (dashed) and test (solid) error as a function of the training epochs for dropout and $L _ { 0 }$ WRNs at CIFAR 10.
+
+As we can observe at Table 2, with a $\lambda$ of $0 . 0 0 1 / N$ the $L _ { 0 }$ regularized wide residual network improves upon the accuracy of the dropout equivalent network on both CIFAR 10 and CIFAR 100. Furthermore, it simultaneously allows for potential training time speedup due to gradually decreasing the number of FLOPs, as we can see in Figures 4a, 4b. This sparsity is also obtained without any “lag" in convergence speed, as at Figure 4c we observe a behaviour that is similar to the dropout network. Finally, we observe that by further increasing $\lambda$ we obtain a model that has a slight error increase but can allow for a larger speedup.
+
+# 5 DISCUSSION
+
+We have described a general recipe that allows for optimizing the $L _ { 0 }$ norm of parametric models in a principled and effective manner. The method is based on smoothing the combinatorial problem with continuous distributions followed by a hard-sigmoid. To this end, we also proposed a novel distribution which we coin as the hard concrete; it is a “stretched” binary concrete distribution, the samples of which are transformed by a hard-sigmoid. This in turn better mimics the binary nature of Bernoulli distributions while still allowing for efficient gradient based optimization. In experiments we have shown that the proposed $L _ { 0 }$ minimization process leads to neural network sparsification that is competitive with current approaches while theoretically allowing for speedup in training. We have further shown that this process can provide a good inductive bias and regularizer, as on the CIFAR experiments with wide residual networks we improved upon dropout.
+
+As for future work; better harnessing the power of conditional computation for efficiently training very large neural networks with learned sparsity patterns is a potential research direction. It would be also interesting to adopt a full Bayesian treatment over the parameters $\pmb \theta$ , such as the one employed at Molchanov et al. (2017); Louizos et al. (2017). This would then allow for further speedup and compression due to the ability of automatically learning the bit precision of each weight. Finally, it would be interesting to explore the behavior of hard concrete r.v.s at binary latent variable models, since they can be used as a drop in replacement that allow us to maintain both the discrete nature as well as the efficient reparametrization gradient optimization.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Taco Cohen, Thomas Kipf, Patrick Forré, and Rianne van den Berg for feedback on an early draft of this paper.
+
+# REFERENCES
+
+Hirotogu Akaike. Information theory and an extension of the maximum likelihood principle. In Selected Papers of Hirotugu Akaike, pp. 199–213. Springer, 1998.
+
+Matthew James Beal. Variational algorithms for approximate Bayesian inference. 2003.
+
+Emmanuel Bengio, Pierre-Luc Bacon, Joelle Pineau, and Doina Precup. Conditional computation in neural networks for faster models. arXiv preprint arXiv:1511.06297, 2015.
+
+Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013.
+
+Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012.
+
+Yarin Gal, Jiri Hron, and Alex Kendall. Concrete dropout. arXiv preprint arXiv:1705.07832, 2017.
+
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015.
+
+Markus Harva and Ata Kabán. Variational learning for rectified factor analysis. Signal Processing, 87(3):509–527, 2007.
+
+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, 2016a.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European Conference on Computer Vision, pp. 630–645. Springer, 2016b.
+
+John R Hershey and Peder A Olsen. Approximating the kullback leibler divergence between gaussian mixture models. In Acoustics, Speech and Signal Processing, 2007. ICASSP 2007. IEEE International Conference on, volume 4, pp. IV–317. IEEE, 2007.
+
+Geoffrey E Hinton and Zoubin Ghahramani. Generative models for discovering sparse distributed representations. Philosophical Transactions of the Royal Society of London B: Biological Sciences, 352(1358):1177–1190, 1997.
+
+Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. arXiv preprint arXiv:1611.01144, 2016.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. International Conference on Learning Representations (ICLR), 2014.
+
+Diederik P Kingma, Tim Salimans, and Max Welling. Variational dropout and the local reparameterization trick. In Advances in Neural Information Processing Systems, pp. 2575–2583, 2015.
+
+Yann LeCun, John S Denker, and Sara A Solla. Optimal brain damage. In Advances in neural information processing systems 2, NIPS 1989, volume 2, pp. 598–605. Morgan-Kaufmann Publishers, 1990.
+
+Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Christos Louizos, Karen Ullrich, and Max Welling. Bayesian compression for deep learning. arXiv preprint arXiv:1705.08665, 2017.
+
+Chris J Maddison, Andriy Mnih, and Yee Whye Teh. The concrete distribution: A continuous relaxation of discrete random variables. arXiv preprint arXiv:1611.00712, 2016.
+
+Toby J Mitchell and John J Beauchamp. Bayesian variable selection in linear regression. Journal of the American Statistical Association, 83(404):1023–1032, 1988.
+
+Andriy Mnih and Karol Gregor. Neural variational inference and learning in belief networks. arXiv preprint arXiv:1402.0030, 2014.
+
+Andriy Mnih and Danilo Rezende. Variational inference for monte carlo objectives. In International Conference on Machine Learning, pp. 2188–2196, 2016.
+
+Dmitry Molchanov, Arsenii Ashukha, and Dmitry Vetrov. Variational dropout sparsifies deep neural networks. arXiv preprint arXiv:1701.05369, 2017.
+
+Kirill Neklyudov, Dmitry Molchanov, Arsenii Ashukha, and Dmitry Vetrov. Structured bayesian pruning via log-normal multiplicative noise. arXiv preprint arXiv:1705.07283, 2017.
+
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21-26 June 2014, pp. 1278–1286, 2014.
+
+Jason Tyler Rolfe. Discrete variational autoencoders. arXiv preprint arXiv:1609.02200, 2016.
+
+Tim Salimans. A structured variational auto-encoder for learning deep hierarchies of sparse features. arXiv preprint arXiv:1602.08734, 2016.
+
+Gideon Schwarz et al. Estimating the dimension of a model. The annals of statistics, 6(2):461–464, 1978.
+
+Suraj Srinivas and R Venkatesh Babu. Generalized dropout. arXiv preprint arXiv:1611.06791, 2016.
+
+Suraj Srinivas, Akshayvarun Subramanya, and R Venkatesh Babu. Training sparse neural networks. In Computer Vision and Pattern Recognition Workshops (CVPRW), 2017 IEEE Conference on, pp. 455–462. IEEE, 2017.
+
+Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1):1929–1958, 2014.
+
+Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), pp. 267–288, 1996.
+
+Jonathan Tompson, Ross Goroshin, Arjun Jain, Yann LeCun, and Christoph Bregler. Efficient object localization using convolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 648–656, 2015.
+
+George Tucker, Andriy Mnih, Chris J Maddison, and Jascha Sohl-Dickstein. Rebar: Low-variance, unbiased gradient estimates for discrete latent variable models. arXiv preprint arXiv:1703.07370, 2017.
+
+Karen Ullrich, Edward Meeds, and Max Welling. Soft weight-sharing for neural network compression. ICLR, 2017.
+
+Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Learning structured sparsity in deep neural networks. In Advances in Neural Information Processing Systems, pp. 2074–2082, 2016.
+
+Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.
+
+# APPENDIX
+
+# A RELATION TO VARIATIONAL INFERENCE
+
+The objective function described in Eq. 3 is in fact a special case of a variational lower bound over the parameters of the network under a spike and slab (Mitchell & Beauchamp, 1988) prior. The spike and slab distribution is the golden standard in sparsity as far as Bayesian inference is concerned and it is defined as a mixture of a delta spike at zero and a continuous distribution over the real line (e.g. a standard normal):
+
+$$
+p ( z ) = { \mathrm { B e r n o u l l i } } ( \pi ) , \qquad p ( \theta | z = 0 ) = \delta ( \theta ) , \qquad p ( \theta | z = 1 ) = { \mathcal { N } } ( \theta | 0 , 1 ) .
+$$
+
+Since the true posterior distribution over the parameters under this prior is intractable, we will use variational inference (Beal, 2003). Let $q ( \theta , z )$ be a spike and slab approximate posterior over the parameters $\theta$ and gate variables $z$ , where we assume that it factorizes over the dimensionality of the parameters $\theta$ . It turns out that we can write the following variational free energy under the spike and slab prior and approximate posterior over a parameter vector $\pmb \theta$ :
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { F } = - \mathrm { E } _ { q ( z ) q ( \theta , z ) } \big [ \log p ( D | \theta ) \big | \Big ] + \displaystyle \sum _ { i = 1 } ^ { \lfloor \theta \rfloor } K L ( q ( z _ { i } ) ) \big | | p ( z _ { i } ) \big ) + } \\ { \displaystyle \quad \quad + \sum _ { j = 1 } ^ { \lfloor \theta \rfloor } \big ( q ( z _ { j } = 1 ) K L ( q ( \theta _ { j } | z _ { j } = 1 ) | | p ( \theta _ { j } | z _ { j } = 1 ) ) + } \\ { \displaystyle \quad \quad + q ( z _ { j } = 0 ) K L ( q ( \theta _ { j } | z _ { j } = 0 ) | | p ( \theta _ { j } | z _ { j } = 0 ) ) \big ) } \\ { = - \mathrm { E } _ { q ( z ) q ( \theta _ { z } ) } \big [ \log p ( D | \theta ) \big | \Big ] + \displaystyle \sum _ { j = 1 } ^ { \lfloor \theta \rfloor } K L ( q ( z _ { j } ) | | p ( z _ { j } ) ) + } \\ { \displaystyle \quad \quad + \sum _ { j = 1 } ^ { \lfloor \theta \rfloor } q ( z _ { j } = 1 ) K L ( q ( \theta _ { j } | z _ { j } = 1 ) | | p ( \theta _ { j } | z _ { j } = 1 ) ) , } \end{array}
+$$
+
+where the last step is due to $K L ( q ( \theta _ { j } | z _ { j } = 0 ) | | p ( \theta _ { j } | z _ { j } = 0 ) ) = 0 ^ { 5 }$ . The term that involves $K L ( q ( z _ { j } ) | | p ( z _ { j } ) )$ corresponds to the KL-divergence from the Bernoulli prior $p ( z _ { j } )$ to the Bernoulli approximate posterior $q ( z _ { j } )$ and $K L ( q ( \theta _ { j } | z _ { j } = 1 ) | | p ( \theta _ { j } | z _ { j } = 1 ) )$ can be interpreted as the “code cost” or else the amount of information the parameter $\theta _ { j }$ contains about the data $\mathcal { D }$ , measured by the KL-divergence from the prior $p ( \theta _ { j } | z _ { j } = 1 )$ ).
+
+Now consider making the assumption that we are optimizing, rather than integrating, over $\pmb { \theta }$ and further assuming that $K L ( q ( \theta _ { j } | z _ { j } = 1 ) | | p ( \theta _ { j } | z _ { j } = \bar { 1 } ) ) = \lambda$ . We can justify this assumption from an empirical Bayesian procedure: there is a hypothetical prior for each parameter $p ( \theta _ { j } | z _ { j } = 1 )$ that adapts to $q ( \bar { \theta } _ { j } | z _ { j } = 1 )$ ) in a way that results into needing, approximately, $\lambda$ nats to transform $p ( \theta _ { j } | z _ { j } = 1 )$ to that particular $q ( \theta _ { j } | z _ { j } = 1 )$ . Those $\lambda$ nats are thus the amount of information the $q ( \theta _ { j } | z _ { j } = 1 )$ ) can encode about the data had we used that $p ( \theta _ { j } | z _ { j } = 1 )$ as the prior. Notice that under this view we can consider $\lambda$ as the amount of flexibility of that hypothetical prior; with $\lambda = 0$ we have a prior that is flexible enough to represent exactly $q ( \theta _ { j } | z _ { j } = 1 )$ , thus resulting into no code cost and possible overfitting. Under this assumption the variational free energy can be re-written as:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { F } = - \mathbb { E } _ { q ( \mathbf { z } ) } [ \log p ( \mathcal { D } | \widetilde { \pmb { \theta } } \odot \mathbf { z } ) ] + \sum _ { j = 1 } ^ { | \theta | } K L ( q ( z _ { j } ) | | p ( z _ { j } ) ) + \lambda \sum _ { j = 1 } ^ { | \theta | } q ( z _ { j } = 1 ) } \\ { \displaystyle \geq - \mathbb { E } _ { q ( \mathbf { z } ) } [ \log p ( \mathcal { D } | \widetilde { \pmb { \theta } } \odot \mathbf { z } ) ] + \lambda \sum _ { j = 1 } ^ { | \theta | } \pi _ { j } , } \end{array}
+$$
+
+where $\tilde { \pmb { \theta } }$ corresponds to the optimized $\pmb { \theta }$ and the last step is due to the positivity of the KL-divergence. Now by taking the negative log-probability of the data to be equal to the loss $\mathcal { L } ( \cdot )$ of Eq. 1 we see that Eq. 22 is the same as Eq. 3. Note that in case that we are interested over the uncertainty of the gates $\mathbf { z }$ , we should optimize Eq. 21, rather than Eq. 22, as this will properly penalize the entropy of $q ( \mathbf { z } )$ . Furthermore, Eq. 21 also allows for the incorporation of prior information about the behavior of the gates (e.g. gates being active $10 \%$ of the time, on average). We have thus shown that the expected $L _ { 0 }$ minimization procedure is in fact a close surrogate to a variational bound involving a spike and slab distribution over the parameters and a fixed coding cost for the parameters when the gates are active.
+
+# B THE HARD CONCRETE DISTRIBUTION
+
+As mentioned in the main text, the hard concrete is a straightforward modification of the binary concrete (Maddison et al., 2016; Jang et al., 2016); let $q _ { s } ( s | \phi )$ be the probability density function
+
+(pdf) and $Q _ { s } ( s | \phi )$ the cumulative distribution function (CDF) of a binary concrete random variable $s$ :
+
+$$
+\begin{array} { l } { { q _ { s } ( s | \phi ) = \displaystyle \frac { \beta \alpha s ^ { - \beta - 1 } ( 1 - s ) ^ { - \beta - 1 } } { ( \alpha s ^ { - \beta } + ( 1 - s ) ^ { - \beta } ) ^ { 2 } } , \qquad } } \\ { { Q _ { s } ( s | \phi ) = \mathrm { S i g m o i d } ( ( \log s - \log ( 1 - s ) ) \beta - \log \alpha ) . } } \end{array}
+$$
+
+Now by stretching this distribution to the $( \gamma , \zeta )$ interval, with $\gamma < 0$ and $\zeta > 1$ we obtain ${ \bar { s } } =$ $s ( \zeta - \gamma ) + \gamma$ with the following pdf and CDF:
+
+$$
+q _ { \bar { s } } ( \bar { s } | \phi ) = \frac { 1 } { | \zeta - \gamma | } q _ { s } \biggl ( \frac { \bar { s } - \gamma } { \zeta - \gamma } \biggr | \phi \biggr ) , \qquad Q _ { \bar { s } } ( \bar { s } | \phi ) = Q _ { s } \biggl ( \frac { \bar { s } - \gamma } { \zeta - \gamma } \biggr | \phi \biggr ) .
+$$
+
+and by further rectifying $\bar { s }$ with the hard-sigmoid, $z = \operatorname* { m i n } ( 1 , \operatorname* { m a x } ( 0 , \bar { s } ) )$ , we obtain the following distribution over $z$ :
+
+$$
+q ( z | \phi ) = Q _ { \bar { s } } ( 0 | \phi ) \delta ( z ) + ( 1 - Q _ { \bar { s } } ( 1 | \phi ) ) \delta ( z - 1 ) + ( Q _ { \bar { s } } ( 1 | \phi ) - Q _ { \bar { s } } ( 0 | \phi ) ) q _ { \bar { s } } ( z | \bar { s } \in ( 0 , 1 ) , \phi ) ,
+$$
+
+which is composed by a delta peak at zero with probability $Q _ { \bar { s } } ( 0 | \phi )$ , a delta peak at one with probability $1 \bar { - Q } _ { \bar { s } } ( 1 | \bar { \phi } )$ , and a truncated version of $q _ { \bar { s } } ( { \bar { s } } | \phi )$ in the $( 0 , 1 )$ range.
+
+# C NEGATIVE KL-DIVERGENCE FOR HARD CONCRETE DISTRIBUTIONS
+
+In case th 21 is to be optimized with a hard concrete $q ( z )$ then we have to compute the KL-divergence from a prior $p ( z )$ to $q ( z )$ . It is necessary for the prior $p ( z )$ to have the same support as $q ( z )$ in order for the KL-divergence to be valid; as a result we can let the prior $p ( z )$ similarly be a hard-sigmoid transformation of an arbitrary continuous distribution $p ( \bar { s } )$ with CDF $P _ { \bar { s } } ( \bar { s } )$ :
+
+$$
+p ( z ) = P _ { \bar { s } } ( 0 ) \delta ( z ) + ( 1 - P _ { \bar { s } } ( 1 ) ) \delta ( z - 1 ) + ( P _ { \bar { s } } ( 1 ) - P _ { \bar { s } } ( 0 ) ) p _ { \bar { s } } ( z | \bar { s } \in ( 0 , 1 ) )
+$$
+
+Since both $q ( z )$ and $p ( z )$ are mixtures with the same number of components we can use the chain rule of relative entropy (Cover & Thomas, 2012; Hershey & Olsen, 2007) in order to compute the KL-divergence:
+
+$$
+\begin{array} { c } { { \displaystyle K L ( q ( z ) | | p ( z ) ) = Q _ { \bar { s } } ( 0 ) \log \frac { Q _ { \bar { s } } ( 0 ) } { P _ { \bar { s } } ( 0 ) } + ( 1 - Q _ { \bar { s } } ( 1 ) ) \log \frac { 1 - Q _ { \bar { s } } ( 1 ) } { 1 - P _ { \bar { s } } ( 1 ) } + } } \\ { { + \left( Q _ { \bar { s } } ( 1 ) - Q _ { \bar { s } } ( 0 ) \right) \mathbb { E } _ { q _ { \bar { s } } ( z | \bar { s } \in ( 0 , 1 ) ) } [ \log q _ { \bar { s } } ( z ) - \log p _ { \bar { s } } ( z ) ] , } } \end{array}
+$$
+
+where $\bar { s }$ corresponds to the the pre-rectified variable. Notice that in case that the integral under the truncated distribution $q ( \bar { s } | \bar { s } \in ( \bar { 0 } , 1 ) )$ is not available in closed form we can still obtain a Monte Carlo estimate by sampling the truncated distribution, on e.g. a $( \gamma , \zeta )$ interval, via the inverse transform method:
+
+$$
+u \sim \mathcal { U } ( 0 , 1 ) , \qquad z = Q _ { \bar { s } } ^ { - 1 } \bigl ( Q _ { \bar { s } } ( \gamma ) + u ( Q _ { \bar { s } } ( \zeta ) - Q _ { \bar { s } } ( \gamma ) ) \bigr ) ,
+$$
+
+where $Q _ { \bar { s } } ^ { - 1 } ( \cdot )$ corresponds to the quantile function and $Q _ { \bar { s } } ( \cdot )$ to the CDF of the random variable $\bar { s }$ . Furthermore, it should be mentioned that $K L ( q ( z ) | | p ( z ) ) \neq K L ( q ( \bar { s } ) | | p ( \bar { s } ) )$ , since the rectifications are not invertible transformations.

parse/train/H1Y8hhg0b/H1Y8hhg0b_content_list.json

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+    {
+        "type": "text",
+        "text": "LEARNING SPARSE NEURAL NETWORKSTHROUGH $L _ { 0 }$ REGULARIZATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Christos Louizos∗ University of Amsterdam TNO, Intelligent Imaging c.louizos@uva.nl ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Max Welling   \nUniversity of Amsterdam CIFAR   \nm.welling@uva.nl ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Diederik P. Kingma OpenAI dpkingma@openai.com ",
+        "bbox": [
+            601,
+            172,
+            790,
+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            266,
+            544,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a practical method for $L _ { 0 }$ norm regularization for neural networks: pruning the network during training by encouraging weights to become exactly zero. Such regularization is interesting since (1) it can greatly speed up training and inference, and (2) it can improve generalization. AIC and BIC, well-known model selection criteria, are special cases of $L _ { 0 }$ regularization. However, since the $L _ { 0 }$ norm of weights is non-differentiable, we cannot incorporate it directly as a regularization term in the objective function. We propose a solution through the inclusion of a collection of non-negative stochastic gates, which collectively determine which weights to set to zero. We show that, somewhat surprisingly, for certain distributions over the gates, the expected $L _ { 0 }$ regularized objective is differentiable with respect to the distribution parameters. We further propose the hard concrete distribution for the gates, which is obtained by “stretching” a binary concrete distribution and then transforming its samples with a hard-sigmoid. The parameters of the distribution over the gates can then be jointly optimized with the original network parameters. As a result our method allows for straightforward and efficient learning of model structures with stochastic gradient descent and allows for conditional computation in a principled way. We perform various experiments to demonstrate the effectiveness of the resulting approach and regularizer. ",
+        "bbox": [
+            233,
+            296,
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            587
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks are flexible function approximators that have been very successful in a broad range of tasks. They can easily scale to millions of parameters while allowing for tractable optimization with mini-batch stochastic gradient descent (SGD), graphical processing units (GPUs) and parallel computation. Nevertheless they do have drawbacks. Firstly, it has been shown in recent works (Han et al., 2015; Ullrich et al., 2017; Molchanov et al., 2017) that they are greatly overparametrized as they can be pruned significantly without any loss in accuracy; this exhibits unnecessary computation and resources. Secondly, they can easily overfit and even memorize random patterns in the data (Zhang et al., 2016), if not properly regularized. This overfitting can lead to poor generalization in practice. ",
+        "bbox": [
+            174,
+            602,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A way to address both of these issues is by employing model compression and sparsification techniques. By sparsifying the model, we can avoid unnecessary computation and resources, since irrelevant degrees of freedom are pruned away and do not need to be computed. Furthermore, we reduce its complexity, thus penalizing memorization and alleviating overfitting. ",
+        "bbox": [
+            176,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A conceptually attractive approach is the $L _ { 0 }$ norm regularization of (blocks of) parameters; this explicitly penalizes parameters for being different than zero with no further restrictions. However, the combinatorial nature of this problem makes for an intractable optimization for large models. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we propose a general framework for surrogate $L _ { 0 }$ regularized objectives. It is realized by smoothing the expected $L _ { 0 }$ regularized objective with continuous distributions in a way that can maintain the exact zeros in the parameters while still allowing for efficient gradient based optimization. This is achieved by transforming continuous random variables (r.v.s) with a hard nonlinearity, the hard-sigmoid. We further propose and employ a novel distribution obtained by this procedure; the hard concrete. It is obtained by “stretching” a binary concrete random variable (Maddison et al., 2016; Jang et al., 2016) and then passing its samples through a hard-sigmoid. We demonstrate the effectiveness of this simple procedure in various experiments. ",
+        "bbox": [
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+    {
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+        "img_path": "images/1ce9b8a3efa0b28dff6f2b5f0d0b5baf383aeb8c996ef35676c1753856894556.jpg",
+        "image_caption": [
+            "Figure 1: $L _ { p }$ norm penalties for a parameter $\\theta$ according to different values of $p$ . It is easily observed that both weight decay and Lasso, $p = 2$ and $p = 1$ respectively, impose shrinkage for large values of $\\theta$ . By gradually allowing $p < 1$ we observe that the shrinkage is reduced and at the limit of $p = 0$ we observe that the penalty is a constant for $\\theta \\neq 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "2 MINIMIZING THE $L _ { 0 }$ NORM OF PARAMETRIC MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            449,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "One way to sparsify parametric models, such as deep neural networks, with the least assumptions about the parameters is the following; let $\\mathcal { D }$ be a dataset consisting of $N$ i.i.d. input output pairs $\\left\\{ ( \\mathbf { x } _ { 1 } , \\mathbf { y } _ { 1 } ) , \\dotsc , ( \\mathbf { x } _ { N } , \\mathbf { y } _ { N } ) \\right\\}$ and consider a regularized empirical risk minimization procedure with an $L _ { 0 }$ regularization on the parameters $\\pmb \\theta$ of a hypothesis (e.g. a neural network) $h ( \\cdot ; \\bar { \\pmb \\theta } ) ^ { 1 }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e7ebd52420fe04825fcd651f17672e37c28a754d0a64aa10e61abb322cce000f.jpg",
+        "text": "$$\n\\mathcal { R } ( \\pmb { \\theta } ) = \\frac { 1 } { N } \\bigg ( \\sum _ { i = 1 } ^ { N } \\mathcal { L } \\big ( h ( \\mathbf { x } _ { i } ; \\pmb { \\theta } ) , \\mathbf { y } _ { i } \\big ) \\bigg ) + \\lambda \\| \\pmb { \\theta } \\| _ { 0 } , \\qquad \\| \\pmb { \\theta } \\| _ { 0 } = \\sum _ { j = 1 } ^ { | \\theta | } \\mathbb { I } [ \\theta _ { j } \\neq 0 ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $| \\theta |$ is the dimensionality of the parameters, $\\lambda$ is a weighting factor for the regularization and $\\mathcal { L } ( \\cdot )$ corresponds to a loss function, e.g. cross-entropy loss for classification or mean-squared error for regression. The $L _ { 0 }$ norm penalizes the number of non-zero entries of the parameter vector and thus encourages sparsity in the final estimates $\\pmb { \\theta } ^ { * }$ . The Akaike Information Criterion (AIC) (Akaike, 1998) and the Bayesian Information Criterion (BIC) (Schwarz et al., 1978), well-known model selection criteria, correspond to specific choices of $\\lambda$ . Notice that the $L _ { 0 }$ norm induces no shrinkage on the actual values of the parameters $\\pmb { \\theta }$ ; this is in contrast to e.g. $L _ { 1 }$ regularization and the Lasso (Tibshirani, 1996), where the sparsity is due to shrinking the actual values of $\\pmb { \\theta }$ . We provide a visualization of this effect in Figure 1. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, optimization under this penalty is computationally intractable due to the nondifferentiability and combinatorial nature of $2 ^ { | \\theta | }$ possible states of the parameter vector $\\pmb \\theta$ . How can we relax the discrete nature of the $L _ { 0 }$ penalty such that we allow for efficient continuous optimization of Eq. 1, while allowing for exact zeros in the parameters? This section will present the necessary details of our approach. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 A GENERAL RECIPE FOR EFFICIENTLY MINIMIZING $L _ { 0 }$ NORMS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider the $L _ { 0 }$ norm under a simple re-parametrization of $\\pmb \\theta$ : ",
+        "bbox": [
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+            128,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e2138f86fc352d8e97cae4e886118d4475f0368cfc624037e92b16061ec331a2.jpg",
+        "text": "$$\n\\theta _ { j } = \\tilde { \\theta } _ { j } z _ { j } , \\qquad z _ { j } \\in \\{ 0 , 1 \\} , \\qquad \\tilde { \\theta } _ { j } \\neq 0 , \\qquad \\lVert \\theta \\rVert _ { 0 } = \\sum _ { j = 1 } ^ { \\lvert \\theta \\rvert } z _ { j } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            150,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the $z _ { j }$ correspond to binary “gates” that denote whether a parameter is present and the $L _ { 0 }$ norm corresponds to the amount of gates being “on”. By letting $q ( z _ { j } | \\pi _ { j } ) = \\mathbf { B } \\mathrm { e r n } ( \\pi _ { j } )$ be a Bernoulli distribution over each gate $z _ { j }$ we can reformulate the minimization of Eq. 1 as penalizing the number of parameters being used, on average, as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b81436dd1e566ce0cf92c486b6b13a4605fd862e9b12838c813a66b20d6cd9e.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\mathcal { R } ( \\tilde { \\pmb { \\theta } } , \\boldsymbol { \\pi } ) = \\mathbb { E } _ { q ( \\mathbf { z } | \\boldsymbol { \\pi } ) } [ \\frac { 1 } { N } \\bigg ( \\displaystyle \\sum _ { i = 1 } ^ { N } \\mathcal { L } \\big ( h ( \\mathbf { x } _ { i } ; \\tilde { \\pmb { \\theta } } \\odot \\mathbf { z } ) , \\mathbf { y } _ { i } \\big ) \\bigg ) ] + \\lambda \\displaystyle \\sum _ { j = 1 } ^ { | \\theta | } \\pi _ { j } , } } \\\\ & { } & { \\tilde { \\pmb { \\theta } } ^ { * } , \\boldsymbol { \\pi } ^ { * } = \\underset { \\tilde { \\pmb { \\theta } } , \\boldsymbol { \\pi } } { \\operatorname { a r g m i n } } \\{ \\mathcal { R } ( \\tilde { \\pmb { \\theta } } , \\boldsymbol { \\pi } ) \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            262,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ corresponds to the elementwise product. The objective described in Eq. 3 is in fact a special case of a variational bound over the parameters involving spike and slab (Mitchell & Beauchamp, 1988) priors and approximate posteriors; we refer interested readers to appendix A. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Now the second term of the r.h.s. of Eq. 3 is straightforward to minimize however the first term is problematic for $\\pi$ due to the discrete nature of $\\mathbf { z }$ , which does not allow for efficient gradient based optimization. While in principle a gradient estimator such as the REINFORCE (Williams, 1992) could be employed, it suffers from high variance and control variates (Mnih & Gregor, 2014; Mnih & Rezende, 2016; Tucker et al., 2017), that require auxiliary models or multiple evaluations of the network, have to be employed. Two simpler alternatives would be to use either the straightthrough (Bengio et al., 2013) estimator as done at Srinivas et al. (2017) or the concrete distribution as e.g. at Gal et al. (2017). Unfortunately both of these approach have drawbacks; the first one provides biased gradients due to ignoring the Heaviside function in the likelihood during the gradient evaluation whereas the second one does not allow for the gates (and hence parameters) to be exactly zero during optimization, thus precluding the benefits of conditional computation (Bengio et al., 2013). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fortunately, there is a simple alternative way to smooth the objective such that we allow for efficient gradient based optimization of the expected $L _ { 0 }$ norm along with zeros in the parameters $\\pmb \\theta$ . Let s be a continuous random variable with a distribution $q ( \\mathbf { s } )$ that has parameters $\\phi$ . We can now let the gates $\\mathbf { z }$ be given by a hard-sigmoid rectification of $\\mathbf { s } ^ { 2 }$ , as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f0e81d617555e3115e249b47fc208855b5924608640199a3dcfc8bdb08984c54.jpg",
+        "text": "$$\n\\begin{array} { l } { \\mathbf { s } \\sim q ( \\mathbf { s } | \\phi ) } \\\\ { \\mathbf { z } = \\operatorname* { m i n } ( \\mathbf { 1 } , \\operatorname* { m a x } ( \\mathbf { 0 } , \\mathbf { s } ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This would then allow the gate to be exactly zero and, due to the underlying continuous random variable s, we can still compute the probability of the gate being non-zero (active). This is easily obtained by the cumulative distribution function (CDF) ${ \\bf { \\bar { Q } } } ( \\cdot )$ of $\\mathbf { s }$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4acd7ff8da7eae11fdf190d3becda6dc534022472450092168e1f03a5c556e01.jpg",
+        "text": "$$\nq ( \\mathbf { z } \\neq 0 | \\phi ) = 1 - Q ( \\mathbf { s } \\leq 0 | \\phi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "i.e. it is the probability of the s variable being positive. We can thus smooth the binary Bernoulli gates $\\mathbf { z }$ appearing in Eq. 3 by employing continuous distributions in the aforementioned way: ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7de715ccf55eecffc7081d101e09bc4e10639bf7191c70dcce42d05e1800171e.jpg",
+        "text": "$$\n\\mathcal { R } ( \\tilde { \\pmb { \\theta } } , \\phi ) = \\mathbb { E } _ { q ( \\{ \\pmb { \\theta } \\} ) } \\left[ \\frac { 1 } { N } \\bigg ( \\sum _ { i = 1 } ^ { N } \\mathcal { L } \\big ( h ( \\mathbf { x } _ { i } ; \\tilde { \\pmb { \\theta } } \\odot g ( \\mathbf { s } ) ) , \\mathbf { y } _ { i } \\big ) \\bigg ) \\right] + \\lambda \\sum _ { j = 1 } ^ { | \\theta | } \\big ( 1 - Q ( s _ { j } \\le 0 | \\phi _ { j } ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Notice that this is a close surrogate to the original objective function in Eq. 3, as we similarly have a cost that explicitly penalizes the probability of a gate being different from zero. Now for continuous ",
+        "bbox": [
+            173,
+            859,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "distributions $q ( \\mathbf { s } )$ that allow for the reparameterization trick (Kingma & Welling, 2014; Rezende et al., 2014) we can express the objective in Eq. 7 as an expectation over a parameter free noise distribution $p ( \\epsilon )$ and a deterministic and differentiable transformation $f ( \\cdot )$ of the parameters $\\phi$ and $\\epsilon$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8061e13e35d36f84e09bfb1cc926a682c2a650004c95afbda2ca13c043b8d0e3.jpg",
+        "text": "$$\n\\mathcal { R } ( \\tilde { \\pmb { \\theta } } , \\phi ) = \\mathbb { E } _ { p ( \\epsilon ) } \\left[ \\frac { 1 } { N } \\bigg ( \\sum _ { i = 1 } ^ { N } \\mathcal { L } \\big ( h ( \\mathbf { x } _ { i } ; \\tilde { \\pmb { \\theta } } \\odot g ( f ( \\phi , \\epsilon ) ) ) , \\mathbf { y } _ { i } \\big ) \\bigg ) \\right] + \\lambda \\sum _ { j = 1 } ^ { | \\theta | } \\big ( 1 - Q ( s _ { j } \\le 0 | \\phi _ { j } ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            154,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which allows us to make the following Monte Carlo approximation to the (generally) intractable expectation over the noise distribution $p ( \\epsilon )$ : ",
+        "bbox": [
+            171,
+            204,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a05bc6216ccab572c97e5041ca1030bbbb6713236e3546ad220489a86247d8d0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { \\mathcal { R } } ( \\tilde { \\pmb { \\theta } } , \\phi ) = \\displaystyle \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\left( \\frac { 1 } { N } \\bigg ( \\sum _ { i = 1 } ^ { N } \\mathcal { L } \\big ( h ( \\mathbf { x } _ { i } ; \\tilde { \\pmb { \\theta } } \\odot \\mathbf { z } ^ { ( l ) } ) , \\mathbf { y } _ { i } \\big ) \\bigg ) \\right) + \\lambda \\sum _ { j = 1 } ^ { | \\boldsymbol { \\theta } | } \\big ( 1 - Q ( s _ { j } \\le 0 | \\phi _ { j } ) \\big ) } \\\\ & { \\qquad = \\mathcal { L } _ { E } ( \\tilde { \\pmb { \\theta } } , \\phi ) + \\lambda \\mathcal { L } _ { C } ( \\phi ) , \\quad \\mathrm { w h e r e ~ } \\ \\mathbf { z } ^ { ( l ) } = g \\big ( f ( \\phi , \\epsilon ^ { ( l ) } ) \\big ) \\ \\mathrm { a n d } \\ \\epsilon ^ { ( l ) } \\sim p ( \\epsilon ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { L } _ { E }$ corresponds to the error loss that measures how well the model is fitting the current dataset whereas $\\mathcal { L } _ { C }$ refers to the complexity loss that measures the flexibility of the model. Crucially, the total cost in Eq. 9 is now differentiable w.r.t. $\\phi$ , thus enabling for efficient stochastic gradient based optimization, while still allowing for exact zeros at the parameters. One price we pay is that now the gradient of the log-likelihood w.r.t. the parameters $\\phi$ of $q ( \\mathbf { s } )$ is sparse due to the rectifications; nevertheless this should not pose an issue considering the prevalence of rectified linear units in neural networks. Furthermore, due to the stochasticity at s the hard-sigmoid gate $\\mathbf { z }$ is smoothed to a soft version on average, thus allowing for gradient based optimization to succeed, even when the mean of s is negative or larger than one. An example visualization can be seen in Figure 2b. It should be noted that a similar argument was also shown at Bengio et al. (2013), where with logistic noise a rectifier nonlinearity was smoothed to a softplus3 on average. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 THE HARD CONCRETE DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The framework described in Section 2.1 gives us the freedom to choose an appropriate smoothing distribution $q ( \\mathbf { s } )$ . A choice that seems to work well in practice is the following; assume that we have a binary concrete (Maddison et al., 2016; Jang et al., 2016) random variable $s$ distributed in the $( 0 , 1 )$ interval with probability density $q _ { s } ( s | \\phi )$ and cumulative density $Q _ { s } ( s | \\phi )$ . The parameters of the distribution are $\\phi = ( \\log \\alpha , \\beta )$ , where $\\log \\alpha$ is the location and $\\beta$ is the temperature. We can “stretch” this distribution to the $( \\gamma , \\zeta )$ interval, with $\\gamma < 0$ and $\\zeta > 1$ , and then apply a hard-sigmoid on its random samples: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d6f0cb49ef4c1c7efe8d80c74f3e2b3450be78038fce0e9a268256ea1f4252f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { u \\sim \\mathcal { U } ( 0 , 1 ) , \\quad s = \\mathrm { S i g m o i d } \\big ( ( \\log u - \\log ( 1 - u ) + \\log \\alpha ) / \\beta \\big ) , \\quad \\bar { s } = s ( \\zeta - \\gamma ) + \\gamma , } \\\\ & { \\qquad z = \\operatorname* { m i n } ( 1 , \\operatorname* { m a x } ( 0 , \\bar { s } ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This would then induce a distribution where the probability mass of $q _ { \\bar { s } } ( { \\bar { s } } | \\phi )$ on the negative values, $Q _ { \\bar { s } } ( 0 | \\phi )$ , is “folded” to a delta peak at zero, the probability mass on values larger than one, $1 -$ $Q _ { \\bar { s } } ( 1 | \\phi )$ , is “folded” to a delta peak at one and the original distribution $q _ { \\bar { s } } ( { \\bar { s } } | \\phi )$ is truncated to the (0, 1) range. We provide more information and the density of the resulting distribution at the appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Notice that a similar behavior would have been obtained even if we passed samples from any other distribution over the real line through a hard-sigmoid. The only requirement of the approach is that we can evaluate the CDF of $\\bar { s }$ at 0 and 1. The main reason for picking the binary concrete is its close ties with Bernoulli r.v.s. It was originally proposed at Maddison et al. (2016); Jang et al. (2016) as a smooth approximation to Bernoulli r.vs, a fact that allows for gradient based optimization of its parameters through the reparametrization trick. The temperature $\\beta$ controls the degree of approximation, as with $\\beta = 0$ we can recover the original Bernoulli r.v. (but lose the differentiable properties) whereas with $0 < \\beta < 1$ we obtain a probability density that concentrates its mass near the endpoints (e.g. as shown in Figure 2a). As a result, the hard concrete also inherits the same theoretical properties w.r.t. the Bernoulli distribution. Furthermore, it can serve as a better approximation of the discrete nature, since it includes $\\{ 0 , 1 \\}$ in its support, while still allowing for (sub)gradient optimization of its parameters due to the continuous probability mass that connects those two values. We can also view this distribution as a “rounded\" version of the original binary concrete, where values larger than $\\frac { 1 - \\gamma } { \\zeta - \\gamma }$ are rounded to one whereas values smaller than $\\frac { - \\gamma } { \\zeta - \\gamma }$ are rounded to zero. We provide an example visualization of the hard concrete distribution in Figure 2a. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/acf2c6fdf4de450085133af4706b8591a0adb311cdedf797b18bc8fa891d39d7.jpg",
+        "image_caption": [
+            "Figure 2: (a) The binary concrete distribution with location $\\log \\alpha = 0$ and temperature $\\beta = 0 . 5$ and the hard concrete equivalent distribution obtained by stretching the concrete distribution to $( \\gamma = - 0 . 1 , \\zeta = 1 . 1 )$ and then applying a hard-sigmoid. Under this specification the hard concrete distribution assigns, roughly, half of its mass to $\\{ 0 , 1 \\}$ and the rest to $( 0 , 1 )$ . (b) The expected value of the afforementioned concrete and hard concrete gate as a function of the location $\\log \\alpha$ , obtained by averaging 10000 samples. We also added the value of the gates obtained by removing the noise entirely. We can see that the noise smooths the hard-sigmoid to a sigmoid on average. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The $L _ { 0 }$ complexity loss of the objective in Eq. 9 under the hard concrete r.v. is conveniently expressed as follows: ",
+        "bbox": [
+            173,
+            443,
+            823,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba4d95045f7971aaf35582a7cff0f8a38a73fe373043be22542f78871fb8b539.jpg",
+        "text": "$$\n\\mathcal { L } _ { C } = \\sum _ { j = 1 } ^ { | \\theta | } \\big ( 1 - Q _ { \\bar { s } _ { j } } ( 0 | \\phi ) \\big ) = \\sum _ { j = 1 } ^ { | \\theta | } \\mathrm { S i g m o i d } \\big ( \\log \\alpha _ { j } - \\beta \\log \\frac { - \\gamma } { \\zeta } \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At test time we use the following estimator for the final parameters $\\pmb { \\theta } ^ { * }$ under a hard concrete gate: ",
+        "bbox": [
+            171,
+            526,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/12b2e3e6d09fc7eb24737b635da47114637ce6be776bd518edd476a7334cda7b.jpg",
+        "text": "$$\n\\hat { \\bf z } = \\mathrm { m i n } ( { \\bf 1 } , \\mathrm { m a x } ( { \\bf 0 } , \\mathrm { S i g m o i d } ( \\mathrm { l o g } { \\pmb \\alpha } ) ( \\zeta - \\gamma ) + \\gamma ) ) , \\qquad { \\pmb \\theta } ^ { \\ast } = \\tilde { \\pmb \\theta } ^ { \\ast } \\odot \\hat { \\bf z } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 COMBINING THE $L _ { 0 }$ NORM WITH OTHER NORMS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            588,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While the $L _ { 0 }$ norm leads to sparse estimates without imposing any shrinkage on $\\pmb \\theta$ it might still be desirable to impose some form of prior assumptions on the values of $\\pmb { \\theta }$ with alternative norms, e.g. impose smoothness with the $L _ { 2 }$ norm (i.e. weight decay). In the following we will show how this combination is feasible for the $L _ { 2 }$ norm. The expected $L _ { 2 }$ norm under the Bernoulli gating mechanism can be conveniently expressed as: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8059fb25bc7172ef56137038b6415f8934ca451517c888fe67131bea39b07e37.jpg",
+        "text": "$$\n\\mathbb { E } _ { q ( \\mathbf { z } | \\pi ) } \\left[ \\lVert \\pmb { \\theta } \\rVert _ { 2 } ^ { 2 } \\right] = \\sum _ { j = 1 } ^ { | \\theta | } \\mathbb { E } _ { q ( z _ { j } | \\pi _ { j } ) } \\left[ z _ { j } ^ { 2 } \\tilde { \\theta } _ { j } ^ { 2 } \\right] = \\sum _ { j = 1 } ^ { | \\theta | } \\pi _ { j } \\tilde { \\theta } _ { j } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            690,
+            668,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\pi _ { j }$ corresponds to the success probability of the Bernoulli gate $z _ { j }$ . To maintain a similar expression with our smoothing mechanism, and avoid extra shrinkage for the gates $z _ { j }$ , we can take into account that the standard $L _ { 2 }$ norm penalty is proportional to the negative log density of a zero mean Gaussian prior with a standard deviation of $\\sigma = 1$ . We will then assume that the $\\sigma$ for each $\\theta$ is governed by $z$ in a way that when $z = 0$ we have that $\\sigma = 1$ and when $z > 0$ we have that $\\sigma = z$ . As a result, we can obtain the following expression for the $L _ { 2 }$ penalty (where $\\begin{array} { r } { \\hat { \\theta } = \\frac { \\theta } { \\sigma } , } \\end{array}$ ): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3232fee9acb061697f08978777916c7b45dd199d8f0c77772e0c457c74812053.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { q ( \\mathbf { z } | \\phi ) } \\left[ \\| \\hat { \\theta } \\| _ { 2 } ^ { 2 } \\right] = \\sum _ { j = 1 } ^ { | \\theta | } \\left( Q _ { \\bar { s } _ { j } } ( 0 | \\phi _ { j } ) \\frac { 0 } { 1 } + \\left( 1 - Q _ { \\bar { s } _ { j } } ( 0 | \\phi _ { j } ) \\right) \\mathbb { E } _ { q ( z _ { j } | \\phi _ { j } , \\bar { s } _ { j } > 0 ) } \\left[ \\frac { \\widetilde { \\theta } _ { j } ^ { 2 } \\chi _ { j } ^ { 2 } } { \\widetilde { \\chi } _ { j } ^ { 2 } } \\right] \\right) } \\\\ { \\displaystyle \\qquad = \\sum _ { j = 1 } ^ { | \\theta | } \\left( 1 - Q _ { \\bar { s } _ { j } } ( 0 | \\phi _ { j } ) \\right) \\widetilde { \\theta } _ { j } ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.4 GROUP SPARSITY UNDER AN $L _ { 0 }$ NORM ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "For reasons of computational efficiency it is usually desirable to perform group sparsity instead of parameter sparsity, as this can allow for practical computation savings. For example, in neural networks speedups can be obtained by employing a dropout (Srivastava et al., 2014) like procedure with neuron sparsity in fully connected layers or feature map sparsity for convolutional layers (Wen et al., 2016; Louizos et al., 2017; Neklyudov et al., 2017). This is straightforward to do with hard concrete gates; simply share the gate between all of the members of the group. The expected $L _ { 0 }$ and, according to section 2.3, $L _ { 2 }$ penalties in this scenario can be rewritten as: ",
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+        "type": "equation",
+        "img_path": "images/4c14ebb9e1b655a3028a6915d58df38926a97e877cea4515350878de377c334e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathbb { E } _ { q ( \\mathbf { z } | \\boldsymbol { \\phi } ) } \\left[ \\| \\pmb { \\theta } \\| _ { 0 } \\right] = \\sum _ { g = 1 } ^ { | G | } | g | \\left( 1 - Q ( s _ { g } \\leq 0 | \\phi _ { g } ) \\right) } \\\\ & { \\displaystyle \\mathbb { E } _ { q ( \\mathbf { z } | \\boldsymbol { \\phi } ) } \\left[ \\| \\hat { \\pmb { \\theta } } \\| _ { 2 } ^ { 2 } \\right] = \\sum _ { g = 1 } ^ { | G | } \\left( \\big ( 1 - Q ( s _ { g } \\leq 0 | \\phi _ { g } ) \\big ) \\sum _ { j = 1 } ^ { | g | } \\tilde { \\theta } _ { j } ^ { 2 } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $| G |$ corresponds to the number of groups and $| g |$ corresponds to the number of parameters of group $g$ . For all of our subsequent experiments we employed neuron sparsity, where we introduced a gate per input neuron for fully connected layers and a gate per output feature map for convolutional layers. Notice that in the interpretation we adopt the gate is shared across all locations of the feature map for convolutional layers, akin to spatial dropout (Tompson et al., 2015). This can lead to practical computation savings while training, a benefit which is not possible with the commonly used independent dropout masks per spatial location (e.g. as at Zagoruyko & Komodakis (2016)). ",
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+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Compression and sparsification of neural networks has recently gained much traction in the deep learning community. The most common and straightforward technique is parameter / neuron pruning (LeCun et al., 1990) according to some criterion. Whereas weight pruning (Han et al., 2015; Ullrich et al., 2017; Molchanov et al., 2017) is in general inefficient for saving computation time, neuron pruning (Wen et al., 2016; Louizos et al., 2017; Neklyudov et al., 2017) can lead to computation savings. Unfortunately, all of the aforementioned methods require training the original dense network thus precluding the benefits we can obtain by having exact sparsity on the computation during training. This is in contrast to our approach where sparsification happens during training, thus theoretically allowing conditional computation to speed-up training (Bengio et al., 2013; 2015). ",
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+    {
+        "type": "text",
+        "text": "Emulating binary r.v.s with rectifications of continuous r.v.s is not a new concept and has been previously done with Gaussian distributions in the context of generative modelling (Hinton & Ghahramani, 1997; Harva & Kabán, 2007; Salimans, 2016) and with logistic distributions at (Bengio et al., 2013) in the context of conditional computation. These distributions can similarly represent the value of exact zero, while still maintaining the tractability of continuous optimization. Nevertheless, they are sub-optimal when we require approximations to binary r.v.s (as is the case for the $L _ { 0 }$ penalty); we cannot represent the bimodal behavior of a Bernoulli r.v. due to the fact that the underlying distribution is unimodal. Another technique that allows for gradient based optimization of discrete r.v.s are the smoothing transformations proposed by Rolfe (2016). There the core idea is that if a model has binary latent variables, then we can smooth them with continuous noise in a way that allows for reparametrization gradients. There are two main differences with the hard concrete distribution we employ here; firstly, the double rectification of the hard concrete r.v.s allows us to represent the values of exact zero and one (instead of just zero) and, secondly, due to the underlying concrete distribution the random samples from the hard concrete will better emulate binary r.v.s. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We validate the effectiveness of our method on two tasks. The first corresponds to the toy classification task of MNIST using a simple multilayer perceptron (MLP) with two hidden layers of size 300 and 100 (LeCun et al., 1998), and a simple convolutional network, the LeNet-5-Caffe4. The second corresponds to the more modern task of CIFAR 10 and CIFAR 100 classification using Wide Residual Networks (Zagoruyko & Komodakis, 2016). For all of our experiments we set $\\gamma = - 0 . 1$ , $\\zeta = 1 . 1$ and, following the recommendations from Maddison et al. (2016), set $\\beta = 2 / 3$ for the concrete distributions. We initialized the locations $\\log \\alpha$ by sampling from a normal distribution with a standard deviation of 0.01 and a mean that yields $\\frac { \\alpha } { \\alpha + 1 }$ to be approximately equal to the original dropout rate employed at each of the networks. We used a single sample of the gate $\\mathbf { z }$ for each minibatch of datapoints during the optimization, even though this can lead to larger variance in the gradients (Kingma et al., 2015). In this way we show that we can obtain the speedups in training with practical implementations, without actually hurting the overall performance of the network. ",
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+    {
+        "type": "text",
+        "text": "4.1 MNIST CLASSIFICATION AND SPARSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "For these experiments we did no further regularization besides the $L _ { 0 }$ norm and optimization was done with Adam (Kingma & Ba, 2014) using the default hyper-parameters and temporal averaging. We can see at Table 1 that our approach is competitive with other methods that tackle neural network compression. However, it is worth noting that all of these approaches prune the network post-training using thresholds while requiring training the full network. We can further see that our approach minimizes the amount of parameters more at layers where the gates affect a larger part of the cost; for the MLP this corresponds to the input layer whereas for the LeNet5 this corresponds to the first fully connected layer. In contrast, the methods with sparsity inducing priors (Louizos et al., 2017; Neklyudov et al., 2017) sparsify parameters irrespective of that extra cost (since they are only encouraged by the prior to move parameters to zero) and as a result they achieve similar sparsity on all of the layers. Nonetheless, it should be mentioned that we can in principle increase the sparsification on specific layers simply by specifying a separate $\\lambda$ for each layer, e.g. by increasing the $\\lambda$ for gates that affect less parameters. We provide such results at the $^ { 6 6 } \\lambda$ sep.” rows. ",
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+    {
+        "type": "table",
+        "img_path": "images/1ae9c61aa51cc2d07ae17dd1f5ead9e7f57c6c2e65fb8717d9c2c914b4924090.jpg",
+        "table_caption": [
+            "Table 1: Comparison of the learned architectures and performance of the baselines from Louizos et al. (2017) and the proposed $L _ { 0 }$ minimization under $L _ { 0 _ { h c } }$ . We show the amount of neurons left after pruning with the estimator in Eq. 13 along with the error in the test set after 200 epochs. $N$ denotes the number of training datapoints. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network&amp; size</td><td>Method</td><td>Pruned architecture</td><td>Error (%)</td></tr><tr><td rowspan=\"4\">MLP 784-300-100</td><td>Sparse VD (Molchanov et al., 2017)</td><td>512-114-72</td><td>1.8</td></tr><tr><td>BC-GNJ (Louizos et al., 2017)</td><td>278-98-13</td><td>1.8</td></tr><tr><td>BC-GHS S (Louizos et al., 2017)</td><td>311-86-14</td><td>1.8</td></tr><tr><td>Lohe,入=0.1/N</td><td>219-214-100</td><td>1.4</td></tr><tr><td rowspan=\"7\">LeNet-5-Caffe 20-50-800-500</td><td>LOne, X sep.</td><td>266-88-33</td><td>1.8</td></tr><tr><td>Sparse VD (Molchanov et al., 2017) GL (Wen et al., 2016)</td><td>14-19-242-131 3-12-192-500</td><td>1.0</td></tr><tr><td>GD (Srinivas &amp; Babu,2016)</td><td>7-13-208-16</td><td>1.0 1.1</td></tr><tr><td>SBP (Neklyudov et al., 2017)</td><td>3-18-284-283</td><td>0.9</td></tr><tr><td>BC-GNJ (Louizos et al., 2017)</td><td>8-13-88-13</td><td>1.0</td></tr><tr><td>BC-GHS (Louizos et al., 2017)</td><td>5-10-76-16</td><td>1.0</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>L0he,λ=0.1/N LOhe, 入 sep.</td><td>20-25-45-462 9-18-65-25</td><td>0.9 1.0</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To get a better idea about the potential speedup we can obtain in training we plot in Figure 3 the expected, under the probability of the gate being active, floating point operations (FLOPs) as a function of the training iterations. We also included the theoretical speedup we can obtain by using dropout (Srivastava et al., 2014) networks. As we can observe, our $L _ { 0 }$ minimization procedure that is targeted towards neuron sparsity can potentially yield significant computational benefits compared to the original or dropout architectures, with minimal or no loss in performance. We further observe that there is a significant difference in the flop count for the LeNet model between the $\\lambda = 0 . 1 / N$ and $\\lambda$ sep. settings. This is because we employed larger values for $\\lambda$ $( 1 0 / N$ and $0 . 5 / N )$ for the convolutional layers (which contribute the most to the computation) in the $\\lambda$ sep. setting. As a result, this setting is more preferable when we are concerned with speedup, rather than network compression (which is affected only by the number of parameters). ",
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a2429c72848dca80dc4c12b8e376e6cbc831d91a74dd589a3bdba2cc13c1c6ac.jpg",
+        "image_caption": [
+            "Figure 3: Expected number of floating point operations (FLOPs) during training for the original, dropout and $L _ { 0 }$ regularized networks. These were computed by assuming one flop for multiplication and one flop for addition. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2 CIFAR CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For WideResNets we apply $L _ { 0 }$ regularization on the weights of the hidden layer of the residual blocks, i.e. where dropout is usually employed. We also employed an $L _ { 2 }$ regularization term as described in Section 2.3 with the weight decay coefficient used in Zagoruyko & Komodakis (2016). For the layers with the hard concrete gates we divided the weight decay coefficient by 0.7 to ensure that a-priori we assume the same length-scale as the 0.3 dropout equivalent network. For optimization we employed the procedure described in Zagoruyko & Komodakis (2016) with a minibatch of 128 datapoints, which was split between two GPUs, and used a single sample for the gates for each GPU. ",
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+    {
+        "type": "table",
+        "img_path": "images/b7175c25c6f422bf8b3bcdb4256db6415f401732e40c177f66aa4f5f6b959dc6.jpg",
+        "table_caption": [
+            "Table 2: Results on the benchmark classification tasks of CIFAR 10 and CIFAR 100. All of the baseline results are taken from Zagoruyko & Komodakis (2016). For the $L _ { 0 }$ regularized WRN we report the median of the error on the test set after 200 epochs over 5 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td>original-ResNet-110 (He et al., 2016a)</td><td>6.43</td><td>25.16</td></tr><tr><td>pre-act-ResNet-110 (He et al., 2016b)</td><td>6.37</td><td>1</td></tr><tr><td>WRN-28-10 (Zagoruyko &amp; Komodakis, 2016)</td><td>4.00</td><td>21.18</td></tr><tr><td>WRN-28-10-dropout (Zagoruyko &amp; Komodakis,2016)</td><td>3.89</td><td>18.85</td></tr><tr><td>WRN-28-10-L0hc,入= 0.001/N</td><td>3.83</td><td>18.75</td></tr><tr><td>WRN-28-10-L0hc,λ = 0.002/N</td><td>3.93</td><td>19.04</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/43e3baa6be118a350690ba6da0ab3f6b2e3caa76fc34dad8b75cdee09d5351f6.jpg",
+        "image_caption": [
+            "Figure 4: (a, b) Expected number of FLOPs during training for the dropout and $L _ { 0 }$ regularized WRNs for CIFAR 10 (a) and CIFAR 100 (b). The original WRN is not shown as it has the same practical FLOPs as the dropout equivalent network. (c) Train (dashed) and test (solid) error as a function of the training epochs for dropout and $L _ { 0 }$ WRNs at CIFAR 10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As we can observe at Table 2, with a $\\lambda$ of $0 . 0 0 1 / N$ the $L _ { 0 }$ regularized wide residual network improves upon the accuracy of the dropout equivalent network on both CIFAR 10 and CIFAR 100. Furthermore, it simultaneously allows for potential training time speedup due to gradually decreasing the number of FLOPs, as we can see in Figures 4a, 4b. This sparsity is also obtained without any “lag\" in convergence speed, as at Figure 4c we observe a behaviour that is similar to the dropout network. Finally, we observe that by further increasing $\\lambda$ we obtain a model that has a slight error increase but can allow for a larger speedup. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have described a general recipe that allows for optimizing the $L _ { 0 }$ norm of parametric models in a principled and effective manner. The method is based on smoothing the combinatorial problem with continuous distributions followed by a hard-sigmoid. To this end, we also proposed a novel distribution which we coin as the hard concrete; it is a “stretched” binary concrete distribution, the samples of which are transformed by a hard-sigmoid. This in turn better mimics the binary nature of Bernoulli distributions while still allowing for efficient gradient based optimization. In experiments we have shown that the proposed $L _ { 0 }$ minimization process leads to neural network sparsification that is competitive with current approaches while theoretically allowing for speedup in training. We have further shown that this process can provide a good inductive bias and regularizer, as on the CIFAR experiments with wide residual networks we improved upon dropout. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As for future work; better harnessing the power of conditional computation for efficiently training very large neural networks with learned sparsity patterns is a potential research direction. It would be also interesting to adopt a full Bayesian treatment over the parameters $\\pmb \\theta$ , such as the one employed at Molchanov et al. (2017); Louizos et al. (2017). This would then allow for further speedup and compression due to the ability of automatically learning the bit precision of each weight. Finally, it would be interesting to explore the behavior of hard concrete r.v.s at binary latent variable models, since they can be used as a drop in replacement that allow us to maintain both the discrete nature as well as the efficient reparametrization gradient optimization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
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+            545
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Taco Cohen, Thomas Kipf, Patrick Forré, and Rianne van den Berg for feedback on an early draft of this paper. ",
+        "bbox": [
+            174,
+            559,
+            823,
+            588
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            608,
+            285,
+            623
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hirotogu Akaike. Information theory and an extension of the maximum likelihood principle. In Selected Papers of Hirotugu Akaike, pp. 199–213. Springer, 1998. ",
+        "bbox": [
+            174,
+            631,
+            825,
+            660
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthew James Beal. Variational algorithms for approximate Bayesian inference. 2003. ",
+        "bbox": [
+            171,
+            666,
+            754,
+            683
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Emmanuel Bengio, Pierre-Luc Bacon, Joelle Pineau, and Doina Precup. Conditional computation in neural networks for faster models. arXiv preprint arXiv:1511.06297, 2015. ",
+        "bbox": [
+            173,
+            689,
+            821,
+            719
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013. ",
+        "bbox": [
+            173,
+            727,
+            823,
+            756
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. ",
+        "bbox": [
+            178,
+            762,
+            812,
+            779
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yarin Gal, Jiri Hron, and Alex Kendall. Concrete dropout. arXiv preprint arXiv:1705.07832, 2017. ",
+        "bbox": [
+            176,
+            785,
+            820,
+            801
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015. ",
+        "bbox": [
+            174,
+            808,
+            823,
+            838
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Markus Harva and Ata Kabán. Variational learning for rectified factor analysis. Signal Processing, 87(3):509–527, 2007. ",
+        "bbox": [
+            174,
+            844,
+            825,
+            873
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "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, 2016a. ",
+        "bbox": [
+            176,
+            881,
+            825,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European Conference on Computer Vision, pp. 630–645. Springer, 2016b. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "John R Hershey and Peder A Olsen. Approximating the kullback leibler divergence between gaussian mixture models. In Acoustics, Speech and Signal Processing, 2007. ICASSP 2007. IEEE International Conference on, volume 4, pp. IV–317. IEEE, 2007. ",
+        "bbox": [
+            174,
+            140,
+            821,
+            184
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey E Hinton and Zoubin Ghahramani. Generative models for discovering sparse distributed representations. Philosophical Transactions of the Royal Society of London B: Biological Sciences, 352(1358):1177–1190, 1997. ",
+        "bbox": [
+            174,
+            191,
+            825,
+            234
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. arXiv preprint arXiv:1611.01144, 2016. ",
+        "bbox": [
+            171,
+            243,
+            825,
+            272
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. ",
+        "bbox": [
+            171,
+            281,
+            823,
+            310
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Max Welling. Auto-encoding variational bayes. International Conference on Learning Representations (ICLR), 2014. ",
+        "bbox": [
+            171,
+            319,
+            825,
+            348
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma, Tim Salimans, and Max Welling. Variational dropout and the local reparameterization trick. In Advances in Neural Information Processing Systems, pp. 2575–2583, 2015. ",
+        "bbox": [
+            176,
+            356,
+            823,
+            386
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yann LeCun, John S Denker, and Sara A Solla. Optimal brain damage. In Advances in neural information processing systems 2, NIPS 1989, volume 2, pp. 598–605. Morgan-Kaufmann Publishers, 1990. ",
+        "bbox": [
+            176,
+            393,
+            826,
+            435
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. ",
+        "bbox": [
+            174,
+            445,
+            821,
+            474
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Christos Louizos, Karen Ullrich, and Max Welling. Bayesian compression for deep learning. arXiv preprint arXiv:1705.08665, 2017. ",
+        "bbox": [
+            176,
+            482,
+            821,
+            512
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chris J Maddison, Andriy Mnih, and Yee Whye Teh. The concrete distribution: A continuous relaxation of discrete random variables. arXiv preprint arXiv:1611.00712, 2016. ",
+        "bbox": [
+            173,
+            520,
+            820,
+            549
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Toby J Mitchell and John J Beauchamp. Bayesian variable selection in linear regression. Journal of the American Statistical Association, 83(404):1023–1032, 1988. ",
+        "bbox": [
+            173,
+            558,
+            823,
+            587
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andriy Mnih and Karol Gregor. Neural variational inference and learning in belief networks. arXiv preprint arXiv:1402.0030, 2014. ",
+        "bbox": [
+            173,
+            594,
+            823,
+            623
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andriy Mnih and Danilo Rezende. Variational inference for monte carlo objectives. In International Conference on Machine Learning, pp. 2188–2196, 2016. ",
+        "bbox": [
+            173,
+            632,
+            823,
+            662
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Dmitry Molchanov, Arsenii Ashukha, and Dmitry Vetrov. Variational dropout sparsifies deep neural networks. arXiv preprint arXiv:1701.05369, 2017. ",
+        "bbox": [
+            174,
+            670,
+            823,
+            699
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kirill Neklyudov, Dmitry Molchanov, Arsenii Ashukha, and Dmitry Vetrov. Structured bayesian pruning via log-normal multiplicative noise. arXiv preprint arXiv:1705.07283, 2017. ",
+        "bbox": [
+            176,
+            707,
+            823,
+            737
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21-26 June 2014, pp. 1278–1286, 2014. ",
+        "bbox": [
+            174,
+            744,
+            826,
+            801
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jason Tyler Rolfe. Discrete variational autoencoders. arXiv preprint arXiv:1609.02200, 2016. ",
+        "bbox": [
+            169,
+            809,
+            789,
+            825
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tim Salimans. A structured variational auto-encoder for learning deep hierarchies of sparse features. arXiv preprint arXiv:1602.08734, 2016. ",
+        "bbox": [
+            174,
+            833,
+            825,
+            863
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gideon Schwarz et al. Estimating the dimension of a model. The annals of statistics, 6(2):461–464, 1978. ",
+        "bbox": [
+            173,
+            871,
+            825,
+            900
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Suraj Srinivas and R Venkatesh Babu. Generalized dropout. arXiv preprint arXiv:1611.06791, 2016. ",
+        "bbox": [
+            173,
+            909,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Suraj Srinivas, Akshayvarun Subramanya, and R Venkatesh Babu. Training sparse neural networks. In Computer Vision and Pattern Recognition Workshops (CVPRW), 2017 IEEE Conference on, pp. 455–462. IEEE, 2017. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1):1929–1958, 2014. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            209
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), pp. 267–288, 1996. ",
+        "bbox": [
+            171,
+            229,
+            825,
+            258
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Tompson, Ross Goroshin, Arjun Jain, Yann LeCun, and Christoph Bregler. Efficient object localization using convolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 648–656, 2015. ",
+        "bbox": [
+            176,
+            279,
+            825,
+            321
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "George Tucker, Andriy Mnih, Chris J Maddison, and Jascha Sohl-Dickstein. Rebar: Low-variance, unbiased gradient estimates for discrete latent variable models. arXiv preprint arXiv:1703.07370, 2017. ",
+        "bbox": [
+            174,
+            342,
+            826,
+            383
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Karen Ullrich, Edward Meeds, and Max Welling. Soft weight-sharing for neural network compression. ICLR, 2017. ",
+        "bbox": [
+            173,
+            405,
+            825,
+            434
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Learning structured sparsity in deep neural networks. In Advances in Neural Information Processing Systems, pp. 2074–2082, 2016. ",
+        "bbox": [
+            174,
+            454,
+            826,
+            497
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992. ",
+        "bbox": [
+            173,
+            517,
+            823,
+            546
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016. ",
+        "bbox": [
+            174,
+            566,
+            825,
+            595
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016. ",
+        "bbox": [
+            174,
+            616,
+            825,
+            645
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            683,
+            263,
+            698
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A RELATION TO VARIATIONAL INFERENCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            720,
+            544,
+            736
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The objective function described in Eq. 3 is in fact a special case of a variational lower bound over the parameters of the network under a spike and slab (Mitchell & Beauchamp, 1988) prior. The spike and slab distribution is the golden standard in sparsity as far as Bayesian inference is concerned and it is defined as a mixture of a delta spike at zero and a continuous distribution over the real line (e.g. a standard normal): ",
+        "bbox": [
+            173,
+            755,
+            825,
+            825
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2eee4c8f85aaadda957afce2d636d77875fbe18a19216f7ac3a84cb3651a79e2.jpg",
+        "text": "$$\np ( z ) = { \\mathrm { B e r n o u l l i } } ( \\pi ) , \\qquad p ( \\theta | z = 0 ) = \\delta ( \\theta ) , \\qquad p ( \\theta | z = 1 ) = { \\mathcal { N } } ( \\theta | 0 , 1 ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            838,
+            751,
+            856
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Since the true posterior distribution over the parameters under this prior is intractable, we will use variational inference (Beal, 2003). Let $q ( \\theta , z )$ be a spike and slab approximate posterior over the parameters $\\theta$ and gate variables $z$ , where we assume that it factorizes over the dimensionality of the parameters $\\theta$ . It turns out that we can write the following variational free energy under the spike and slab prior and approximate posterior over a parameter vector $\\pmb \\theta$ : ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            589,
+            119
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a034657348c41723b36c45755a9242c9a83266cbdb342bdcb270129cac469abd.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { F } = - \\mathrm { E } _ { q ( z ) q ( \\theta , z ) } \\big [ \\log p ( D | \\theta ) \\big | \\Big ] + \\displaystyle \\sum _ { i = 1 } ^ { \\lfloor \\theta \\rfloor } K L ( q ( z _ { i } ) ) \\big | | p ( z _ { i } ) \\big ) + } \\\\ { \\displaystyle \\quad \\quad + \\sum _ { j = 1 } ^ { \\lfloor \\theta \\rfloor } \\big ( q ( z _ { j } = 1 ) K L ( q ( \\theta _ { j } | z _ { j } = 1 ) | | p ( \\theta _ { j } | z _ { j } = 1 ) ) + } \\\\ { \\displaystyle \\quad \\quad + q ( z _ { j } = 0 ) K L ( q ( \\theta _ { j } | z _ { j } = 0 ) | | p ( \\theta _ { j } | z _ { j } = 0 ) ) \\big ) } \\\\ { = - \\mathrm { E } _ { q ( z ) q ( \\theta _ { z } ) } \\big [ \\log p ( D | \\theta ) \\big | \\Big ] + \\displaystyle \\sum _ { j = 1 } ^ { \\lfloor \\theta \\rfloor } K L ( q ( z _ { j } ) | | p ( z _ { j } ) ) + } \\\\ { \\displaystyle \\quad \\quad + \\sum _ { j = 1 } ^ { \\lfloor \\theta \\rfloor } q ( z _ { j } = 1 ) K L ( q ( \\theta _ { j } | z _ { j } = 1 ) | | p ( \\theta _ { j } | z _ { j } = 1 ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            131,
+            697,
+            340
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where the last step is due to $K L ( q ( \\theta _ { j } | z _ { j } = 0 ) | | p ( \\theta _ { j } | z _ { j } = 0 ) ) = 0 ^ { 5 }$ . The term that involves $K L ( q ( z _ { j } ) | | p ( z _ { j } ) )$ corresponds to the KL-divergence from the Bernoulli prior $p ( z _ { j } )$ to the Bernoulli approximate posterior $q ( z _ { j } )$ and $K L ( q ( \\theta _ { j } | z _ { j } = 1 ) | | p ( \\theta _ { j } | z _ { j } = 1 ) )$ can be interpreted as the “code cost” or else the amount of information the parameter $\\theta _ { j }$ contains about the data $\\mathcal { D }$ , measured by the KL-divergence from the prior $p ( \\theta _ { j } | z _ { j } = 1 )$ ). ",
+        "bbox": [
+            173,
+            352,
+            825,
+            424
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Now consider making the assumption that we are optimizing, rather than integrating, over $\\pmb { \\theta }$ and further assuming that $K L ( q ( \\theta _ { j } | z _ { j } = 1 ) | | p ( \\theta _ { j } | z _ { j } = \\bar { 1 } ) ) = \\lambda$ . We can justify this assumption from an empirical Bayesian procedure: there is a hypothetical prior for each parameter $p ( \\theta _ { j } | z _ { j } = 1 )$ that adapts to $q ( \\bar { \\theta } _ { j } | z _ { j } = 1 )$ ) in a way that results into needing, approximately, $\\lambda$ nats to transform $p ( \\theta _ { j } | z _ { j } = 1 )$ to that particular $q ( \\theta _ { j } | z _ { j } = 1 )$ . Those $\\lambda$ nats are thus the amount of information the $q ( \\theta _ { j } | z _ { j } = 1 )$ ) can encode about the data had we used that $p ( \\theta _ { j } | z _ { j } = 1 )$ as the prior. Notice that under this view we can consider $\\lambda$ as the amount of flexibility of that hypothetical prior; with $\\lambda = 0$ we have a prior that is flexible enough to represent exactly $q ( \\theta _ { j } | z _ { j } = 1 )$ , thus resulting into no code cost and possible overfitting. Under this assumption the variational free energy can be re-written as: ",
+        "bbox": [
+            173,
+            429,
+            826,
+            555
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f20f84b53bafdc8004c079e54407263c41e469caeb48a6ae18348554af93b030.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { F } = - \\mathbb { E } _ { q ( \\mathbf { z } ) } [ \\log p ( \\mathcal { D } | \\widetilde { \\pmb { \\theta } } \\odot \\mathbf { z } ) ] + \\sum _ { j = 1 } ^ { | \\theta | } K L ( q ( z _ { j } ) | | p ( z _ { j } ) ) + \\lambda \\sum _ { j = 1 } ^ { | \\theta | } q ( z _ { j } = 1 ) } \\\\ { \\displaystyle \\geq - \\mathbb { E } _ { q ( \\mathbf { z } ) } [ \\log p ( \\mathcal { D } | \\widetilde { \\pmb { \\theta } } \\odot \\mathbf { z } ) ] + \\lambda \\sum _ { j = 1 } ^ { | \\theta | } \\pi _ { j } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            566,
+            741,
+            661
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\tilde { \\pmb { \\theta } }$ corresponds to the optimized $\\pmb { \\theta }$ and the last step is due to the positivity of the KL-divergence. Now by taking the negative log-probability of the data to be equal to the loss $\\mathcal { L } ( \\cdot )$ of Eq. 1 we see that Eq. 22 is the same as Eq. 3. Note that in case that we are interested over the uncertainty of the gates $\\mathbf { z }$ , we should optimize Eq. 21, rather than Eq. 22, as this will properly penalize the entropy of $q ( \\mathbf { z } )$ . Furthermore, Eq. 21 also allows for the incorporation of prior information about the behavior of the gates (e.g. gates being active $10 \\%$ of the time, on average). We have thus shown that the expected $L _ { 0 }$ minimization procedure is in fact a close surrogate to a variational bound involving a spike and slab distribution over the parameters and a fixed coding cost for the parameters when the gates are active. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B THE HARD CONCRETE DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            813,
+            519,
+            829
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As mentioned in the main text, the hard concrete is a straightforward modification of the binary concrete (Maddison et al., 2016; Jang et al., 2016); let $q _ { s } ( s | \\phi )$ be the probability density function ",
+        "bbox": [
+            176,
+            847,
+            825,
+            877
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(pdf) and $Q _ { s } ( s | \\phi )$ the cumulative distribution function (CDF) of a binary concrete random variable $s$ : ",
+        "bbox": [
+            169,
+            103,
+            825,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/190462a48bce7d0a40b045dca6d2849655fd5acb6e438d3bef8edfd9dcce6e1a.jpg",
+        "text": "$$\n\\begin{array} { l } { { q _ { s } ( s | \\phi ) = \\displaystyle \\frac { \\beta \\alpha s ^ { - \\beta - 1 } ( 1 - s ) ^ { - \\beta - 1 } } { ( \\alpha s ^ { - \\beta } + ( 1 - s ) ^ { - \\beta } ) ^ { 2 } } , \\qquad } } \\\\ { { Q _ { s } ( s | \\phi ) = \\mathrm { S i g m o i d } ( ( \\log s - \\log ( 1 - s ) ) \\beta - \\log \\alpha ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            125,
+            674,
+            180
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Now by stretching this distribution to the $( \\gamma , \\zeta )$ interval, with $\\gamma < 0$ and $\\zeta > 1$ we obtain ${ \\bar { s } } =$ $s ( \\zeta - \\gamma ) + \\gamma$ with the following pdf and CDF: ",
+        "bbox": [
+            173,
+            184,
+            825,
+            214
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a7c7221cf855043297e2d8f09a87edc0e0b8935e239f1a95f5805d0a35b36a9d.jpg",
+        "text": "$$\nq _ { \\bar { s } } ( \\bar { s } | \\phi ) = \\frac { 1 } { | \\zeta - \\gamma | } q _ { s } \\biggl ( \\frac { \\bar { s } - \\gamma } { \\zeta - \\gamma } \\biggr | \\phi \\biggr ) , \\qquad Q _ { \\bar { s } } ( \\bar { s } | \\phi ) = Q _ { s } \\biggl ( \\frac { \\bar { s } - \\gamma } { \\zeta - \\gamma } \\biggr | \\phi \\biggr ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            219,
+            720,
+            256
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "and by further rectifying $\\bar { s }$ with the hard-sigmoid, $z = \\operatorname* { m i n } ( 1 , \\operatorname* { m a x } ( 0 , \\bar { s } ) )$ , we obtain the following distribution over $z$ : ",
+        "bbox": [
+            174,
+            261,
+            823,
+            290
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/22cabdfc56032e70e2fbd5c805f4922445aa5f3e61dbc4dc877b87e7b7abefdc.jpg",
+        "text": "$$\nq ( z | \\phi ) = Q _ { \\bar { s } } ( 0 | \\phi ) \\delta ( z ) + ( 1 - Q _ { \\bar { s } } ( 1 | \\phi ) ) \\delta ( z - 1 ) + ( Q _ { \\bar { s } } ( 1 | \\phi ) - Q _ { \\bar { s } } ( 0 | \\phi ) ) q _ { \\bar { s } } ( z | \\bar { s } \\in ( 0 , 1 ) , \\phi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            296,
+            805,
+            315
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "which is composed by a delta peak at zero with probability $Q _ { \\bar { s } } ( 0 | \\phi )$ , a delta peak at one with probability $1 \\bar { - Q } _ { \\bar { s } } ( 1 | \\bar { \\phi } )$ , and a truncated version of $q _ { \\bar { s } } ( { \\bar { s } } | \\phi )$ in the $( 0 , 1 )$ range. ",
+        "bbox": [
+            174,
+            335,
+            826,
+            366
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C NEGATIVE KL-DIVERGENCE FOR HARD CONCRETE DISTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            385,
+            769,
+            401
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In case th 21 is to be optimized with a hard concrete $q ( z )$ then we have to compute the KL-divergence from a prior $p ( z )$ to $q ( z )$ . It is necessary for the prior $p ( z )$ to have the same support as $q ( z )$ in order for the KL-divergence to be valid; as a result we can let the prior $p ( z )$ similarly be a hard-sigmoid transformation of an arbitrary continuous distribution $p ( \\bar { s } )$ with CDF $P _ { \\bar { s } } ( \\bar { s } )$ : ",
+        "bbox": [
+            173,
+            415,
+            825,
+            473
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/593fa8bb352012f7a321bb88c93f382a182757490f6187bf784a0ce7d7e2fcbb.jpg",
+        "text": "$$\np ( z ) = P _ { \\bar { s } } ( 0 ) \\delta ( z ) + ( 1 - P _ { \\bar { s } } ( 1 ) ) \\delta ( z - 1 ) + ( P _ { \\bar { s } } ( 1 ) - P _ { \\bar { s } } ( 0 ) ) p _ { \\bar { s } } ( z | \\bar { s } \\in ( 0 , 1 ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            478,
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+            497
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Since both $q ( z )$ and $p ( z )$ are mixtures with the same number of components we can use the chain rule of relative entropy (Cover & Thomas, 2012; Hershey & Olsen, 2007) in order to compute the KL-divergence: ",
+        "bbox": [
+            173,
+            502,
+            825,
+            545
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efbb3871ba1d8962b6f02ac1ab5b340ef86a970b7824e03e1cbb28ecfb50dce5.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\displaystyle K L ( q ( z ) | | p ( z ) ) = Q _ { \\bar { s } } ( 0 ) \\log \\frac { Q _ { \\bar { s } } ( 0 ) } { P _ { \\bar { s } } ( 0 ) } + ( 1 - Q _ { \\bar { s } } ( 1 ) ) \\log \\frac { 1 - Q _ { \\bar { s } } ( 1 ) } { 1 - P _ { \\bar { s } } ( 1 ) } + } } \\\\ { { + \\left( Q _ { \\bar { s } } ( 1 ) - Q _ { \\bar { s } } ( 0 ) \\right) \\mathbb { E } _ { q _ { \\bar { s } } ( z | \\bar { s } \\in ( 0 , 1 ) ) } [ \\log q _ { \\bar { s } } ( z ) - \\log p _ { \\bar { s } } ( z ) ] , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            551,
+            758,
+            607
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\bar { s }$ corresponds to the the pre-rectified variable. Notice that in case that the integral under the truncated distribution $q ( \\bar { s } | \\bar { s } \\in ( \\bar { 0 } , 1 ) )$ is not available in closed form we can still obtain a Monte Carlo estimate by sampling the truncated distribution, on e.g. a $( \\gamma , \\zeta )$ interval, via the inverse transform method: ",
+        "bbox": [
+            176,
+            611,
+            821,
+            666
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3d48ba145cff5922efca9fc3b80b4f7014fd639369aa1b980a3bda586fe289d.jpg",
+        "text": "$$\nu \\sim \\mathcal { U } ( 0 , 1 ) , \\qquad z = Q _ { \\bar { s } } ^ { - 1 } \\bigl ( Q _ { \\bar { s } } ( \\gamma ) + u ( Q _ { \\bar { s } } ( \\zeta ) - Q _ { \\bar { s } } ( \\gamma ) ) \\bigr ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            671,
+            692,
+            691
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $Q _ { \\bar { s } } ^ { - 1 } ( \\cdot )$ corresponds to the quantile function and $Q _ { \\bar { s } } ( \\cdot )$ to the CDF of the random variable $\\bar { s }$ . Furthermore, it should be mentioned that $K L ( q ( z ) | | p ( z ) ) \\neq K L ( q ( \\bar { s } ) | | p ( \\bar { s } ) )$ , since the rectifications are not invertible transformations. ",
+        "bbox": [
+            173,
+            699,
+            826,
+            742
+        ],
+        "page_idx": 12
+    }
+]

parse/train/S11KBYclx/S11KBYclx.md

@@ -0,0 +1,258 @@
+# LEARNING CURVE PREDICTION WITH BAYESIAN NEURAL NETWORKS
+
+Aaron Klein, Stefan Falkner, Jost Tobias Springenberg & Frank Hutter   
+Department of Computer Science   
+University of Freiburg   
+{kleinaa,sfalkner,springj,fh}@cs.uni-freiburg.de
+
+# ABSTRACT
+
+Different neural network architectures, hyperparameters and training protocols lead to different performances as a function of time. Human experts routinely inspect the resulting learning curves to quickly terminate runs with poor hyperparameter settings and thereby considerably speed up manual hyperparameter optimization. The same information can be exploited in automatic hyperparameter optimization by means of a probabilistic model of learning curves across hyperparameter settings. Here, we study the use of Bayesian neural networks for this purpose and improve their performance by a specialized learning curve layer.
+
+# 1 INTRODUCTION
+
+Deep learning has celebrated many successes, but its performance relies crucially on good hyperparameter settings. Bayesian optimization (e.g, Brochu et al. (2010); Snoek et al. (2012); Shahriari et al. (2016)) is a powerful method for optimizing the hyperparameters of deep neural networks (DNNs). However, its traditional treatment of DNN performance as a black box poses fundamental limitations for large and computationally expensive data sets, for which training a single model can take weeks. Human experts go beyond this blackbox notion in their manual tuning and exploit cheaper signals about which hyperparameter settings work well: they estimate overall performance based on runs using subsets of the data or initial short runs to weed out bad parameter settings; armed with these tricks, human experts can often outperform Bayesian optimization.
+
+Recent extensions of Bayesian optimization and multi-armed bandits therefore also drop the limiting blackbox assumption and exploit the performance of short runs (Swersky et al., 2014; Domhan et al., 2015; Li et al., 2017), performance on small subsets of the data (Klein et al., 2017), and performance on other, related data sets (Swersky et al., 2013; Feurer et al., 2015).
+
+While traditional solutions for scalable Bayesian optimization include approximate Gaussian process models (e.g., Hutter et al.; Swersky et al. (2014)) and random forests (Hutter et al., 2011), a recent trend is to exploit the flexible model class of neural networks for this purpose (Snoek et al., 2015; Springenberg et al., 2016). In this paper, we study this model class for the prediction of learning curves. Our contributions in this paper are:
+
+1. We study how well Bayesian neural networks can fit learning curves for various architectures and hyperparameter settings, and how reliable their uncertainty estimates are. 2. Building on the parametric learning curve models of Domhan et al. (2015), we develop a specialized neural network architecture with a learning curve layer that improves learning curve predictions. 3. We compare different ways to generate Bayesian neural networks: probabilistic back propagation (Hernández-Lobato and Adams, 2015) and two different stochastic gradient based Markov Chain Monte Carlo (MCMC) methods – stochastic gradient Langevin dynamics (SGLD (Welling and Teh, 2011)) and stochastic gradient Hamiltonian MCMC (SGHMC (Chen et al., 2014)) – for standard Bayesian neural networks and our specialized architecture and show that SGHMC yields better uncertainty estimates.
+
+4. We evaluate the predictive quality for both completely new learning curves and for extrapolating partially-observed curves, showing better performance than the parametric function approach by Domhan et al. (2015) at stages were learning curves have not yet converged.
+
+5. We extend the recent multi-armed bandit strategy Hyperband (Li et al., 2017) by sampling using our model rather than uniformly at random, thereby enabling it to approach nearoptimal configurations faster than traditional Bayesian optimization.
+
+![](images/75f4c03df9a714254c23d4bb03fe878176b7232705146d1096ed5c21651906ad.jpg)  
+Figure 1: Example learning curves of random hyperparameter configurations of 4 different iterative machine learning methods: convolutional neural network (CNN), fully connected neural network (FCNet), logistic regression (LR), and variational auto-encoder (VAE). Although different configurations lead to different learning curves, they usually share some characteristics for a certain algorithm and dataset (but vary across these).
+
+# 2 PROBABILISTIC PREDICTION OF LEARNING CURVES
+
+In this section, we describe a general framework to model learning curves of iterative machine learning methods. We first describe the approach by Domhan et al. (2015) which we will dub LCExtrapolation from here on. Afterwards, we discuss a more general joint model across time steps and hyperparameter values that can exploit similarities between hyperparameter configurations and predict for unobserved learning curves. We also study the observation noise of different hyperparameter configurations and show how we can adapt our model to capture this noise.
+
+# 2.1 LEARNING CURVE PREDICTION WITH BASIS FUNCTION
+
+An intuitive model for learning curves proposed by Domhan et al. (2015) uses a set of $k$ different parametric functions $\phi _ { i } ( \pmb { \theta } _ { i } , t ) \in \{ \phi _ { 1 } ( \pmb { \theta } _ { 1 } , t ) , . . . , \phi _ { k } ( \pmb { \theta } _ { k } , t ) \}$ to extrapolate learning curves $\left( y _ { 1 } , \ldots , y _ { n } \right)$ from the first $n$ time steps. Each parametric function $\phi _ { i }$ depends on a time step $t \in [ 1 , T ]$ and on a parameter vector $\theta _ { i }$ . The individual functions are combined into a single model by a weighted linear combination
+
+$$
+\hat { f } ( t | \Theta , { \pmb w } ) = \sum _ { i = 1 } ^ { k } w _ { i } \phi _ { i } ( t , \pmb \theta _ { i } ) ,
+$$
+
+where $\boldsymbol { \Theta } = ( \theta _ { 1 } , \ldots , \theta _ { k } )$ denotes the combined vector of all parameters $\theta _ { 1 } , \ldots , \theta _ { k }$ , and $\textbf { \em w } =$ $( w _ { 1 } , \ldots , w _ { k } )$ is the concatenated vector of the respective weights of each function. Assuming observation noise around the true but unknown value $f ( t )$ , i.e., assuming $y _ { t } \sim \mathcal { N } ( \hat { f } ( t | \Theta , w ) , \sigma ^ { 2 } )$ , Domhan et al. (2015) define a prior for all parameters ${ \dot { P } } ( { \bf \dot { \Theta } } , w , \sigma ^ { 2 } )$ and use a gradient-free MCMC method (Foreman-Mackey et al., 2013) to obtain $S$ samples, $( \Theta _ { 1 } , \dot { w } _ { 1 } , \sigma _ { 1 } ^ { 2 } ) , \dots , ( \Theta _ { S } , { w } _ { S } , \sigma _ { S } ^ { 2 } )$ , from the posterior
+
+$$
+P ( \Theta , w , \sigma ^ { 2 } | y _ { 1 } \dots , y _ { n } ) \propto P ( y _ { 1 } , \dots , y _ { n } | \Theta , w , \sigma ^ { 2 } ) P ( \Theta , w , \sigma ^ { 2 } )
+$$
+
+using the likelihood
+
+$$
+P ( y _ { 1 } , \dots , y _ { n } | \Theta , w , \sigma ^ { 2 } ) = \prod _ { t = 1 } ^ { n } \mathcal { N } ( y _ { t } ; \hat { f } ( t | \Theta , w ) , \sigma ^ { 2 } ) .
+$$
+
+These samples then yield probabilistic extrapolations of the learning curve for future time steps $m$ with mean and variance predictions
+
+$$
+\begin{array} { c } { \displaystyle \hat { y } _ { m } = \mathbb { E } [ y _ { m } | y _ { 1 } , \dots y _ { n } ] \approx \frac { 1 } { S } \sum _ { s = 1 } ^ { S } \hat { f } ( m | \Theta _ { s } , w _ { s } ) , \mathrm { a n d } } \\ { \displaystyle \mathrm { v a r } ( \hat { y } _ { m } ) \approx \frac { 1 } { S } \sum _ { s = 1 } ^ { S } ( \hat { f } ( m | \Theta _ { s } , w _ { s } ) - \hat { y } _ { m } ) ^ { 2 } + \sum _ { s = 1 } ^ { S } \sigma _ { s } ^ { 2 } . } \end{array}
+$$
+
+For our experiments, we use the original implementation by Domhan et al. (2015) with one modification: the original code included a term in the likelihood that enforced the prediction at $t = T$ to be strictly greater than the last value of that particular curve. This biases the estimation to never underestimate the accuracy at the asymptote. We found that in some of our benchmarks, this led to instabilities, especially with very noisy learning curves. Removing it cured that problem, and we did not observe any performance degradation on any of the other benchmarks.
+
+The ability to include arbitrary parametric functions makes this model very flexible, and Domhan et al. (2015) used it successfully to terminate evaluations of poorly-performing hyperparameters early for various different architectures of neural networks (thereby speeding up Bayesian optimization by a factor of two). However, the model’s major disadvantage is that it does not use previously evaluated hyperparameters at all and therefore can only make useful predictions after observing a substantial initial fraction of the learning curve.
+
+# 2.2 LEARNING CURVE PREDICTION WITH BAYESIAN NEURAL NETWORKS
+
+In practice, similar hyperparameter configurations often lead to similar learning curves, and modelling this dependence would allow predicting learning curves for new configurations without the need to observe their initial performance. Swersky et al. (2014) followed this approach based on an approximate Gaussian process model. Their Freeze-Thaw method showed promising results for finding good hyperparameters of iterative machine learning algorithms using learning curve prediction to allocate most resources for well-performing configurations during the optimization. The method introduces a special covariance function corresponding to exponentially decaying functions to model the learning curves. This results in an analytically tractable model, but using different functions to account for cases where the learning curves do not converge exponentially is not trivial.
+
+Here, we formulate the problem using Bayesian neural networks. We aim to model the validation accuracy $g ( { \pmb x } , t )$ of a configuration $\pmb { x } ~ \in ~ \pmb { \chi } ~ \subset ~ \mathbb { R } ^ { d }$ at time step $t ~ \in ~ ( 0 , 1 ]$ based on noisy observations ${ y ( { \pmb x } , t ) } \ \sim \ \bar { \mathcal { N } } ( { g ( { \pmb x } , t ) , \sigma ^ { 2 } } )$ . For each configuration $_ { \textbf { \em x } }$ trained for $T _ { x }$ time steps, we obtain $T _ { x }$ data points for our model; denoting the combined data by $\mathcal { D } =$ $\{ ( { \pmb x } _ { 1 } , t _ { 1 } , y _ { 1 1 } ) ) , ( { \pmb x } _ { 1 } , t _ { 2 } , y _ { 1 2 } ) , \dots , ( { \pmb x } _ { n } , T _ { x _ { n } } , y _ { n T _ { x _ { n } } } ) \}$ we can then write the joint probability of the data $\mathcal { D }$ and the network weights $W$ as
+
+$$
+P ( \mathcal { D } , W ) = P ( W ) P ( \sigma ^ { 2 } ) \prod _ { i = 1 } ^ { | D | } \mathcal { N } ( y _ { i } ; \hat { g } ( { \pmb x } _ { i } , t _ { i } | W ) , \sigma ^ { 2 } ) .
+$$
+
+where $\hat { g } ( \pmb { x } _ { i } , t _ { i } | W )$ is the prediction of a neural network. It is intractable to compute the posterior weight distribution $p ( W | \mathcal { D } )$ , but we can use MCMC to sample it, in particular stochastic gradient
+
+MCMC methods, such as SGLD (Welling and Teh, 2011) or SGHMC (Chen et al., 2014). Given $M$ samples $W ^ { 1 } , \dots , W ^ { M }$ , we can then obtain the mean and variance of the predictive distribution $p ( g | \mathbf { x } , \bar { t } , \mathcal { D } )$ as
+
+$$
+\begin{array} { r l } & { \displaystyle \hat { \mu } ( \pmb { x } , t | \mathcal { D } ) = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \hat { g } ( \pmb { x } , t | W ^ { i } ) , \mathrm { a n d } } \\ & { \displaystyle \widehat { \sigma ^ { 2 } } ( \pmb { x } , t | \mathcal { D } ) = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \left( \hat { g } ( \pmb { x } , t | W ^ { i } ) - \hat { \mu } ( \pmb { x } , t | \mathcal { D } ) \right) ^ { 2 } , } \end{array}
+$$
+
+respectively. We will write these shorthand as $\hat { \mu } ( { \pmb x } , t )$ and $\widehat { \sigma ^ { 2 } } ( \pmb { x } , t )$ . This is similar to Eqs. 4 and exactly the model that Springenberg et al. (2016) used for (blackbox) Bayesian optimization with Bayesian neural networks; the only difference is in the input to the model: here, there is a data point for every time step of the curve, whereas Springenberg et al. (2016) only used a single data point per curve (for its final time step).
+
+![](images/202a87c34ea79e6348a09850e5f1523257a1dfe8dac0f5c24a32f22b678f8616.jpg)  
+Figure 2: Example functions generated with our $k = 5$ basis functions (formulas for which are given in Appendix B). For each function, we drew 50 different parameters $\theta _ { i }$ uniformly at random in the output domain of the hidden layer(s) of our model. This illustrates the type of functions used to model the learning curves.
+
+# 2.3 HETEROSCEDASTIC NOISE OF HYPERPARAMETER CONFIGURATION
+
+In the model described above, we assume homoscedastic noise across hyperparameter configurations. To evaluate how realistic this assumption is, we sampled 40 configurations of a fully connected network (see Section 3.1 for a more detailed description of how the data was generated and Table 2 for the list of hyperparameters) and evaluated each configuration $R = 1 0$ times with different pseudorandom number seeds (see the right panel of Figure 3 for some examples). Figure 3 (left) shows on the vertical axis the noise $\begin{array} { r } { \bar { \sigma ^ { 2 } } ( { \pmb x } , t ) = \frac { 1 } { R } \sum _ { r = 1 } ^ { R } ( y ( { \pmb x } , t ) - \bar { \mu } ( { \pmb x } , t ) ) ^ { 2 } } \end{array}$ and on the horizontal axis the rank of each configuration based on their asymptotic sample mean performance $\begin{array} { r } { 1 ) = \frac { 1 } { R } \sum _ { r = 1 } ^ { R } y ( \pmb { x } , t = 1 ) } \end{array}$ .
+
+![](images/6b6df89bcd43e5b7b654913621e73c7b16dbca16d689949396f7f0678b2a6bed.jpg)  
+Figure 3: On the left, we plot the noise estimate of 40 different configurations on the FCNet benchmark sorted by their asymptotic mean performance at $t = 1$ . The color indicates the time step $t$ , darker meaning a larger value. On the right, we show the learning curves of 5 different configurations each evaluated 10 times (indicated with the same color) with a different seed.
+
+Maybe not surprisingly, the noise seems to correlate with the asymptotic performance of a configuration. The fact that the noise between different configurations varies on different orders of magnitudes suggests a heteroscedastic noise model to best describe this behavior. We incorporate this observation by making the noise dependent on the input data to allow to predict different noise levels for different hyperparameters. In principle, one could also model a $t$ dependent noise, but we could not find the same trend across different datasets.
+
+# 2.4 NEW BASIS FUNCTION LAYER FOR LEARNING CURVE PREDICTION WITH BAYESIAN NEURAL NETWORKS
+
+We now combine Bayesian neural networks with the parametric functions to incorporate more knowledge about learning curves into the network itself. Instead of obtaining the parameters $\Theta$ and $\pmb { w }$ by sampling from the posterior, we use a Bayesian neural network to learn several mappings simultaneously:
+
+1. $\hat { \mu } _ { \infty }$ : $\mathcal X  \mathbb R$ , the asymptotic value of the learning curve   
+2. $\Theta$ : $\mathcal { X } \to \mathbb { R } ^ { K }$ , the parameters of a parametric function model (see Figure 2 for some example curves from our basis functions)   
+3. $v \colon \mathcal { X } \to \mathbb { R } ^ { k }$ , the corresponding weights for each function in the model   
+4. $\widehat { \sigma ^ { 2 } }$ : $\mathcal { X } \to \mathbb { R } ^ { + }$ , the observational noise for this hyperparameter configuration
+
+![](images/99302961fa6a852f3b59d323516058817487587445569220835086049107ec27.jpg)  
+Figure 4: Our neural network architecture to model learning curves. A common hidden layer is used to simultaneously model $\hat { \mu } _ { \infty }$ , the parameters $\Theta$ of the basis functions, their respective weights $\pmb { w }$ , and the noise $\widehat { \sigma ^ { 2 } }$ .
+
+With these quantities, we can compute the likelihood in (3) which allows training the network.
+
+Phrased differently, we use a neural network to predict the model parameters $\Theta$ and weights $\pmb { w }$ of our parametric functions, yielding the following form for our network’s mean predictions:
+
+$$
+\begin{array} { r } { \hat { g } ( \pmb { x } _ { i } , t _ { i } | W ) = \hat { f } ( t _ { i } | \Theta ( \pmb { x } _ { i } , W ) , \pmb { w } ( \pmb { x } _ { i } , W ) ) . } \end{array}
+$$
+
+A schematic of this is shown in Fig. 4. For training, we will use the same MCMC methods, namely SGLD and SGHMC mentioned above.
+
+# 3 EXPERIMENTS
+
+We now empirically evaluate the predictive performance of Bayesian neural networks, with and without our special learning curve layer. For both networks, we used a 3-layer architecture with tanh activations and 64 units per layer. We also evaluate two different sampling methods for both types of networks: stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian MCMC (SGHMC), following the approach of Springenberg et al. (2016) to automatically adapt the noise estimate and the preconditioning of the gradients.
+
+As baselines, we compare to other approaches suitable for this task. Besides the aforementioned work by Domhan et al. (2015), we also compare against random forests Breimann (2001) with empirical variance estimates (Hutter et al., 2014), a Gaussian process (GP) using the learning curve kernel from Swersky et al. (2014), another Bayesian neural network technique called probabilistic back propagation (PBP) (Hernández-Lobato and Adams, 2015), and the simple heuristic of using the last seen value (LastSeenValue) of each learning curve for extrapolation. The last model has been successfully used by Li et al. (2017) despite its simplicity.
+
+![](images/b05c7c6c26b75f30900e84c89472ea378803ec4a95d98ad870c9d2681a5043ab.jpg)  
+Figure 5: Qualitative comparison of the different models. The left panel shows the mean predictions of different methods on the CNN benchmark. All models observed the validation error of the first 12 epochs of the true learning curve (black). On the right, the posterior distributions over the value at 40 epochs is plotted.
+
+# 3.1 DATASETS
+
+For our empirical evaluation, we generated the following four datasets of learning curves, in each case sampling hyperparameter configurations at random from the hyperparameter spaces detailed in Table 2 in the appendix (see also Section D for some characteristic of these datasets):
+
+• CNN: We sampled 256 configurations of 5 different hyperparameters of a 3-layer convolutional neural network (CNN) and trained each of them for 40 epochs on the CIFAR10 (Krizhevsky, 2009) benchmark.   
+FCNet: We sampled 4096 configurations of 10 hyperparameters of a 2-layer feed forward neural network (FCNet) on MNIST (LeCun et al., 2001), with batch normalization, dropout and ReLU activation functions, annealing the learning rate over time according to a power function. We trained the neural network for 100 epochs.   
+• LR: We sampled 1024 configurations of the 4 hyperparameters of logistic regression (LR) and also trained it for 100 epochs on MNIST. VAE: We sampled 1024 configuration of the 4 hyperparameters of a variational autoencoder (VAE) (Kingma and Welling, 2014). We trained the VAE on MNIST, optimizing the approximation of the lower bound for 300 epochs.
+
+# .2 PREDICTING ASYMPTOTIC VALUES OF PARTIALLY OBSERVED CURVES
+
+We first study the problem of predicting the asymptotic values of partially-observed learning curves tackled by Domhan et al. (2015). The LC-Extrapolation method by Domhan et al. (2015), the GP, and the last seen value work on individual learning curves and do not allow to model performance across hyperparameter configurations. Thus, we trained them separately on individual partial learning curves. The other models, including our Bayesian neural networks, on the other hand, can use training data from different hyperparameter configurations. Here, we used training data with the same number of epochs for every partial learning curve.1
+
+The left panel of Figure 5 visualizes the extrapolation task, showing a learning curve from the CNN dataset and the prediction of the various models trained only using the first 12 of 40 epochs of the learning curve. The right panel shows the corresponding predictive distributions obtained with these models. LastSeenValue does not yield a distribution and uncertainties are not defined.
+
+For a more quantitative evaluation, we used all models to predict the asymptotic value of all learning curves, evaluating predictions based on observing between $10 \%$ and $90 \%$ of the learning curves.
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=2>CNNMSE·10² ALL</td><td rowspan=1 colspan=2>FCNetMSE-102 ALL</td><td rowspan=1 colspan=1>MSE·10²</td><td rowspan=1 colspan=1>LRALL</td><td rowspan=1 colspan=2>VAEMSE·104 ALL</td></tr><tr><td rowspan=1 colspan=1>SGLD</td><td rowspan=1 colspan=1>1.8±0.9</td><td rowspan=1 colspan=1>0.60±0.25</td><td rowspan=1 colspan=1>4.5±0.6</td><td rowspan=1 colspan=1>0.13±0.05</td><td rowspan=1 colspan=1>1.1 ± 0.3</td><td rowspan=1 colspan=1>0.77±0.08</td><td rowspan=1 colspan=1>4.7 ± 1.7</td><td rowspan=1 colspan=1>2.16 ±0.07</td></tr><tr><td rowspan=1 colspan=1>SGLD-LC</td><td rowspan=1 colspan=1>1.4 ± 0.9</td><td rowspan=1 colspan=1>1.09 ± 0.22</td><td rowspan=1 colspan=1>3.3 ±0.7</td><td rowspan=1 colspan=1>0.54± 0.06</td><td rowspan=1 colspan=1>1.3 ± 0.7</td><td rowspan=1 colspan=1>0.94 ±0.09</td><td rowspan=1 colspan=1>3.1 ± 1.1</td><td rowspan=1 colspan=1>1.55 ± 0.01</td></tr><tr><td rowspan=1 colspan=1>SGHMC</td><td rowspan=1 colspan=1>0.8±0.6</td><td rowspan=1 colspan=1>0.96± 0.15</td><td rowspan=1 colspan=1>1.9 ±0.3</td><td rowspan=1 colspan=1>0.50± 0.04</td><td rowspan=1 colspan=1>0.5±0.2</td><td rowspan=1 colspan=1>1.08±0.05</td><td rowspan=1 colspan=1>3.6± 1.5</td><td rowspan=1 colspan=1>2.34± 0.07</td></tr><tr><td rowspan=1 colspan=1>SGHMC-LC</td><td rowspan=1 colspan=1>0.7±0.6</td><td rowspan=1 colspan=1>1.15 ± 0.34</td><td rowspan=1 colspan=1>2.0±0.3</td><td rowspan=1 colspan=1>0.80±0.05</td><td rowspan=1 colspan=1>0.5±0.2</td><td rowspan=1 colspan=1>1.17 ± 0.08</td><td rowspan=1 colspan=1>1.9 ± 1.1</td><td rowspan=1 colspan=1>1.39 ± 0.65</td></tr></table>
+
+Table 1: In each column we report the mean squared error (MSE) and the average log-likelihood (ALL) of the 16 fold CV learning curve prediction for a neural network without learning curve prediction layer (SGLD and SGHMC) and with our new layer (SGLD-LC and SGHMC-LC).
+
+Figure 6 shows the mean squared error between the true asymptotic value and the models’ predictions (left) and the average log-likelihood of the true value given each model as a function of how much of the learning curves has been observed. Note that we removed some methods from the plots to avoid clutter; the same plots with all methods can be found in the supplementary material. We notice several patterns:
+
+1. Throughout, our specialized network architecture performs better than the standard Bayesian neural networks, and SGHMC outperformed SGLD (see Appendix C).   
+2. PBP shows mixed results for the extrapolated mean, and does not provide reliable uncertainties (also see Appendix C). We hypothesize that this may be due to its linear activation functions.   
+3. If no uncertainties are required, LastSeenValue is hard to beat. This is because many configurations approach their final performance quite quickly.   
+4. The GP mean predictions are very competitive, but the average log-likelihood indicates overconfidence, especially for short learning curves. We assume that the prior assumption of an exponential function is not flexible enough in practice and that after observing some data points the Gaussian process becomes too certain of its predictions.   
+5. The random forest’s mean predictions almost match the quality of the GP. The uncertainty estimates are better, but still too aggressive when only a small fraction of the learning curve was observed. Random forests do not extrapolate and similar to LastSeenValue also achieve a very good mean prediction if the learning curve has almost converged.   
+6. Local models for single curves clearly outperform global models for almost complete learning curves. Global models, like our BNN approach, on the other hand, are trained on many configurations and need to generalize across these, yielding somewhat worse performance for this (arguably easy) task.2
+
+# 3.3 PREDICTING UNOBSERVED LEARNING CURVES
+
+As mentioned before, training a joint model across hyperparameters and time steps allows us to make predictions for completely unobserved learning curves of new configurations. To estimate how well Bayesian neural networks perform in this task, we used the datasets from Section 3.1 and split all of them into 16 folds, allowing us to perform cross-validation of the predictive performance. For each fold, we trained all models on the full learning curves in the training set and let them predict the held-out learning curves.
+
+Table 1 shows the mean squared error and the average log-likelihood (both computed for all points in each learning curve) across the 16 folds. We make two observations: firstly, both neural network architectures lead to a reasonable mean squared error and average log-likelihood, for both SGLD and SGHMC. Except on the VAE dataset our learning curve layer seems to improve mean squared error and average log likelihood. Secondly, SGHMC performed better than SGLD, with the latter resulting in predictions with too small variances. Figure 7 visualizes the results for our CNN dataset in more detail, showing that true and predicted accuracies correlate quite strongly.
+
+![](images/6e4fa59fccc6c9bb52ff215e81ce2021db8ce82be943c99bcc576401e2759c52.jpg)  
+Figure 6: Assessment of the predictive quality based on partially observed learning curves on all 4 benchmarks. The panels on the left show the mean squared error of the predicted asymptotic value (y-axis) after observing all learning curves up to a given fraction of maximum number of epochs (x-axis). The panels on the right show the average log-likelihood based on the predictive mean and variance of the asymptotic value. Note that LastSeenValue does not provide a predictive variance.
+
+![](images/d32af6b872e17abef5b865d8314e51b5dd5b5d4b6363445170512d6be7971da8.jpg)  
+Figure 7: On the horizontal axis, we plot the true value and on the vertical axis the predicted values. Each point is colored by its log-likelihood (the brighter the higher). Our learning curve BNN trained with SGHMC leads to the best mean predictions and assigns the highest likelihood to the test points.
+
+# 3.4 MODEL-BASED HYPERBAND
+
+In this section, we show how our model can be used to improve hyperparameter optimization of iterative machine learning algorithms. For this, we extended the multi-armed bandit strategy Hyperband (Li et al., 2017), which in each iteration $i$ first samples $N _ { i }$ hyperparameter configurations $\dot { C } \dot { = } \{ { \pmb x } _ { 1 } , \ldots , { \pmb x } _ { N _ { i } } \}$ and then uses successive halving (Jamieson and Talwalkar, 2016) to iteratively discard poorly-performing configurations from $C$ . While the original Hyperband method samples configurations $C$ from a uniform distribution $\mathcal { U }$ over hyperparameter configurations, our extension instead samples them based on our model, with all other parts remaining unchanged. More precisely, we sample $M _ { i } \ > > \ N _ { i }$ configurations uniformly, $\hat { C } = \{ \pmb { x } _ { 1 } , \dotsc , \pmb { x } _ { M _ { i } } \} \sim \mathcal { U }$ , and pick the $N _ { i }$ configurations with the highest predicted asymptotic mean $a ( \pmb { x } ) = \hat { \mu } ( \pmb { x } , t = 1 )$ or upper confidence value $a ( { \pmb x } ) = \hat { \mu } ( { \pmb x } , t = 1 ) \hat { \bf \mu } + \hat { \sigma } ( \hat { \bf \ x } , t = 1 )$ .
+
+For a thorough empirical evaluation and to reduce computational requirements we reused the data from Section 3.1 to construct a surrogate benchmark based on a random forest (see Appendix E for more details). After each iteration we report the final performance of the best observed configuration so far, along with the wall clock time that would have been needed for optimizing the true objective function. Reporting the (predicted) wall clock time (rather than the number of iterations) also takes our optimizer’s overhead into account. This is important since one may worry that this overhead may be substantial due to our use of a model; however, as we show it does not harm performance.
+
+![](images/8c9350f5faf2edc1a98eff87c848e6b701aa9a5d8f1cc9a9753b8328db64be00.jpg)  
+Figure 8: Comparison of Hyperband, Hyperband with our model, and standard Bayesian optimization on the CNN benchmark. Hyperband finds a good configuration faster than standard Bayesian optimization, but it only approaches the global optimum quickly when extended with our model.
+
+Figure 8 shows the immediate regret on the CNN benchmark as a function of wallclock time, for three optimizers: Hyperband, our model-based extension of Hyperband, as well as standard Bayesian optimization with a Gaussian process (a comparison to additional baselines can be found in Appendix E). Standard Bayesian optimization does not make use of learning curves and thus needs to evaluate each configuration for the full amount of epochs. Nevertheless, since training one configuration to the end takes less time than one round of successive halving, Bayesian optimization produces its first results earlier than Hyperband. In accordance with results by Li et al. (2017), in this experiment Hyperband found a configuration with good performance faster than standard Bayesian optimization, but its random sampling did not suffice to quickly approach the best configuration; given enough time Bayesian optimization performed better. However, extended by our model, Hyperband both found a good configuration fast and approached the global optimum fastest.
+
+We would like to emphasize that our model-based extension of Hyperband is not limited to our particular Bayesian neural networks as the underlying model. Since Hyperband stops the majority of the configurations very early, the setting is quite similar to that of very short partially observed learning curves in Section 3.2, with the differences that some configurations have been evaluated for longer and that the model now has to generalize to unseen configurations. For example, for the CNN and LR benchmarks, the experiments in Section 3.2 showed that random forests achieve strong average log-likelihoods for very short partial learning curves, and we would therefore also expect them to work well when combined with Hyperband. However, given our model’s more robust likelihoods and mean squared error values we believe it to be less sensitive to the underlying data. Additionally, we can incorporate more prior knowledge about the general shape of the curves into our model, something that is not easily done for many other model classes.
+
+# 4 CONCLUSION
+
+We studied Bayesian neural networks for modelling the learning curves of iterative machine learning methods, such as stochastic gradient descent for convolutional neural networks. Based on the parametric learning curve models of Domhan et al. (2015), we also developed a specialized neural network architecture with a learning curve layer that improves learning curve predictions. In future work, we aim to study recurrent neural networks for predicting learning curves and will extend Bayesian optimization methods with Bayesian neural networks (Springenberg et al., 2016) based on our learning curve models.
+
+# ACKNOWLEDGMENT
+
+This work has partly been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant no. 716721, by the European Commission under grant no. H2020-ICT-645403-ROBDREAM, and by the German Research Foundation (DFG) under Priority Programme Autonomous Learning (SPP 1527, grant HU 1900/3-1).
+
+# REFERENCES
+
+E. Brochu, V. Cora, and N. de Freitas. A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. CoRR, 2010.   
+J. Snoek, H. Larochelle, and R. P. Adams. Practical Bayesian optimization of machine learning algorithms. In Proc. of NIPS’12, 2012.   
+B. Shahriari, K. Swersky, Z. Wang, R. Adams, and N. de Freitas. Taking the human out of the loop: A Review of Bayesian Optimization. Proc. of the IEEE, (1), 12/2015 2016.   
+K. Swersky, J. Snoek, and R. Adams. Freeze-thaw Bayesian optimization. CoRR, 2014.   
+T. Domhan, J. T. Springenberg, and F. Hutter. Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. In Q. Yang and M. Wooldridge, editors, Proc. of IJCAI’15, 2015.   
+L. Li, K. Jamieson, G. DeSalvo, A. Rostamizadeh, and A. Talwalkar. Hyperband: Bandit-based configuration evaluation for hyperparameter optimization. In Proc. of ICLR’17, 2017.   
+A. Klein, S. Falkner, S. Bartels, P. Hennig, and F. Hutter. Fast Bayesian optimization of machine learning hyperparameters on large datasets. In Proc. of AISTATS’17, 2017.   
+K. Swersky, J. Snoek, and R. Adams. Multi-task Bayesian optimization. In Proc. of NIPS’13, 2013.   
+M. Feurer, T. Springenberg, and F. Hutter. Initializing Bayesian hyperparameter optimization via meta-learning. In Proc. of AAAI’15, 2015.   
+F. Hutter, H. Hoos, K. Leyton-Brown, and K. Murphy. Time-bounded sequential parameter optimization.   
+F. Hutter, H. Hoos, and K. Leyton-Brown. Sequential model-based optimization for general algorithm configuration. In LION’11, 2011.   
+J. Snoek, O. Rippel, K. Swersky, R. Kiros, N. Satish, N. Sundaram, M. M. A. Patwary, Prabhat, and R. P. Adams. Scalable Bayesian optimization using deep neural networks. In Proc. of ICML’15, 2015.   
+J. Springenberg, A. Klein, S. Falkner, and F. Hutter. Bayesian optimization with robust Bayesian neural networks. In Proc. of NIPS’16, 2016.   
+J. Hernández-Lobato and R. Adams. Probabilistic backpropagation for scalable learning of Bayesian neural networks. In Proc. of ICML’15, 2015.   
+M. Welling and Y. Teh. Bayesian learning via stochastic gradient Langevin dynamics. In Proc. of ICML’11, 2011.   
+T. Chen, E.B. Fox, and C. Guestrin. Stochastic gradient Hamiltonian Monte Carlo. In Proc. of ICML’14, 2014.   
+D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman. emcee $:$ The MCMC Hammer. PASP, 2013.   
+L. Breimann. Random forests. MLJ, 2001.   
+F. Hutter, L. Xu, H. Hoos, and K. Leyton-Brown. Algorithm runtime prediction: Methods and evaluation. AIJ, 2014.   
+A. Krizhevsky. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.   
+Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. In S. Haykin and B. Kosko, editors, Intelligent Signal Processing. IEEE Press, 2001. URL http://www.iro.umontreal.ca/\~lisa/pointeurs/lecun-01a.pdf.   
+D. Kingma and M. Welling. Auto-encoding variational bayes. In Proc. of ICLR’14, 2014.   
+K. Jamieson and A. Talwalkar. Non-stochastic best arm identification and hyperparameter optimization. In Proc. of AISTATS’16, 2016.   
+K. Eggensperger, F. Hutter, H.H. Hoos, and K. Leyton-Brown. Efficient benchmarking of hyperparameter optimizers via surrogates. In Proc. of AAAI’15, 2015.
+
+# A EXPERIMENTAL SETUP – DETAILS
+
+Table 2 shows the hyperparameters of our 4 benchmarks described in Section 3.1 and their ranges.   
+Table 2: Hyperparameter configuration space of the four different iterative methods. For the FCNet we decayed the learning rate by a $\alpha _ { d e c a y } = ( 1 + \gamma * t ) ^ { - \kappa }$ and also sampled different values for $\gamma$ and $\kappa$ .   
+
+<table><tr><td></td><td>Name</td><td>Range</td><td>log scale</td></tr><tr><td rowspan="5">CNN</td><td>batch size</td><td>[32, 512]</td><td>1</td></tr><tr><td>number of units layer 1</td><td>[24,210]</td><td>√</td></tr><tr><td>number of units layer 2</td><td>[24,210]</td><td>√</td></tr><tr><td>number of units layer 3</td><td>[24,210]</td><td>&lt;</td></tr><tr><td>learning rate</td><td>[10-6,10-0]</td><td>√</td></tr><tr><td rowspan="9">FCNet</td><td>initial learning rate</td><td>[10-6,10°]</td><td>√</td></tr><tr><td>L2 regularization</td><td>[10-8,10-1]</td><td></td></tr><tr><td>batch size</td><td>[32,512]</td><td>√</td></tr><tr><td>Y</td><td>[-3,-1]</td><td></td></tr><tr><td>K</td><td>[0,1]</td><td></td></tr><tr><td>momentum</td><td>[0.3,0.999]</td><td></td></tr><tr><td>number units 1</td><td>[25,122]</td><td></td></tr><tr><td>number units 2</td><td>[25,212]</td><td></td></tr><tr><td>dropout rate layer 1</td><td>[0.0,0.99]</td><td></td></tr><tr><td rowspan="4">LR</td><td>dropout rate layer 2</td><td>[0.0,0.99]</td><td></td></tr><tr><td>learning rate</td><td>[10-6,100]</td><td>√</td></tr><tr><td>L2</td><td>[0.0,1.0]</td><td></td></tr><tr><td>batch size</td><td>[20,2000]</td><td></td></tr><tr><td rowspan="5">VAE</td><td>dropout rate on inputs L</td><td>[0.0,0.75]</td><td></td></tr><tr><td>number of hidden units</td><td>[1,3]</td><td></td></tr><tr><td>batch size</td><td>[32,2048] [16,512]</td><td></td></tr><tr><td>z dimension</td><td></td><td></td></tr><tr><td></td><td>[2,200]</td><td></td></tr></table>
+
+# B DESCRIPTION OF THE BASIS FUNCTIONS
+
+To reduce complexity, we used a subset of the basis function from Domhan et al. (2015) which we found to be sufficient for learning curve prediction. We adapted these functions to model the residual between the asymptotic value $y _ { \infty }$ and we scaled the parameters $\Theta$ to be in $[ 0 , 1 ]$ . Table 3 shows the exact equations we used.
+
+# C PREDICTING ASYMPTOTIC VALUES OF PARTIALLY OBSERVED CURVES
+
+Figure 9 shows the mean squared error and the average log-likelihood of predicting the asymptotic value after observing different amounts of the learning curve for all methods. See Section 3.2 in the main text for more details.
+
+# D DATASET CHARACTERISTICS
+
+Figure 10 shows the distributions over runtimes for all random configurations of different benchmarks. As it can be seen there is a high variance of the runtime between different configurations for all benchmarks and some configurations need order of magnitudes longer than others.
+
+![](images/b9eb85e592aeb151f0d1358c063601fdde2beaf21d7bc5491a06006ea960ff12.jpg)  
+Figure 9: Assessment of the predictive quality based on partially observed learning curves on all 4 benchmarks. The panels on the left show the mean squared error of the predicted asymptotic value (y-axis) after observing all learning curves up to a given fraction of maximum number of epochs (x-axis). The panels on the right show the average log-likelihood based on the predictive mean and variance of the asymptotic value.
+
+Table 3: The formulas of our 5 basis functions.   
+
+<table><tr><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Formula</td></tr><tr><td rowspan=1 colspan=1>vapor pressure</td><td rowspan=1 colspan=1>Φ(t,a,b,c)=e-a-0.1-b.t-11c-lg(t)e--01b</td></tr><tr><td rowspan=1 colspan=1>pow</td><td rowspan=1 colspan=1>(t,a,b)=a·(tb-1)</td></tr><tr><td rowspan=1 colspan=1>log power</td><td rowspan=1 colspan=1>Φ(ta,b,c)=2·[()-(((t)</td></tr><tr><td rowspan=1 colspan=1>exponential</td><td rowspan=1 colspan=1>Φ(t,a,b)=a·[e-10-b -e-10·b·t]</td></tr><tr><td rowspan=1 colspan=1>hill-3</td><td rowspan=1 colspan=1>(t,a,b,c)=a.[（cb.t-b+1)-1-(cb+1)-1]</td></tr></table>
+
+Figure 11 shows the empirical cumulative distribution of the asymptotic performance of random configurations. These plots give an intuition about the difficulty of a benchmark as they show how many random configurations one has to sample in order to achieve a good performance.
+
+# E OPTIMIZATION ON SURROGATE BENCHMARKS
+
+We follow the approach by Eggensperger et al. (2015) and used the generated datasets from Section 3.1 to build surrogates of the objective functions. This allows us to compare different optimizers on this benchmark very efficiently while (approximately) preserving the characteristics of the underlying benchmark. Using surrogates instead of optimizing the real benchmarks allows us to carry out a more thorough empirical evaluation (since single function evaluations are cheap and we therefore can afford more and longer optimization runs). We used random forests as surrogates following Eggensperger et al. (2015) since they do not introduce a bias for our approach (in contrast to, e.g., surrogates based on standard feed forward neural networks trained with stochastic gradient descent ). For each benchmark, we used all configurations to train a random forest predicting the validation accuracy of a configuration at a certain time step as well as the wall clock time for its evaluation.
+
+After each round of successive halving, we return the current best observed configuration and its asymptotic performance. For each function evaluation, we predict the accuracy, and the runtime. By adding the optimization overhead, we can predict the wallclock time necessary to optimize the real objective function. Reporting the wallclock time rather than the number of iterations also takes the additional overhead from running our method into account. Furthermore, we argue that the wallclock time is more interesting in practice.
+
+As baselines, we compare to Gaussian process based Bayesian optimization with the upper confidence bound acquisition function (GP-BO-UCB) and the (log) expected improvement acquisition function (GP-BO-Log-EI), as well as Bayesian optimization with Bayesian neural networks (BOHAMIANN) (Springenberg et al., 2016). As an additional baseline, we use standard Bayesian neural networks (SGHMC-BNN) inside Hyperband instead of our specialized neural network architecture (SGHMCLCNet). We compare our model-based version of Hyperband against all these baselines with sampling from the asymptotic mean (SGHMC-LCNet-mean-sampling) and the asymptotic upper confidence bound (SGHMC-LCNet-ucb-sampling). Figure 12 shows that our specialized neural network architecture helps to speed up the convergence of Hyperband on the CNN and the FCNet benchmark. For the LR benchmarks our model still improves over Hyperband but does not perform better than Bayesian optimization. Hyperband as well as our method find the optimum already after the first round of successive halving for the VAE benchmark, which leads to an regret of 0.
+
+![](images/69e670c6d4b34b667a4f1d53ed7b0093390e81320c8387179f73cd8fbade8c21.jpg)  
+Figure 10: Distributions over runtimes for different random configurations for the CNN, FCNet, LR and VAE benchmark described in Section 3.One can see that all distributions are long tailed and that especially on the FCNet benchmark some configuration need order of magnitudes longer than others.
+
+![](images/287fb6fd47d5d135b71d04a2ea9a7fe7652e8baa67ff6bc8d75524fa463094f1.jpg)  
+Figure 11: Empirical cumulative distributions for the CNN, FCNet, LR and VAE benchmark.
+
+![](images/c028c1defddf9f22f04dff6d49c0efd0fe68546444bd755785852f09afbbf434.jpg)  
+Figure 12: Different optimizers on a random forests based suggorate of all collected learning curves on different benchmarks. Note how sampling from the model improves Hyperband’s performance by converging to the optimum faster.

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+    {
+        "type": "text",
+        "text": "LEARNING CURVE PREDICTION WITH BAYESIAN NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Aaron Klein, Stefan Falkner, Jost Tobias Springenberg & Frank Hutter   \nDepartment of Computer Science   \nUniversity of Freiburg   \n{kleinaa,sfalkner,springj,fh}@cs.uni-freiburg.de ",
+        "bbox": [
+            184,
+            169,
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Different neural network architectures, hyperparameters and training protocols lead to different performances as a function of time. Human experts routinely inspect the resulting learning curves to quickly terminate runs with poor hyperparameter settings and thereby considerably speed up manual hyperparameter optimization. The same information can be exploited in automatic hyperparameter optimization by means of a probabilistic model of learning curves across hyperparameter settings. Here, we study the use of Bayesian neural networks for this purpose and improve their performance by a specialized learning curve layer. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            440,
+            336,
+            457
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning has celebrated many successes, but its performance relies crucially on good hyperparameter settings. Bayesian optimization (e.g, Brochu et al. (2010); Snoek et al. (2012); Shahriari et al. (2016)) is a powerful method for optimizing the hyperparameters of deep neural networks (DNNs). However, its traditional treatment of DNN performance as a black box poses fundamental limitations for large and computationally expensive data sets, for which training a single model can take weeks. Human experts go beyond this blackbox notion in their manual tuning and exploit cheaper signals about which hyperparameter settings work well: they estimate overall performance based on runs using subsets of the data or initial short runs to weed out bad parameter settings; armed with these tricks, human experts can often outperform Bayesian optimization. ",
+        "bbox": [
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+            599
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent extensions of Bayesian optimization and multi-armed bandits therefore also drop the limiting blackbox assumption and exploit the performance of short runs (Swersky et al., 2014; Domhan et al., 2015; Li et al., 2017), performance on small subsets of the data (Klein et al., 2017), and performance on other, related data sets (Swersky et al., 2013; Feurer et al., 2015). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While traditional solutions for scalable Bayesian optimization include approximate Gaussian process models (e.g., Hutter et al.; Swersky et al. (2014)) and random forests (Hutter et al., 2011), a recent trend is to exploit the flexible model class of neural networks for this purpose (Snoek et al., 2015; Springenberg et al., 2016). In this paper, we study this model class for the prediction of learning curves. Our contributions in this paper are: ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. We study how well Bayesian neural networks can fit learning curves for various architectures and hyperparameter settings, and how reliable their uncertainty estimates are. 2. Building on the parametric learning curve models of Domhan et al. (2015), we develop a specialized neural network architecture with a learning curve layer that improves learning curve predictions. 3. We compare different ways to generate Bayesian neural networks: probabilistic back propagation (Hernández-Lobato and Adams, 2015) and two different stochastic gradient based Markov Chain Monte Carlo (MCMC) methods – stochastic gradient Langevin dynamics (SGLD (Welling and Teh, 2011)) and stochastic gradient Hamiltonian MCMC (SGHMC (Chen et al., 2014)) – for standard Bayesian neural networks and our specialized architecture and show that SGHMC yields better uncertainty estimates. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "4. We evaluate the predictive quality for both completely new learning curves and for extrapolating partially-observed curves, showing better performance than the parametric function approach by Domhan et al. (2015) at stages were learning curves have not yet converged. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "5. We extend the recent multi-armed bandit strategy Hyperband (Li et al., 2017) by sampling using our model rather than uniformly at random, thereby enabling it to approach nearoptimal configurations faster than traditional Bayesian optimization. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/75f4c03df9a714254c23d4bb03fe878176b7232705146d1096ed5c21651906ad.jpg",
+        "image_caption": [
+            "Figure 1: Example learning curves of random hyperparameter configurations of 4 different iterative machine learning methods: convolutional neural network (CNN), fully connected neural network (FCNet), logistic regression (LR), and variational auto-encoder (VAE). Although different configurations lead to different learning curves, they usually share some characteristics for a certain algorithm and dataset (but vary across these). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2 PROBABILISTIC PREDICTION OF LEARNING CURVES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we describe a general framework to model learning curves of iterative machine learning methods. We first describe the approach by Domhan et al. (2015) which we will dub LCExtrapolation from here on. Afterwards, we discuss a more general joint model across time steps and hyperparameter values that can exploit similarities between hyperparameter configurations and predict for unobserved learning curves. We also study the observation noise of different hyperparameter configurations and show how we can adapt our model to capture this noise. ",
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+    {
+        "type": "text",
+        "text": "2.1 LEARNING CURVE PREDICTION WITH BASIS FUNCTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "An intuitive model for learning curves proposed by Domhan et al. (2015) uses a set of $k$ different parametric functions $\\phi _ { i } ( \\pmb { \\theta } _ { i } , t ) \\in \\{ \\phi _ { 1 } ( \\pmb { \\theta } _ { 1 } , t ) , . . . , \\phi _ { k } ( \\pmb { \\theta } _ { k } , t ) \\}$ to extrapolate learning curves $\\left( y _ { 1 } , \\ldots , y _ { n } \\right)$ from the first $n$ time steps. Each parametric function $\\phi _ { i }$ depends on a time step $t \\in [ 1 , T ]$ and on a parameter vector $\\theta _ { i }$ . The individual functions are combined into a single model by a weighted linear combination ",
+        "bbox": [
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+        "img_path": "images/5bbc0f72dee9822a1a6d553c9bac970c54e3e46ac92abdeb3e9f7494cfbb19b6.jpg",
+        "text": "$$\n\\hat { f } ( t | \\Theta , { \\pmb w } ) = \\sum _ { i = 1 } ^ { k } w _ { i } \\phi _ { i } ( t , \\pmb \\theta _ { i } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { \\Theta } = ( \\theta _ { 1 } , \\ldots , \\theta _ { k } )$ denotes the combined vector of all parameters $\\theta _ { 1 } , \\ldots , \\theta _ { k }$ , and $\\textbf { \\em w } =$ $( w _ { 1 } , \\ldots , w _ { k } )$ is the concatenated vector of the respective weights of each function. Assuming observation noise around the true but unknown value $f ( t )$ , i.e., assuming $y _ { t } \\sim \\mathcal { N } ( \\hat { f } ( t | \\Theta , w ) , \\sigma ^ { 2 } )$ , Domhan et al. (2015) define a prior for all parameters ${ \\dot { P } } ( { \\bf \\dot { \\Theta } } , w , \\sigma ^ { 2 } )$ and use a gradient-free MCMC method (Foreman-Mackey et al., 2013) to obtain $S$ samples, $( \\Theta _ { 1 } , \\dot { w } _ { 1 } , \\sigma _ { 1 } ^ { 2 } ) , \\dots , ( \\Theta _ { S } , { w } _ { S } , \\sigma _ { S } ^ { 2 } )$ , from the posterior ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/519497d9ab4cc589f85f3b90a3452ecd2ced616f5305d102367ba1514dcd25b1.jpg",
+        "text": "$$\nP ( \\Theta , w , \\sigma ^ { 2 } | y _ { 1 } \\dots , y _ { n } ) \\propto P ( y _ { 1 } , \\dots , y _ { n } | \\Theta , w , \\sigma ^ { 2 } ) P ( \\Theta , w , \\sigma ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "using the likelihood ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6da3ff64c5bc740bd5ac5ce9e579075b0418430a0182a01f20ee2ae8e8767e81.jpg",
+        "text": "$$\nP ( y _ { 1 } , \\dots , y _ { n } | \\Theta , w , \\sigma ^ { 2 } ) = \\prod _ { t = 1 } ^ { n } \\mathcal { N } ( y _ { t } ; \\hat { f } ( t | \\Theta , w ) , \\sigma ^ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "These samples then yield probabilistic extrapolations of the learning curve for future time steps $m$ with mean and variance predictions ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e125cf145da5a9b620f63f1d2d838bf7587a6e2f9bff0a6c996c12f31d3f2ef7.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle \\hat { y } _ { m } = \\mathbb { E } [ y _ { m } | y _ { 1 } , \\dots y _ { n } ] \\approx \\frac { 1 } { S } \\sum _ { s = 1 } ^ { S } \\hat { f } ( m | \\Theta _ { s } , w _ { s } ) , \\mathrm { a n d } } \\\\ { \\displaystyle \\mathrm { v a r } ( \\hat { y } _ { m } ) \\approx \\frac { 1 } { S } \\sum _ { s = 1 } ^ { S } ( \\hat { f } ( m | \\Theta _ { s } , w _ { s } ) - \\hat { y } _ { m } ) ^ { 2 } + \\sum _ { s = 1 } ^ { S } \\sigma _ { s } ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For our experiments, we use the original implementation by Domhan et al. (2015) with one modification: the original code included a term in the likelihood that enforced the prediction at $t = T$ to be strictly greater than the last value of that particular curve. This biases the estimation to never underestimate the accuracy at the asymptote. We found that in some of our benchmarks, this led to instabilities, especially with very noisy learning curves. Removing it cured that problem, and we did not observe any performance degradation on any of the other benchmarks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The ability to include arbitrary parametric functions makes this model very flexible, and Domhan et al. (2015) used it successfully to terminate evaluations of poorly-performing hyperparameters early for various different architectures of neural networks (thereby speeding up Bayesian optimization by a factor of two). However, the model’s major disadvantage is that it does not use previously evaluated hyperparameters at all and therefore can only make useful predictions after observing a substantial initial fraction of the learning curve. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.2 LEARNING CURVE PREDICTION WITH BAYESIAN NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In practice, similar hyperparameter configurations often lead to similar learning curves, and modelling this dependence would allow predicting learning curves for new configurations without the need to observe their initial performance. Swersky et al. (2014) followed this approach based on an approximate Gaussian process model. Their Freeze-Thaw method showed promising results for finding good hyperparameters of iterative machine learning algorithms using learning curve prediction to allocate most resources for well-performing configurations during the optimization. The method introduces a special covariance function corresponding to exponentially decaying functions to model the learning curves. This results in an analytically tractable model, but using different functions to account for cases where the learning curves do not converge exponentially is not trivial. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here, we formulate the problem using Bayesian neural networks. We aim to model the validation accuracy $g ( { \\pmb x } , t )$ of a configuration $\\pmb { x } ~ \\in ~ \\pmb { \\chi } ~ \\subset ~ \\mathbb { R } ^ { d }$ at time step $t ~ \\in ~ ( 0 , 1 ]$ based on noisy observations ${ y ( { \\pmb x } , t ) } \\ \\sim \\ \\bar { \\mathcal { N } } ( { g ( { \\pmb x } , t ) , \\sigma ^ { 2 } } )$ . For each configuration $_ { \\textbf { \\em x } }$ trained for $T _ { x }$ time steps, we obtain $T _ { x }$ data points for our model; denoting the combined data by $\\mathcal { D } =$ $\\{ ( { \\pmb x } _ { 1 } , t _ { 1 } , y _ { 1 1 } ) ) , ( { \\pmb x } _ { 1 } , t _ { 2 } , y _ { 1 2 } ) , \\dots , ( { \\pmb x } _ { n } , T _ { x _ { n } } , y _ { n T _ { x _ { n } } } ) \\}$ we can then write the joint probability of the data $\\mathcal { D }$ and the network weights $W$ as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/98131cd18eef463840d444a0f54f964f33933e84b3a86f5ff2e6d7a3932e92c5.jpg",
+        "text": "$$\nP ( \\mathcal { D } , W ) = P ( W ) P ( \\sigma ^ { 2 } ) \\prod _ { i = 1 } ^ { | D | } \\mathcal { N } ( y _ { i } ; \\hat { g } ( { \\pmb x } _ { i } , t _ { i } | W ) , \\sigma ^ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { g } ( \\pmb { x } _ { i } , t _ { i } | W )$ is the prediction of a neural network. It is intractable to compute the posterior weight distribution $p ( W | \\mathcal { D } )$ , but we can use MCMC to sample it, in particular stochastic gradient ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "MCMC methods, such as SGLD (Welling and Teh, 2011) or SGHMC (Chen et al., 2014). Given $M$ samples $W ^ { 1 } , \\dots , W ^ { M }$ , we can then obtain the mean and variance of the predictive distribution $p ( g | \\mathbf { x } , \\bar { t } , \\mathcal { D } )$ as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d14bff20189bfc649a6b88404c0e822854cc1e2b8127fc3b8cf58cc385f0726.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\hat { \\mu } ( \\pmb { x } , t | \\mathcal { D } ) = \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\hat { g } ( \\pmb { x } , t | W ^ { i } ) , \\mathrm { a n d } } \\\\ & { \\displaystyle \\widehat { \\sigma ^ { 2 } } ( \\pmb { x } , t | \\mathcal { D } ) = \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\left( \\hat { g } ( \\pmb { x } , t | W ^ { i } ) - \\hat { \\mu } ( \\pmb { x } , t | \\mathcal { D } ) \\right) ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "respectively. We will write these shorthand as $\\hat { \\mu } ( { \\pmb x } , t )$ and $\\widehat { \\sigma ^ { 2 } } ( \\pmb { x } , t )$ . This is similar to Eqs. 4 and exactly the model that Springenberg et al. (2016) used for (blackbox) Bayesian optimization with Bayesian neural networks; the only difference is in the input to the model: here, there is a data point for every time step of the curve, whereas Springenberg et al. (2016) only used a single data point per curve (for its final time step). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/202a87c34ea79e6348a09850e5f1523257a1dfe8dac0f5c24a32f22b678f8616.jpg",
+        "image_caption": [
+            "Figure 2: Example functions generated with our $k = 5$ basis functions (formulas for which are given in Appendix B). For each function, we drew 50 different parameters $\\theta _ { i }$ uniformly at random in the output domain of the hidden layer(s) of our model. This illustrates the type of functions used to model the learning curves. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 HETEROSCEDASTIC NOISE OF HYPERPARAMETER CONFIGURATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            502,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the model described above, we assume homoscedastic noise across hyperparameter configurations. To evaluate how realistic this assumption is, we sampled 40 configurations of a fully connected network (see Section 3.1 for a more detailed description of how the data was generated and Table 2 for the list of hyperparameters) and evaluated each configuration $R = 1 0$ times with different pseudorandom number seeds (see the right panel of Figure 3 for some examples). Figure 3 (left) shows on the vertical axis the noise $\\begin{array} { r } { \\bar { \\sigma ^ { 2 } } ( { \\pmb x } , t ) = \\frac { 1 } { R } \\sum _ { r = 1 } ^ { R } ( y ( { \\pmb x } , t ) - \\bar { \\mu } ( { \\pmb x } , t ) ) ^ { 2 } } \\end{array}$ and on the horizontal axis the rank of each configuration based on their asymptotic sample mean performance $\\begin{array} { r } { 1 ) = \\frac { 1 } { R } \\sum _ { r = 1 } ^ { R } y ( \\pmb { x } , t = 1 ) } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b6df89bcd43e5b7b654913621e73c7b16dbca16d689949396f7f0678b2a6bed.jpg",
+        "image_caption": [
+            "Figure 3: On the left, we plot the noise estimate of 40 different configurations on the FCNet benchmark sorted by their asymptotic mean performance at $t = 1$ . The color indicates the time step $t$ , darker meaning a larger value. On the right, we show the learning curves of 5 different configurations each evaluated 10 times (indicated with the same color) with a different seed. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            660,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Maybe not surprisingly, the noise seems to correlate with the asymptotic performance of a configuration. The fact that the noise between different configurations varies on different orders of magnitudes suggests a heteroscedastic noise model to best describe this behavior. We incorporate this observation by making the noise dependent on the input data to allow to predict different noise levels for different hyperparameters. In principle, one could also model a $t$ dependent noise, but we could not find the same trend across different datasets. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.4 NEW BASIS FUNCTION LAYER FOR LEARNING CURVE PREDICTION WITH BAYESIAN NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now combine Bayesian neural networks with the parametric functions to incorporate more knowledge about learning curves into the network itself. Instead of obtaining the parameters $\\Theta$ and $\\pmb { w }$ by sampling from the posterior, we use a Bayesian neural network to learn several mappings simultaneously: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. $\\hat { \\mu } _ { \\infty }$ : $\\mathcal X  \\mathbb R$ , the asymptotic value of the learning curve   \n2. $\\Theta$ : $\\mathcal { X } \\to \\mathbb { R } ^ { K }$ , the parameters of a parametric function model (see Figure 2 for some example curves from our basis functions)   \n3. $v \\colon \\mathcal { X } \\to \\mathbb { R } ^ { k }$ , the corresponding weights for each function in the model   \n4. $\\widehat { \\sigma ^ { 2 } }$ : $\\mathcal { X } \\to \\mathbb { R } ^ { + }$ , the observational noise for this hyperparameter configuration ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/99302961fa6a852f3b59d323516058817487587445569220835086049107ec27.jpg",
+        "image_caption": [
+            "Figure 4: Our neural network architecture to model learning curves. A common hidden layer is used to simultaneously model $\\hat { \\mu } _ { \\infty }$ , the parameters $\\Theta$ of the basis functions, their respective weights $\\pmb { w }$ , and the noise $\\widehat { \\sigma ^ { 2 } }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With these quantities, we can compute the likelihood in (3) which allows training the network. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Phrased differently, we use a neural network to predict the model parameters $\\Theta$ and weights $\\pmb { w }$ of our parametric functions, yielding the following form for our network’s mean predictions: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d37460c31a7b90daf52eafefe7d984c19bf4bc7886c169c28be4aed11ffd096d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { g } ( \\pmb { x } _ { i } , t _ { i } | W ) = \\hat { f } ( t _ { i } | \\Theta ( \\pmb { x } _ { i } , W ) , \\pmb { w } ( \\pmb { x } _ { i } , W ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A schematic of this is shown in Fig. 4. For training, we will use the same MCMC methods, namely SGLD and SGHMC mentioned above. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now empirically evaluate the predictive performance of Bayesian neural networks, with and without our special learning curve layer. For both networks, we used a 3-layer architecture with tanh activations and 64 units per layer. We also evaluate two different sampling methods for both types of networks: stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian MCMC (SGHMC), following the approach of Springenberg et al. (2016) to automatically adapt the noise estimate and the preconditioning of the gradients. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As baselines, we compare to other approaches suitable for this task. Besides the aforementioned work by Domhan et al. (2015), we also compare against random forests Breimann (2001) with empirical variance estimates (Hutter et al., 2014), a Gaussian process (GP) using the learning curve kernel from Swersky et al. (2014), another Bayesian neural network technique called probabilistic back propagation (PBP) (Hernández-Lobato and Adams, 2015), and the simple heuristic of using the last seen value (LastSeenValue) of each learning curve for extrapolation. The last model has been successfully used by Li et al. (2017) despite its simplicity. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/b05c7c6c26b75f30900e84c89472ea378803ec4a95d98ad870c9d2681a5043ab.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative comparison of the different models. The left panel shows the mean predictions of different methods on the CNN benchmark. All models observed the validation error of the first 12 epochs of the true learning curve (black). On the right, the posterior distributions over the value at 40 epochs is plotted. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            387,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For our empirical evaluation, we generated the following four datasets of learning curves, in each case sampling hyperparameter configurations at random from the hyperparameter spaces detailed in Table 2 in the appendix (see also Section D for some characteristic of these datasets): ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• CNN: We sampled 256 configurations of 5 different hyperparameters of a 3-layer convolutional neural network (CNN) and trained each of them for 40 epochs on the CIFAR10 (Krizhevsky, 2009) benchmark.   \nFCNet: We sampled 4096 configurations of 10 hyperparameters of a 2-layer feed forward neural network (FCNet) on MNIST (LeCun et al., 2001), with batch normalization, dropout and ReLU activation functions, annealing the learning rate over time according to a power function. We trained the neural network for 100 epochs.   \n• LR: We sampled 1024 configurations of the 4 hyperparameters of logistic regression (LR) and also trained it for 100 epochs on MNIST. VAE: We sampled 1024 configuration of the 4 hyperparameters of a variational autoencoder (VAE) (Kingma and Welling, 2014). We trained the VAE on MNIST, optimizing the approximation of the lower bound for 300 epochs. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": ".2 PREDICTING ASYMPTOTIC VALUES OF PARTIALLY OBSERVED CURVES",
+        "text_level": 1,
+        "bbox": [
+            184,
+            666,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first study the problem of predicting the asymptotic values of partially-observed learning curves tackled by Domhan et al. (2015). The LC-Extrapolation method by Domhan et al. (2015), the GP, and the last seen value work on individual learning curves and do not allow to model performance across hyperparameter configurations. Thus, we trained them separately on individual partial learning curves. The other models, including our Bayesian neural networks, on the other hand, can use training data from different hyperparameter configurations. Here, we used training data with the same number of epochs for every partial learning curve.1 ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The left panel of Figure 5 visualizes the extrapolation task, showing a learning curve from the CNN dataset and the prediction of the various models trained only using the first 12 of 40 epochs of the learning curve. The right panel shows the corresponding predictive distributions obtained with these models. LastSeenValue does not yield a distribution and uncertainties are not defined. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For a more quantitative evaluation, we used all models to predict the asymptotic value of all learning curves, evaluating predictions based on observing between $10 \\%$ and $90 \\%$ of the learning curves. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/cd56c586d0278dffe482a193a97da0db4a13d985a44aa50e24b585c0219f8060.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=2>CNNMSE·10² ALL</td><td rowspan=1 colspan=2>FCNetMSE-102 ALL</td><td rowspan=1 colspan=1>MSE·10²</td><td rowspan=1 colspan=1>LRALL</td><td rowspan=1 colspan=2>VAEMSE·104 ALL</td></tr><tr><td rowspan=1 colspan=1>SGLD</td><td rowspan=1 colspan=1>1.8±0.9</td><td rowspan=1 colspan=1>0.60±0.25</td><td rowspan=1 colspan=1>4.5±0.6</td><td rowspan=1 colspan=1>0.13±0.05</td><td rowspan=1 colspan=1>1.1 ± 0.3</td><td rowspan=1 colspan=1>0.77±0.08</td><td rowspan=1 colspan=1>4.7 ± 1.7</td><td rowspan=1 colspan=1>2.16 ±0.07</td></tr><tr><td rowspan=1 colspan=1>SGLD-LC</td><td rowspan=1 colspan=1>1.4 ± 0.9</td><td rowspan=1 colspan=1>1.09 ± 0.22</td><td rowspan=1 colspan=1>3.3 ±0.7</td><td rowspan=1 colspan=1>0.54± 0.06</td><td rowspan=1 colspan=1>1.3 ± 0.7</td><td rowspan=1 colspan=1>0.94 ±0.09</td><td rowspan=1 colspan=1>3.1 ± 1.1</td><td rowspan=1 colspan=1>1.55 ± 0.01</td></tr><tr><td rowspan=1 colspan=1>SGHMC</td><td rowspan=1 colspan=1>0.8±0.6</td><td rowspan=1 colspan=1>0.96± 0.15</td><td rowspan=1 colspan=1>1.9 ±0.3</td><td rowspan=1 colspan=1>0.50± 0.04</td><td rowspan=1 colspan=1>0.5±0.2</td><td rowspan=1 colspan=1>1.08±0.05</td><td rowspan=1 colspan=1>3.6± 1.5</td><td rowspan=1 colspan=1>2.34± 0.07</td></tr><tr><td rowspan=1 colspan=1>SGHMC-LC</td><td rowspan=1 colspan=1>0.7±0.6</td><td rowspan=1 colspan=1>1.15 ± 0.34</td><td rowspan=1 colspan=1>2.0±0.3</td><td rowspan=1 colspan=1>0.80±0.05</td><td rowspan=1 colspan=1>0.5±0.2</td><td rowspan=1 colspan=1>1.17 ± 0.08</td><td rowspan=1 colspan=1>1.9 ± 1.1</td><td rowspan=1 colspan=1>1.39 ± 0.65</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: In each column we report the mean squared error (MSE) and the average log-likelihood (ALL) of the 16 fold CV learning curve prediction for a neural network without learning curve prediction layer (SGLD and SGHMC) and with our new layer (SGLD-LC and SGHMC-LC). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows the mean squared error between the true asymptotic value and the models’ predictions (left) and the average log-likelihood of the true value given each model as a function of how much of the learning curves has been observed. Note that we removed some methods from the plots to avoid clutter; the same plots with all methods can be found in the supplementary material. We notice several patterns: ",
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+    {
+        "type": "text",
+        "text": "1. Throughout, our specialized network architecture performs better than the standard Bayesian neural networks, and SGHMC outperformed SGLD (see Appendix C).   \n2. PBP shows mixed results for the extrapolated mean, and does not provide reliable uncertainties (also see Appendix C). We hypothesize that this may be due to its linear activation functions.   \n3. If no uncertainties are required, LastSeenValue is hard to beat. This is because many configurations approach their final performance quite quickly.   \n4. The GP mean predictions are very competitive, but the average log-likelihood indicates overconfidence, especially for short learning curves. We assume that the prior assumption of an exponential function is not flexible enough in practice and that after observing some data points the Gaussian process becomes too certain of its predictions.   \n5. The random forest’s mean predictions almost match the quality of the GP. The uncertainty estimates are better, but still too aggressive when only a small fraction of the learning curve was observed. Random forests do not extrapolate and similar to LastSeenValue also achieve a very good mean prediction if the learning curve has almost converged.   \n6. Local models for single curves clearly outperform global models for almost complete learning curves. Global models, like our BNN approach, on the other hand, are trained on many configurations and need to generalize across these, yielding somewhat worse performance for this (arguably easy) task.2 ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 PREDICTING UNOBSERVED LEARNING CURVES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As mentioned before, training a joint model across hyperparameters and time steps allows us to make predictions for completely unobserved learning curves of new configurations. To estimate how well Bayesian neural networks perform in this task, we used the datasets from Section 3.1 and split all of them into 16 folds, allowing us to perform cross-validation of the predictive performance. For each fold, we trained all models on the full learning curves in the training set and let them predict the held-out learning curves. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows the mean squared error and the average log-likelihood (both computed for all points in each learning curve) across the 16 folds. We make two observations: firstly, both neural network architectures lead to a reasonable mean squared error and average log-likelihood, for both SGLD and SGHMC. Except on the VAE dataset our learning curve layer seems to improve mean squared error and average log likelihood. Secondly, SGHMC performed better than SGLD, with the latter resulting in predictions with too small variances. Figure 7 visualizes the results for our CNN dataset in more detail, showing that true and predicted accuracies correlate quite strongly. ",
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+    {
+        "type": "image",
+        "img_path": "images/6e4fa59fccc6c9bb52ff215e81ce2021db8ce82be943c99bcc576401e2759c52.jpg",
+        "image_caption": [
+            "Figure 6: Assessment of the predictive quality based on partially observed learning curves on all 4 benchmarks. The panels on the left show the mean squared error of the predicted asymptotic value (y-axis) after observing all learning curves up to a given fraction of maximum number of epochs (x-axis). The panels on the right show the average log-likelihood based on the predictive mean and variance of the asymptotic value. Note that LastSeenValue does not provide a predictive variance. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/d32af6b872e17abef5b865d8314e51b5dd5b5d4b6363445170512d6be7971da8.jpg",
+        "image_caption": [
+            "Figure 7: On the horizontal axis, we plot the true value and on the vertical axis the predicted values. Each point is colored by its log-likelihood (the brighter the higher). Our learning curve BNN trained with SGHMC leads to the best mean predictions and assigns the highest likelihood to the test points. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3.4 MODEL-BASED HYPERBAND ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we show how our model can be used to improve hyperparameter optimization of iterative machine learning algorithms. For this, we extended the multi-armed bandit strategy Hyperband (Li et al., 2017), which in each iteration $i$ first samples $N _ { i }$ hyperparameter configurations $\\dot { C } \\dot { = } \\{ { \\pmb x } _ { 1 } , \\ldots , { \\pmb x } _ { N _ { i } } \\}$ and then uses successive halving (Jamieson and Talwalkar, 2016) to iteratively discard poorly-performing configurations from $C$ . While the original Hyperband method samples configurations $C$ from a uniform distribution $\\mathcal { U }$ over hyperparameter configurations, our extension instead samples them based on our model, with all other parts remaining unchanged. More precisely, we sample $M _ { i } \\ > > \\ N _ { i }$ configurations uniformly, $\\hat { C } = \\{ \\pmb { x } _ { 1 } , \\dotsc , \\pmb { x } _ { M _ { i } } \\} \\sim \\mathcal { U }$ , and pick the $N _ { i }$ configurations with the highest predicted asymptotic mean $a ( \\pmb { x } ) = \\hat { \\mu } ( \\pmb { x } , t = 1 )$ or upper confidence value $a ( { \\pmb x } ) = \\hat { \\mu } ( { \\pmb x } , t = 1 ) \\hat { \\bf \\mu } + \\hat { \\sigma } ( \\hat { \\bf \\ x } , t = 1 )$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For a thorough empirical evaluation and to reduce computational requirements we reused the data from Section 3.1 to construct a surrogate benchmark based on a random forest (see Appendix E for more details). After each iteration we report the final performance of the best observed configuration so far, along with the wall clock time that would have been needed for optimizing the true objective function. Reporting the (predicted) wall clock time (rather than the number of iterations) also takes our optimizer’s overhead into account. This is important since one may worry that this overhead may be substantial due to our use of a model; however, as we show it does not harm performance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/8c9350f5faf2edc1a98eff87c848e6b701aa9a5d8f1cc9a9753b8328db64be00.jpg",
+        "image_caption": [
+            "Figure 8: Comparison of Hyperband, Hyperband with our model, and standard Bayesian optimization on the CNN benchmark. Hyperband finds a good configuration faster than standard Bayesian optimization, but it only approaches the global optimum quickly when extended with our model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            611,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 8 shows the immediate regret on the CNN benchmark as a function of wallclock time, for three optimizers: Hyperband, our model-based extension of Hyperband, as well as standard Bayesian optimization with a Gaussian process (a comparison to additional baselines can be found in Appendix E). Standard Bayesian optimization does not make use of learning curves and thus needs to evaluate each configuration for the full amount of epochs. Nevertheless, since training one configuration to the end takes less time than one round of successive halving, Bayesian optimization produces its first results earlier than Hyperband. In accordance with results by Li et al. (2017), in this experiment Hyperband found a configuration with good performance faster than standard Bayesian optimization, but its random sampling did not suffice to quickly approach the best configuration; given enough time Bayesian optimization performed better. However, extended by our model, Hyperband both found a good configuration fast and approached the global optimum fastest. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We would like to emphasize that our model-based extension of Hyperband is not limited to our particular Bayesian neural networks as the underlying model. Since Hyperband stops the majority of the configurations very early, the setting is quite similar to that of very short partially observed learning curves in Section 3.2, with the differences that some configurations have been evaluated for longer and that the model now has to generalize to unseen configurations. For example, for the CNN and LR benchmarks, the experiments in Section 3.2 showed that random forests achieve strong average log-likelihoods for very short partial learning curves, and we would therefore also expect them to work well when combined with Hyperband. However, given our model’s more robust likelihoods and mean squared error values we believe it to be less sensitive to the underlying data. Additionally, we can incorporate more prior knowledge about the general shape of the curves into our model, something that is not easily done for many other model classes. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            438,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We studied Bayesian neural networks for modelling the learning curves of iterative machine learning methods, such as stochastic gradient descent for convolutional neural networks. Based on the parametric learning curve models of Domhan et al. (2015), we also developed a specialized neural network architecture with a learning curve layer that improves learning curve predictions. In future work, we aim to study recurrent neural networks for predicting learning curves and will extend Bayesian optimization methods with Bayesian neural networks (Springenberg et al., 2016) based on our learning curve models. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            585,
+            318,
+            598
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work has partly been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant no. 716721, by the European Commission under grant no. H2020-ICT-645403-ROBDREAM, and by the German Research Foundation (DFG) under Priority Programme Autonomous Learning (SPP 1527, grant HU 1900/3-1). ",
+        "bbox": [
+            176,
+            608,
+            825,
+            664
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            686,
+            285,
+            702
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "E. Brochu, V. Cora, and N. de Freitas. A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. CoRR, 2010.   \nJ. Snoek, H. Larochelle, and R. P. Adams. Practical Bayesian optimization of machine learning algorithms. In Proc. of NIPS’12, 2012.   \nB. Shahriari, K. Swersky, Z. Wang, R. Adams, and N. de Freitas. Taking the human out of the loop: A Review of Bayesian Optimization. Proc. of the IEEE, (1), 12/2015 2016.   \nK. Swersky, J. Snoek, and R. Adams. Freeze-thaw Bayesian optimization. CoRR, 2014.   \nT. Domhan, J. T. Springenberg, and F. Hutter. Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. In Q. Yang and M. Wooldridge, editors, Proc. of IJCAI’15, 2015.   \nL. Li, K. Jamieson, G. DeSalvo, A. Rostamizadeh, and A. Talwalkar. Hyperband: Bandit-based configuration evaluation for hyperparameter optimization. In Proc. of ICLR’17, 2017.   \nA. Klein, S. Falkner, S. Bartels, P. Hennig, and F. Hutter. Fast Bayesian optimization of machine learning hyperparameters on large datasets. In Proc. of AISTATS’17, 2017.   \nK. Swersky, J. Snoek, and R. Adams. Multi-task Bayesian optimization. In Proc. of NIPS’13, 2013.   \nM. Feurer, T. Springenberg, and F. Hutter. Initializing Bayesian hyperparameter optimization via meta-learning. In Proc. of AAAI’15, 2015.   \nF. Hutter, H. Hoos, K. Leyton-Brown, and K. Murphy. Time-bounded sequential parameter optimization.   \nF. Hutter, H. Hoos, and K. Leyton-Brown. Sequential model-based optimization for general algorithm configuration. In LION’11, 2011.   \nJ. Snoek, O. Rippel, K. Swersky, R. Kiros, N. Satish, N. Sundaram, M. M. A. Patwary, Prabhat, and R. P. Adams. Scalable Bayesian optimization using deep neural networks. In Proc. of ICML’15, 2015.   \nJ. Springenberg, A. Klein, S. Falkner, and F. Hutter. Bayesian optimization with robust Bayesian neural networks. In Proc. of NIPS’16, 2016.   \nJ. Hernández-Lobato and R. Adams. Probabilistic backpropagation for scalable learning of Bayesian neural networks. In Proc. of ICML’15, 2015.   \nM. Welling and Y. Teh. Bayesian learning via stochastic gradient Langevin dynamics. In Proc. of ICML’11, 2011.   \nT. Chen, E.B. Fox, and C. Guestrin. Stochastic gradient Hamiltonian Monte Carlo. In Proc. of ICML’14, 2014.   \nD. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman. emcee $:$ The MCMC Hammer. PASP, 2013.   \nL. Breimann. Random forests. MLJ, 2001.   \nF. Hutter, L. Xu, H. Hoos, and K. Leyton-Brown. Algorithm runtime prediction: Methods and evaluation. AIJ, 2014.   \nA. Krizhevsky. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.   \nY. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. In S. Haykin and B. Kosko, editors, Intelligent Signal Processing. IEEE Press, 2001. URL http://www.iro.umontreal.ca/\\~lisa/pointeurs/lecun-01a.pdf.   \nD. Kingma and M. Welling. Auto-encoding variational bayes. In Proc. of ICLR’14, 2014.   \nK. Jamieson and A. Talwalkar. Non-stochastic best arm identification and hyperparameter optimization. In Proc. of AISTATS’16, 2016.   \nK. Eggensperger, F. Hutter, H.H. Hoos, and K. Leyton-Brown. Efficient benchmarking of hyperparameter optimizers via surrogates. In Proc. of AAAI’15, 2015. ",
+        "bbox": [
+            173,
+            712,
+            826,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            101,
+            828,
+            547
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A EXPERIMENTAL SETUP – DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            496,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/7e449a0a88731e8979f2396eba3d048c2211df1dbecd70063a1c0d8483b7391b.jpg",
+        "table_caption": [
+            "Table 2 shows the hyperparameters of our 4 benchmarks described in Section 3.1 and their ranges. ",
+            "Table 2: Hyperparameter configuration space of the four different iterative methods. For the FCNet we decayed the learning rate by a $\\alpha _ { d e c a y } = ( 1 + \\gamma * t ) ^ { - \\kappa }$ and also sampled different values for $\\gamma$ and $\\kappa$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Name</td><td>Range</td><td>log scale</td></tr><tr><td rowspan=\"5\">CNN</td><td>batch size</td><td>[32, 512]</td><td>1</td></tr><tr><td>number of units layer 1</td><td>[24,210]</td><td>√</td></tr><tr><td>number of units layer 2</td><td>[24,210]</td><td>√</td></tr><tr><td>number of units layer 3</td><td>[24,210]</td><td>&lt;</td></tr><tr><td>learning rate</td><td>[10-6,10-0]</td><td>√</td></tr><tr><td rowspan=\"9\">FCNet</td><td>initial learning rate</td><td>[10-6,10°]</td><td>√</td></tr><tr><td>L2 regularization</td><td>[10-8,10-1]</td><td></td></tr><tr><td>batch size</td><td>[32,512]</td><td>√</td></tr><tr><td>Y</td><td>[-3,-1]</td><td></td></tr><tr><td>K</td><td>[0,1]</td><td></td></tr><tr><td>momentum</td><td>[0.3,0.999]</td><td></td></tr><tr><td>number units 1</td><td>[25,122]</td><td></td></tr><tr><td>number units 2</td><td>[25,212]</td><td></td></tr><tr><td>dropout rate layer 1</td><td>[0.0,0.99]</td><td></td></tr><tr><td rowspan=\"4\">LR</td><td>dropout rate layer 2</td><td>[0.0,0.99]</td><td></td></tr><tr><td>learning rate</td><td>[10-6,100]</td><td>√</td></tr><tr><td>L2</td><td>[0.0,1.0]</td><td></td></tr><tr><td>batch size</td><td>[20,2000]</td><td></td></tr><tr><td rowspan=\"5\">VAE</td><td>dropout rate on inputs L</td><td>[0.0,0.75]</td><td></td></tr><tr><td>number of hidden units</td><td>[1,3]</td><td></td></tr><tr><td>batch size</td><td>[32,2048] [16,512]</td><td></td></tr><tr><td>z dimension</td><td></td><td></td></tr><tr><td></td><td>[2,200]</td><td></td></tr></table>",
+        "bbox": [
+            248,
+            169,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B DESCRIPTION OF THE BASIS FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            630,
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+            647
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To reduce complexity, we used a subset of the basis function from Domhan et al. (2015) which we found to be sufficient for learning curve prediction. We adapted these functions to model the residual between the asymptotic value $y _ { \\infty }$ and we scaled the parameters $\\Theta$ to be in $[ 0 , 1 ]$ . Table 3 shows the exact equations we used. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C PREDICTING ASYMPTOTIC VALUES OF PARTIALLY OBSERVED CURVES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            747,
+            785,
+            762
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 9 shows the mean squared error and the average log-likelihood of predicting the asymptotic value after observing different amounts of the learning curve for all methods. See Section 3.2 in the main text for more details. ",
+        "bbox": [
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+            780,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D DATASET CHARACTERISTICS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            848,
+            447,
+            863
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 shows the distributions over runtimes for all random configurations of different benchmarks. As it can be seen there is a high variance of the runtime between different configurations for all benchmarks and some configurations need order of magnitudes longer than others. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/b9eb85e592aeb151f0d1358c063601fdde2beaf21d7bc5491a06006ea960ff12.jpg",
+        "image_caption": [
+            "Figure 9: Assessment of the predictive quality based on partially observed learning curves on all 4 benchmarks. The panels on the left show the mean squared error of the predicted asymptotic value (y-axis) after observing all learning curves up to a given fraction of maximum number of epochs (x-axis). The panels on the right show the average log-likelihood based on the predictive mean and variance of the asymptotic value. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            85,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/ed990c634f31461f5acb942a37b16a7d7e768b9ce90c24e2ecd39c8d40276f59.jpg",
+        "table_caption": [
+            "Table 3: The formulas of our 5 basis functions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Formula</td></tr><tr><td rowspan=1 colspan=1>vapor pressure</td><td rowspan=1 colspan=1>Φ(t,a,b,c)=e-a-0.1-b.t-11c-lg(t)e--01b</td></tr><tr><td rowspan=1 colspan=1>pow</td><td rowspan=1 colspan=1>(t,a,b)=a·(tb-1)</td></tr><tr><td rowspan=1 colspan=1>log power</td><td rowspan=1 colspan=1>Φ(ta,b,c)=2·[()-(((t)</td></tr><tr><td rowspan=1 colspan=1>exponential</td><td rowspan=1 colspan=1>Φ(t,a,b)=a·[e-10-b -e-10·b·t]</td></tr><tr><td rowspan=1 colspan=1>hill-3</td><td rowspan=1 colspan=1>(t,a,b,c)=a.[（cb.t-b+1)-1-(cb+1)-1]</td></tr></table>",
+        "bbox": [
+            214,
+            101,
+            784,
+            315
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 11 shows the empirical cumulative distribution of the asymptotic performance of random configurations. These plots give an intuition about the difficulty of a benchmark as they show how many random configurations one has to sample in order to achieve a good performance. ",
+        "bbox": [
+            174,
+            366,
+            823,
+            409
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E OPTIMIZATION ON SURROGATE BENCHMARKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            429,
+            593,
+            445
+        ],
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+    {
+        "type": "text",
+        "text": "We follow the approach by Eggensperger et al. (2015) and used the generated datasets from Section 3.1 to build surrogates of the objective functions. This allows us to compare different optimizers on this benchmark very efficiently while (approximately) preserving the characteristics of the underlying benchmark. Using surrogates instead of optimizing the real benchmarks allows us to carry out a more thorough empirical evaluation (since single function evaluations are cheap and we therefore can afford more and longer optimization runs). We used random forests as surrogates following Eggensperger et al. (2015) since they do not introduce a bias for our approach (in contrast to, e.g., surrogates based on standard feed forward neural networks trained with stochastic gradient descent ). For each benchmark, we used all configurations to train a random forest predicting the validation accuracy of a configuration at a certain time step as well as the wall clock time for its evaluation. ",
+        "bbox": [
+            174,
+            460,
+            825,
+            599
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "After each round of successive halving, we return the current best observed configuration and its asymptotic performance. For each function evaluation, we predict the accuracy, and the runtime. By adding the optimization overhead, we can predict the wallclock time necessary to optimize the real objective function. Reporting the wallclock time rather than the number of iterations also takes the additional overhead from running our method into account. Furthermore, we argue that the wallclock time is more interesting in practice. ",
+        "bbox": [
+            174,
+            606,
+            825,
+            690
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As baselines, we compare to Gaussian process based Bayesian optimization with the upper confidence bound acquisition function (GP-BO-UCB) and the (log) expected improvement acquisition function (GP-BO-Log-EI), as well as Bayesian optimization with Bayesian neural networks (BOHAMIANN) (Springenberg et al., 2016). As an additional baseline, we use standard Bayesian neural networks (SGHMC-BNN) inside Hyperband instead of our specialized neural network architecture (SGHMCLCNet). We compare our model-based version of Hyperband against all these baselines with sampling from the asymptotic mean (SGHMC-LCNet-mean-sampling) and the asymptotic upper confidence bound (SGHMC-LCNet-ucb-sampling). Figure 12 shows that our specialized neural network architecture helps to speed up the convergence of Hyperband on the CNN and the FCNet benchmark. For the LR benchmarks our model still improves over Hyperband but does not perform better than Bayesian optimization. Hyperband as well as our method find the optimum already after the first round of successive halving for the VAE benchmark, which leads to an regret of 0. ",
+        "bbox": [
+            173,
+            696,
+            825,
+            863
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/69e670c6d4b34b667a4f1d53ed7b0093390e81320c8387179f73cd8fbade8c21.jpg",
+        "image_caption": [
+            "Figure 10: Distributions over runtimes for different random configurations for the CNN, FCNet, LR and VAE benchmark described in Section 3.One can see that all distributions are long tailed and that especially on the FCNet benchmark some configuration need order of magnitudes longer than others. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            106,
+            813,
+            458
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/287fb6fd47d5d135b71d04a2ea9a7fe7652e8baa67ff6bc8d75524fa463094f1.jpg",
+        "image_caption": [
+            "Figure 11: Empirical cumulative distributions for the CNN, FCNet, LR and VAE benchmark. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            540,
+            812,
+            882
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/c028c1defddf9f22f04dff6d49c0efd0fe68546444bd755785852f09afbbf434.jpg",
+        "image_caption": [
+            "Figure 12: Different optimizers on a random forests based suggorate of all collected learning curves on different benchmarks. Note how sampling from the model improves Hyperband’s performance by converging to the optimum faster. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+            617
+        ],
+        "page_idx": 15
+    }
+]

parse/train/r1Chut9xl/r1Chut9xl.md

@@ -0,0 +1,342 @@
+# INFERENCE & INTROSPECTION IN DEEP GENERATIVE MODELS OF SPARSE DATA
+
+Rahul G. Krishnan New York University rahul@cs.nyu.edu
+
+Matthew Hoffman Adobe Research matthoffm@adobe.com
+
+# ABSTRACT
+
+Deep generative models such as deep latent Gaussian models (DLGMs) are powerful and popular density estimators. However, they have been applied almost exclusively to dense data such as images; DLGMs are rarely applied to sparse, high-dimensional integer data such as word counts or product ratings. One reason is that the standard training procedures find poor local optima when applied to such data. We propose two techniques that alleviate this problem, significantly improving our ability to fit DLGMs to sparse, high-dimensional data. Having fit these models, we are faced with another challenge: how to use and interpret the representation that we have learned? To that end, we propose a method that extracts distributed representations of features via a simple linearization of the model.
+
+# 1 INTRODUCTION
+
+Deep latent Gaussian models (DLGMs, a.k.a. variational autoencoders; Rezende et al., 2014; Kingma et al., 2014) have led a resurgence in the use of deep generative models for density estimation. DLGMs assume that observed vectors $x$ are generated by applying a nonlinear transformation (defined by a neural network with parameters $\theta$ ) to a vector of Gaussian random variables $z$ .
+
+Learning in DLGMs proceeds by approximately maximizing the average marginal likelihood $p ( x ) \equiv$ $\begin{array} { r } { \int _ { z } p ( z ) \bar { p ( x | z ) } d z } \end{array}$ of the observations $x$ . Computing the true marginal likelihood is intractable, so we resort to variational expectation-maximization (Bishop, 2006), an approximation to maximumlikelihood estimation. To learn the parameters $\theta$ of the generative model, the procedure needs to find a distribution $q ( z | x )$ that approximates the posterior distribution $p ( z | x )$ of the latent vector $z$ given the observations $x$ . In the past, such $q$ distributions were fit using iterative optimization procedures (e.g., Hoffman et al., 2013). But Rezende et al. (2014) and Kingma et al. (2014) showed that $q ( z | x )$ can be parameterized by a feedforward “inference network” with parameters $\phi$ , speeding up learning. This inference network is trained jointly with the generative model; as training proceeds, the inference network learns to approximate posterior inference on the generative model, and the generative model improves itself using the output of the inference network.
+
+Embedded within this procedure, however, lies a potential problem: both the inference network and the generative model are initialized randomly. Early on in learning, the inference network’s $q ( z | x )$ distributions will be poor approximations to the true posterior $p ( z | x )$ , and the gradients used to update the parameters of the generative model will therefore be poor approximations to the gradients of the true log-likelihood $\log p ( x )$ . Previous stochastic variational inference methods (Hoffman et al., 2013) were slower, but suffered less from this problem since for every data-point, a set of variational parameters was optimized within the inner loop of learning. In this work, we investigate blending the two methodologies for learning models of sparse data. In particular, we use the parameters predicted by the inference network as an initialization and optimize them further during learning. When modeling high-dimensional sparse data, we show that updating the local variational parameters yields generative models with better held-out likelihood, particularly for deeper generative models.
+
+What purpose is served by fitting bigger, deeper, more powerful generative models? Breiman (2001) argues that statistical discriminative modeling falls into two schools of thought: the data modeling culture and the algorithmic modeling culture. The former advocates the use of predictive models that assume interpretable, mechanistic processes while the latter advocates the use of black box techniques with an emphasis on prediction accuracy. Breiman’s arguments also ring true about the divide between deep generative models with complex conditional distributions and simpler, more interpretable statistical models. Consider a classic model such as Latent Dirichlet Allocation (Blei et al., 2003). It is outperformed in held-out likelihood (Miao et al., 2016) by deeper generative models and assumes a simple probabilistic process for data generation that is unlikely to hold in reality. Yet, its generative semantics lend it a distinct advantage: interpretability. The word-topic matrix in the model allows practitioners to read off what the model has learned about the data. Is there a natural way to interpret the generative model when the conditional distributions are parameterized by a deep neural network?
+
+Our second contribution is to introduce a simple, easy to implement method to interpret what is being learned by generative models such as DLGMs whose conditional probabilities are parameterized by deep neural networks. Our hope is to narrow the perceived gulf between a complex generative model’s representational power and its interpretability. We use the Jacobian of the conditional distribution with respect to latent variables in the Bayesian network to form embeddings (or Jacobian vectors) of the observations. We investigate the properties of the Jacobian vectors obtained from deeper, more non-linear generative models.
+
+# 2 BACKGROUND
+
+Generative Model: We consider learning in generative models of the form shown in Figure 1. We observe a set of $D$ word count vectors $x _ { 1 : D }$ , where $x _ { d v }$ denotes the number of times that word index $v \in \{ 1 , \ldots , V \}$ appears in document $d$ . We assume we are given the total number of words per document $\begin{array} { r } { N _ { d } \equiv \sum _ { v } x _ { d v } } \end{array}$ , and that $x _ { d }$ was generated via the following generative process:
+
+$$
+\mathbf { \Psi } _ { d } \sim { \mathcal { N } } ( 0 , I ) ; \ { \gamma } ( z _ { d } ) \equiv \mathrm { M L P } ( z _ { d } ; \theta ) ; \ \mu ( z _ { d } ) \equiv \frac { \exp \{ \gamma ( z _ { d } ) \} } { \sum _ { v } \exp \{ \gamma ( z _ { d } ) _ { v } \} } ; \ x _ { d } \sim \mathrm { M u l t i n o m i a l } ( \mu ( z _ { d } ) , N _ { d } ) .
+$$
+
+That is, we draw a Gaussian random vector, pass it through a multilayer perceptron (MLP) with parameters $\theta$ , pass the resulting vector through the softmax (a.k.a. multinomial logistic) function, and sample $N _ { d }$ times from the resulting distribution over the vocabulary.1
+
+Variational Learning: For ease of exposition notation we drop the subscript on $x _ { d }$ to form $x$ referring to a single data point. We need to approximate the intractable posterior distribution $p ( z | x )$ during learning. Using the well-known variational principle, we can obtain the lower bound on the log marginal likelihood of the data (or $\mathcal { L } ( x ; \theta , \phi ) )$ in Eq. 2. where the inequality is by Jensen’s inequality.
+
+$$
+\begin{array} { r } { \log p ( x ; \theta ) \geq \underset { q _ { \phi } ( z | x ) } { \mathbb { E } } \left[ \log p _ { \theta } ( x | z ) ) \right] - \mathrm { K L } \big ( q _ { \phi } ( z | x ) | | p ( z ) \big ) = \mathcal { L } ( x ; \theta , \phi ) , } \end{array}
+$$
+
+We leverage an inference network or recognition network (Hinton et al., 1995), a neural network which approximates the intractable posterior, during learning. This is a parametric conditional distribution that is optimized to perform inference. Kingma $\&$ Welling (2014); Rezende et al. (2014) use a neural net (with parameters $\phi$ ) to parameterize $q _ { \phi } ( z | x )$ . The challenge in the resulting optimization problem is that the lower bound (2) includes an expectation w.r.t. $q _ { \phi } ( z | x )$ , which implicitly depends on the network parameters $\phi$ . This difficulty is overcome by using stochastic backpropagation.
+
+With a normal distribution as our variational approximation we have that $q _ { \phi } ( z | x ) \sim$ $\mathcal { N } \left( \mu _ { \phi } ( x ) , \Sigma _ { \phi } ( x ) \right)$ . $\mu _ { \phi } ( x ) , \Sigma _ { \phi } ( x )$ are functions of the observation $x$ , and we denote by $\psi ( x ) : = { \frac { } { } }$ $\{ \mu _ { \phi } ( x ) , \Sigma _ { \phi } ( x ) \}$ the local variational parameters predicted by the inference network. A simple transformation allows one to obtain unbiased Monte Carlo estimates of the gradients of $\mathbb { E } _ { q _ { \phi } ( z | x ) } [ \log p _ { \theta } ( x | z ) ) ]$ with respect to $\phi$ . If we assume the prior $p ( z )$ is also normally distributed, the KL and its gradients may be obtained analytically. Throughout this paper we will use $\theta$ to denote the parameters of the generative model, and $\phi$ to denote the parameters of the inference network.
+
+# 3 METHODOLOGY
+
+Inference with Global Information: Sparse data typically exhibits long tails and learning in the presence of rare features is challenging. Inference networks learn to regress to the optimal posterior parameters for every data point and global information about the relative frequencies of the individual features in the training distribution may present valuable information during learning.
+
+The simplest way to incorporate first order statistics across the training data into the inferential process is to condition on tf-idf (Baeza-Yates et al., 1999) features instead of the raw-counts. tf-idf is one of the most widely used techniques in information retrieval. In the context of building bag-of-words representations for documents, tf-idf re-weight features to increase the influence of rarer words while decreasing the influence of common words appearing in all documents. The tf-idf-transformed word-count vector is $\begin{array} { r } { \tilde { x } _ { d v } \equiv x _ { d v } \log \frac { D } { \sum _ { d ^ { \prime } } \operatorname* { m i n } \{ x _ { d ^ { \prime } v } , 1 \} } } \end{array}$ . After applying transform, the resulting vector $\tilde { x }$ is normalized by its L2 norm. It’s worthwhile to note that leveraging first-order statistics for inference is difficult in the traditional paradigm of tracking variational parameters for each data point but is easy with inference networks.
+
+Optimizing Local Variational Parameters: The inference network initially comprises a randomly initialized neural network. The predictions of the inference network early in optimization are suboptimal variational parameters used to derive gradients of the parameters of the generative model. This induces noise and bias to the gradients used to update the parameters of the generative model; this noise and bias may push the generative model towards a poor local optimum. Previous work has suggested that deep neural networks (which form the conditional probability distributions $p _ { \theta } ( x | z ) )$ are sensitive to initialization (Glorot & Bengio, 2010; Larochelle et al., 2009).
+
+To avoid these issues, we only use the local variational parameters $\psi ( x )$ predicted by the inference network to initialize an iterative optimizer that maximizes the ELBO with respect to $\psi$ ; we use the optimized variational parameters $\hat { \psi ( x ) }$ to derive gradients for the generative model. We then train the inference network using stochastic backpropagation and gradient descent, holding the parameters of the generative model $\theta$ fixed. Our procedure is detailed in Algorithm 1.
+
+Algorithm 1 Pseudocode for Learning: We evaluate expectations in $\mathcal { L } ( x )$ (see Eq. 2) using a single sample from the variational distribution and aggregate gradients across mini-batches. $M = 1$ corresponds to performing no additional optimization to the variational parameters We update $\theta , \psi ( x ) , \phi$ using stochastic gradient descent with adaptive learning rates $\eta _ { \theta } , \eta _ { \psi ( x ) } , \eta _ { \phi }$ obtained via ADAM (Kingma & Ba, 2015)
+
+Inputs: Dataset $\mathcal { D } : = [ x _ { 1 } , \dots , x _ { D } ] ,$ , Inference Model: $q _ { \phi } ( z | x )$ , Generative Model: $p _ { \theta } ( x | z ) , p ( z )$   
+while notConverged() do 1. Sample datapoint: $x \sim \mathcal { D }$ 2. Estimate local variational parameters $\psi ( x ) _ { 1 }$ using $q _ { \phi } ( z | x )$ 3. Estimate $\begin{array} { r } { \psi ( x ) _ { M } \approx \hat { \psi ( x ) } = \arg \operatorname* { m a x } _ { \psi ( x ) } \mathcal { L } ( x ; \theta ; \psi ( x ) ) } \end{array}$ via SGD as: $\begin{array} { r } { m = 1 , \dots , M , \psi ( x ) _ { m + 1 } = \psi ( x ) _ { m } + \eta _ { \psi ( x ) } \frac { \partial \mathcal { L } ( x ; \theta , \psi ( x ) _ { m } ) } { \partial \psi ( x ) _ { m } } } \end{array}$ 4. Update $\theta$ as: $\theta \gets \theta + \eta _ { \theta } \nabla _ { \theta } \mathcal { L } ( x ; \theta , \psi ( x ) _ { M } )$ 5. Update $\phi$ as: $\phi  \phi + \eta _ { \phi } \nabla _ { \phi } \mathcal { L } ( x ; \theta , \psi ( x ) )$   
+end while
+
+Introspection: Linear models are inherently interpretable. Consider linear regression, factor analysis (Spearman, 1904), and latent Dirichlet allocation (LDA; Blei et al., 2003), which (standardizing notation) assume the following relationships:
+
+Regression: $\mathbb { E } [ y | x ] = W x + b$ ; Factor Analysis: $x \sim \mathcal { N } ( 0 , I )$ ; $\mathbb { E } [ y | x ] = W x + b ;$ Latent Dirichlet Allocation: $\begin{array} { r } { x \sim \mathrm { D i r i c h l e t } ( \alpha ) ; \quad \mathbb { E } [ y | x ] = W x } \end{array}$ .
+
+In each case, we need only inspect the parameter matrix $W$ to answer the question “what happens to $y$ if we increase $x _ { k }$ a little?” The answer is clear— $y$ moves in the direction of the $k$ th row of $W$ We can ask this question differently and get the same answer: “what is the derivative ∂E[y|x] ?” The answer is simply the parameter matrix $W$ .
+
+For models as in $\mathrm { F i g ~ 1 }$ , the variability in the training data is assumed to be due to the single latent state $z$ . The relationship between latent variables $z$ and observations $x$ cannot be quickly read off of the parameters $\theta$ . But we can still ask what happens if we perturb $z$ by some small $d z$ —this is simply the directional derivative ∂E[x|z]∂z dz. We can interpret this Jacobian matrix in much the same way we would a factor loading matrix, with two main differences. First, the Jacobian matrix ∂E[x|z]∂z varies with z—the interpretation of z may change significantly depending on context. Second, DLGMs exhibit rotational symmetry—the prior on $z$ is rotationally symmetric, and the MLP can apply arbitrary rotations to $z$ before applying any nonlinearities, so a priori there is no “natural” set of basis vectors for $z$ . For a given Jacobian matrix, however, we can find the most significant directions via a singular value decomposition (SVD).
+
+Jacobian Vectors: We present our method to generate embeddings from Bayesian networks of the form Figure 1. We consider three variants of Jacobian embedding vectors, based on the unnormalized potentials from the MLP, logarithmic probabilities, and linear probabilities respectively:
+
+$$
+\mathcal { I } ( z ) ^ { \mathrm { p o t } } = \frac { \partial \gamma ( z ) } { \partial z } \qquad \mathcal { I } ( z ) ^ { \mathrm { l o g } } = \frac { \partial \log \mu ( z ) } { \partial z } \qquad \mathcal { I } ( z ) ^ { \mathrm { p r o b } } = \frac { \partial \mu ( z ) } { \partial z }
+$$
+
+For any $z$ , $\{ \mathcal { I } ( z ) ^ { \mathrm { l o g } } , \mathcal { I } ( z ) ^ { \mathrm { p o t } } , \mathcal { I } ( z ) ^ { \mathrm { p r o b } } \} \in \mathbb { R } ^ { V \times K }$ where $K$ is the latent dimension and $V$ is the dimensionality of the observations. It is this matrix that we use to form embeddings. We denote by $u _ { i }$ the Jacobian vector obtained from the row of the Jacobian matrix. When not referring to a particular variant, we use $\mathcal { I } ( z )$ to denote the Jacobian matrix. $\mathcal { I } ( z )$ is a function of $z$ leaving open the choice of where to evaluate this function. The semantics of our generative model suggest a natural choice: $\mathcal { T } _ { \mathrm { m e a n } } : = \mathbb { E } _ { p ( z ) } [ \mathcal { T } ( z ) ]$ . This set of embeddings captures the variation in the output distribution with respect to the latent state across the prior distribution of the generative model. Additionally, one may also evaluate the Jacobian at the approximate posterior corresponding to an observation $x$ . We study how this may be used to obtain contextual word-vectors.
+
+In frameworks that support automatic differentiation (e.g., Theano; Theano Development Team, 2016), $\mathcal { I } ( z )$ is readily available and we estimate $\mathcal { I } _ { \bf m e a n }$ via Monte-Carlo sampling from the prior.
+
+Deriving Jacobian Vectors: For simplicity, we derive the functional form of the Jacobian in a linear model i.e where $\gamma ( z _ { d } ) = W z _ { d }$ (c.f Eq 1). We drop the subscript $d$ and denote by $\gamma _ { i } ( z )$ , the ith element of the vector $\gamma ( z )$ .
+
+$$
+p ( x _ { i } = 1 | z ) = { \frac { \exp ( \gamma _ { i } ( z ) ) } { \sum _ { j } \exp ( \gamma _ { j } ( z ) ) } } \qquad { \mathrm { a n d } } \quad \gamma _ { i } ( z ) = w _ { i } ^ { T } z
+$$
+
+For linear models, $\nabla _ { z } \gamma _ { i } ( z ) = w _ { i }$ directly corresponds to $\mathcal { I } ( z ) ^ { \mathrm { p o t } }$ . Noting that $\nabla _ { z } \exp ( \gamma _ { i } ( z ) ) =$ $\exp ( \gamma _ { i } ( z ) ) \nabla _ { z } \gamma _ { i } ( z )$ and $\begin{array} { r } { \dot { \nabla } _ { z } \sum _ { j } \exp ( \gamma _ { j } ( z ) ) = \sum _ { j } \exp ( \gamma _ { j } ( z ) ) \dot { \nabla } _ { z } \gamma _ { j } ( z ) } \end{array}$ , we estimate $\mathcal { I } ( z ) ^ { \mathrm { p r o b } }$ as:
+
+$$
+\begin{array} { l l l } { \displaystyle \nabla _ { z } p ( x _ { i } = 1 | z ) = \nabla _ { z } \frac { \exp ( \gamma _ { i } ( z ) ) } { \sum _ { j } \exp ( \gamma _ { j } ( z ) ) } = \frac { \sum _ { j } \exp ( \gamma _ { j } ( z ) ) \nabla _ { z } \exp ( \gamma _ { i } ( z ) ) - \exp ( \gamma _ { i } ( z ) ) \nabla _ { z } \sum _ { j } \exp ( \gamma _ { j } ( z ) ) } { ( \sum _ { j } \exp ( \gamma _ { j } ( z ) ) ) ^ { 2 } } } \\ { \displaystyle } & { \qquad = \frac { \sum _ { j } \exp ( \gamma _ { j } ( z ) ) \exp ( \gamma _ { i } ( z ) ) w _ { i } - \exp ( \gamma _ { i } ( z ) ) \sum _ { j } \exp ( \gamma _ { j } ( z ) ) w _ { j } } { ( \sum _ { j } \exp ( \gamma _ { j } ( z ) ) ) ^ { 2 } } } \\ { \displaystyle } & { \qquad = p ( x _ { i } = 1 | z ) w _ { i } - p ( x _ { i } = 1 | z ) \sum _ { j } p ( x _ { j } = 1 | z ) w _ { j } } \\ { \displaystyle } & { \qquad = p ( x _ { i } = 1 | z ) ( w _ { i } - \sum _ { j } p ( x _ { j } = 1 | z ) w _ { j } ) } \end{array}
+$$
+
+Similarly, we may compute $\mathcal { I } ( z ) ^ { \mathrm { l o g } }$ :
+
+$$
+\nabla _ { z } \log p ( x _ { i } = 1 | z ) = w _ { i } - \sum _ { j } p ( x _ { j } = 1 | z ) w _ { j } = \sum _ { j } p ( x _ { j } = 1 | z ) ( w _ { i } - w _ { j } )
+$$
+
+Denoting a word-pair vector as $w _ { i } - w _ { j }$ , where $w _ { i } , w _ { j }$ are columns of the matrix $W$ . If we define the set of all word-pair vectors as $s$ , then Eq 5 captures the idea that the vector representation for a word $i$ lies in the convex hull of $s$ . Furthermore, the word vector’s location in $\mathrm { C O N V } ( S )$ is determined by the likelihood of the pairing word $( x _ { j } )$ under the model $p ( x _ { j } = 1 | z )$ .
+
+When we use a non-linear conditional probability distribution $\mathcal { I } ( z ) ^ { \mathrm { l o g } }$ becomes: $\nabla _ { z } \log  p ( x _ { i } = $ $\begin{array} { r } { 1 | z ) = \sum _ { j } p ( x _ { j } = 1 | z ) ( \nabla _ { z } \gamma _ { i } ( z ) - \nabla _ { z } \hat { \gamma _ { j } } ( z ) ) } \end{array}$ where $\nabla _ { z } \gamma _ { i } ( z )$ is a non-linear function of $z$ . To the best of our knowledge, Jacobian Vectors and their properties have not been studied.
+
+# 4 RELATED WORK
+
+Learning in Deep Generative Models: Salakhutdinov & Larochelle (2010) optimize the local variational parameters obtained from an inference network when learning deep Boltzmann machines. For DLGMs, Hjelm et al. (2016) also consider the optimization of the local variational parameters, though their exposition focuses on deriving an importance-sampling-based bound to use during learning in deep generative models with discrete latent variables. Their experimental results suggest the procedure does not improve performance much on the binarized MNIST dataset. This is consistent with our experience—we found that our secondary optimization procedure helped more when modeling sparse, high-dimensional text data than when modeling MNIST.
+
+Leveraging Gradient Information: The algorithmic procedure for obtaining Jacobian Vectors that we propose resembles that used to derive Fisher Score features. For an data point $X$ under a parameteric distribution $p ( X ; \theta )$ , the Fisher scores is defined as $U _ { X } = \nabla _ { \boldsymbol { \theta } } \log p ( \bar { X } ; \boldsymbol { \theta } )$ . Jaakkola & Haussler (2007) similarly use $U _ { X }$ to form a kernel function for subsequent use in a discriminative classifier. The intuition behind such methods is to note that the derivative of the log-probability with respect to the parameters of the generative model encodes all the variability in the input under the generative process. We rely on a related intuition, although our motivations are different; we are interested in characterizing isolated features such as words, not vector observations such as documents. Also, we consider Jacobians with respect to per-observation latent variables $z$ , rather than globally shared parameters $\theta$ .
+
+In the context of discriminative modeling, (Erhan et al., 2009) use gradient information to study the patterns with which neurons are activated in a deep neural networks while (Wang et al., 2016) use the spectra of the Jacobian to study the complexity of the functions learned by neural networks.
+
+Introspection via Embeddings: Landauer et al. (1998) proposed latent semantic analysis, one of the earliest works to create vector space representations of documents. Bengio et al. (2003); Mikolov & Dean (2016) propose log-linear models to create word-representations from document corpora in an unsupervised fashion. Rudolph et al. (2016) describe a family of models to create contextual embeddings where the conditional distributions that lie in the exponential family. Finally, Choi et al. (2016) propose a variant of Word2Vec to create representations of diagnosis codes from temporal Electronic Health Record data. The models above explicitly condition the probability of a word on its nearby context. In contrast, our model models the probability of a word as it appears in the document (or rather, conditioned on its global context). Augmenting the generative model in Figure 1 to incorporate local context is a possible direction for future work.
+
+Miao et al. (2016) learn a shallow log-linear model on text data and obtain embeddings for words from the weight matrix that parameterize their generative model. Li et al. (2016) propose a modification to LDA that explicitly models representations for word in addition to modeling the word-topic structure.
+
+# 5 EVALUATION
+
+Text Data: We study the effect of further optimization of the variational parameters and inference with tf-idf features on the two datasets of varying size: the smaller 20Newsgroups (Lang, 2008) (train/valid/test: 10768/500/7505, V : 2000) and the larger RCV2 (Lewis et al., 2004) dataset (train/valid/test: 789414/5000/10000, V : 10000). We follow the preprocessing procedure defined in (Miao et al., 2016) for both datasets. We also train models on the Wikipedia corpus used in (Huang et al., 2012). We remove stop words, words appearing less than ten times in the dataset. and select our vocabulary to comprise the union of the top 20000 words in the corpus, the words in the WordSim353 (Finkelstein et al., 2001) and the words in the Stanford Contextual Word Similarity Dataset (SCWS) (Huang et al., 2012). The resulting dataset is of size train/valid: 1212781/1000 and V :20253.
+
+EHR Data: We train shallow and deep generative models on a dataset constructed from Electronic Medical Records. The dataset comprises 185000 patients where each patient’s data across time was aggregated to create a bag-of-diagnosis-codes representation of the patient. The vocabulary comprises four different kinds of medical diagnosis codes: ICD9 (diagnosis), LOINC (laboratory tests), NDC (prescription medication), CPT (procedures). For a single patient, we have 51321 diagnosis codes.
+
+Training Procedure: On all datasets, we train shallow log-linear models $( \gamma ( z ) = W z + b )$ and deeper three-layer DLGMs $( \gamma ( z ) = \mathrm { M L P } ( z ; \theta ) )$ . We vary the number of secondary optimization steps $M = 1 , 1 0 0$ (cf. Algorithm 1) to study the effect of optimization on $\psi ( x )$ with ADAM (Kingma & Ba, 2015). We use a mini-batch size of 500, a learning rate of 0.01 for $\psi ( x )$ and 0.0008 for $\theta , \phi$ . The inference network was fixed to a two-layer MLP whose intermediate hidden layer $h ( x )$ was used to parameterize the mean and diagonal log-variance $\mu ( x ) , \log { \Sigma ( x ) }$ . To evaluate the quality of the learned generative models, we report an upper bound on perplexity (Mnih & Gregor, 2014) given by $\begin{array} { r } { \exp ( - \frac { 1 } { N } \sum _ { i } \frac { 1 } { N _ { i } } \log p ( x _ { i } ) ) } \end{array}$ where $\log p ( x _ { i } )$ is replaced by $\operatorname { E q } 2$ . The notation 3-M100-tfidf indicates a model where the MLP parameterizing $\gamma ( z )$ has three hidden layers, the local variational parameters are updated 100 times before an update of $\theta$ and tf-idf features were used in the inference network.
+
+Improving Learning: Table 1 depicts our results on 20newsgroups and RCV2. On the smaller dataset, we find that the deeper models overfit quickly and are outperformed by shallow generative models. On the larger datasets, the deeper models’ capacity is more readily utilized yielding better generalization. The use of tf-idf features always helps learning on smaller datasets. On larger datasets, the benefits are smaller when we also optimize $\psi ( x )$ . Finally, the optimization of the local variational parameters appears to help most on the larger datasets. To investigate how this occurs, we plot the held-out likelihood versus epochs. For models trained on the larger RCV2 (Figure 2a) and Wikipedia (Figure 2b) datasets, the larger deep generative models converge to better solutions (and in fewer passes through the data) with the additional optimization of $\psi ( x )$ .
+
+To study the effect of investigate where optimizing $\psi ( x )$ is particularly effective on, we train a three layer model on different subsets of the Wikipedia dataset. The subsets are created by selecting the top $K$ most frequently occurring features in the data. Our rationale is that by holding everything fixed and varying the level of sparsity (datasets with smaller values of $K$ are less sparse) in the data, we can begin to understand when our method is most helpful. On held-out data, we compute the difference between the perplexity when the model is trained with $M = 1$ (denoted $P _ { M 1 , }$ ) and $M = 1 0 0$ (denoted $P _ { M 1 0 0 . }$ ) and compute the relative decrease in perplexity obtained as $\frac { P _ { M 1 } - P _ { M 1 0 0 } } { P _ { M 1 0 0 } }$ The results are depicted in Figure 2c where we see that our method improves learning as a function of the dimensionality of the data.
+
+In Table 5 in the supplementary material, we study the effect of varying the parameters of the inference network. There, we perform a small grid search over the hidden dimension and the number of layers in the inference network and find that optimizing the variational parameters continues to produces models with lower overall perplexity.
+
+Jacobian Vectors: Our first avenue for introspection into the learned generative model is using log-singular values of the Jacobian matrix. Since the Jacobian matrix precisely encodes how sensitive the outputs are with respect to the inputs, the log-singular value spectrum of this matrix directly captures the amount of variance in the data explained by the latent space. Said differently, we can read off the number of active units in the DLGM or VAE by counting the number of log-singular values larger than zero. Furthermore, this method of introspection depends only on the parameters of the generative model. In Figure 2d, 2e, we see that for larger models continuing to optimize the variational parameters allows us to learn models that use many more of the available latent dimensions. This suggests that, when fit to text data, DLGMs may be particularly susceptible to the overpruning phenomenon noted by Burda et al. (2015). In Figure 2, the lower held-out perplexity and the increased utilization of the latent space suggest that the continued optimization of the variational parameters yields more powerful generative models.
+
+We investigate how the Jacobian matrix may be used for model introspection by studying the qualitative properties of $\mathcal { T } _ { \mathbf { m e a n } } ^ { \mathrm { l o g } }$ on DLGMs (of type “3-M100-tfidf”) trained on two, diverse sets of data. We form a Monte Carlo estimate of $\mathcal { T } _ { \mathbf { m e a n } } ^ { \mathrm { l o g } }$ using 400 samples. The cosine distance is used to define neighbors of words in the embedding space of the Jacobian and spectral clustering (Von Luxburg, 2007) is used to form clusters.
+
+<table><tr><td>Model</td><td>K</td><td>20News</td><td>RCV1-v2</td></tr><tr><td>LDA</td><td>50</td><td>1091</td><td rowspan="4">1437 1142 988</td></tr><tr><td>LDA</td><td>200</td><td>1058</td></tr><tr><td>RSM</td><td>50</td><td>953</td></tr><tr><td>SBN fDARN</td><td>50 50</td><td>909 784 917 724</td></tr><tr><td>fDARN</td><td>200</td><td>一</td><td>598</td></tr><tr><td>NVDM</td><td>50</td><td>836</td><td>563</td></tr><tr><td>NVDM</td><td>200</td><td>852</td><td>550</td></tr></table>
+
+Table 1: Test Perplexity: Left: Baselines Results on the 20newsgroups and RCV1-v2 dataset Legend: LDA (Blei et al., 2003), Replicated Softmax (RSM) (Hinton & Salakhutdinov, 2009), Sigmoid Belief Networks (SBN) and Deep Autoregressive Networks (DARN) (Mnih & Gregor, 2014), Neural Variational Document Model (Miao et al., 2016). $K$ denotes the latent dimension in our notation. Right: DLGMs on text data with $K = 1 0 0$ We vary the features presented to the inference network $q _ { \phi } ( z | x )$ during learning between: normalized count vectors $\textstyle ( { \frac { x } { \sum _ { i = 1 } ^ { V } x _ { i } } }$ , denoted “norm”) and normalized tf-idf (denoted “tf-idf”) features.   
+
+<table><tr><td rowspan="2">DLGM</td><td colspan="3">20News RCV1-v2</td></tr><tr><td>M1 M100</td><td>M1</td><td>M11</td></tr><tr><td>1-M1-norm 1-M100-norm</td><td>964 1182</td><td>816 831 866</td><td>498 479 485 453</td></tr><tr><td>3-M1-norm 3-M100-norm</td><td>1040 1341</td><td>408</td><td>360</td></tr><tr><td></td><td>894</td><td>378</td><td>329</td></tr><tr><td>1-M1-tfidf</td><td>895</td><td>785 475</td><td>453</td></tr><tr><td>1-M100-tfidf</td><td>917</td><td>792</td><td>480 451</td></tr><tr><td>3-M1-tfidf</td><td></td><td></td><td></td></tr><tr><td></td><td>1027</td><td>852 391</td><td>346</td></tr><tr><td>3-M100-tfidf</td><td></td><td>377</td><td></td></tr><tr><td></td><td>1029</td><td>833</td><td>327</td></tr></table>
+
+![](images/8a8574d0645fbc71d40da1bfa0c6d0c358d4900d5453571e6cfbe973797ed523.jpg)  
+Figure 2: Mechanics of Learning: Validation Perplexity and Log-singular Values of $\mathcal { T } _ { m e a n } ^ { l o g }$ : Best viewed in color. For the RCV2 and Wikipedia (large) datasets, we visualize the validation perplexity as a function of epochs. The solid lines indicate the validation perplexity for $M = 1$ and the dotted lines indicate $M = 1 0 0$ . The $\mathbf { X }$ -axis is not directly comparable on running times since larger values of $M$ take longer during training. We find that learning with $\dot { M } = \bar { 1 } 0 0$ takes approximately 15 times as long per mini-batch of size 500 on the text datasets. Figure $2 \mathrm { c }$ compares relative differences in the final held-out perplexity, denoted $P$ , between models trained using $M = 1$ and $M = 1 0 0$ . On the $\mathbf { X }$ -axis, we vary the number of features used in the dataset. Figure 2d, 2e depict the sorted log singular values of $\mathcal { T } _ { \mathbf { m e a n } } ^ { \mathrm { l o g } }$ .
+
+In Table 2a, we visualize some of the nearest neighbors of words using $\mathcal { T } _ { \mathbf { m e a n } } ^ { \mathrm { l o g } }$ obtained from models trained on the Wikipedia dataset. The neighbors are semantically sensible. Instead of evaluating the Jacobian at $L$ points $z _ { 1 : L } \sim p ( z )$ , one may instead evaluate it at $z _ { 1 : L } \sim q ( z | x )$ for some $x$ . In Table 2b, we select three polysemous query words alongside “context words” that disambiguate the query’s meaning. For each word-context pair, we create a document comprising a subset of words in the the (a) Word Embeddings (Nearest Neighbors): We visualize nearest neighbors of word embeddings. We exclude plurals of the query and other words in the neighborhood.
+
+<table><tr><td rowspan=1 colspan=1>Query</td><td rowspan=1 colspan=1>Neighborhood</td></tr><tr><td rowspan=1 colspan=1>intelligence</td><td rowspan=1 colspan=1>espionage,secrecy,interrogation, counterterrorism</td></tr><tr><td rowspan=1 colspan=1>zen</td><td rowspan=1 colspan=1>dharma,buddhism,buddhas,meditation,yoga</td></tr><tr><td rowspan=1 colspan=1>artificial</td><td rowspan=1 colspan=1>artificially,molecules,synthetic,soluble</td></tr><tr><td rowspan=1 colspan=1>military</td><td rowspan=1 colspan=1>civilian,armys,commanders,infantry</td></tr></table>
+
+(b) Word Embeddings (Polysemy): We visualize the nearest neighbors under the Jacobian vector induced by the posterior distribution of a document created based on the context word.
+
+<table><tr><td rowspan=1 colspan=1>Word</td><td rowspan=1 colspan=1>Context</td><td></td></tr><tr><td rowspan=1 colspan=1>crane</td><td rowspan=1 colspan=1>constructionbird</td><td rowspan=1 colspan=1>lifting,usaaf,spanned, crushed, lifterected, parkland,locally, farmland,causeway</td></tr><tr><td rowspan=1 colspan=1>bank</td><td rowspan=1 colspan=1>rivermoney</td><td rowspan=1 colspan=1>watershed, footpath,confluence,drains,tributarybanking,government, bankers,comptroller, fiscal</td></tr><tr><td rowspan=1 colspan=1>fires</td><td rowspan=1 colspan=1>burnlayoff</td><td rowspan=1 colspan=1>ignition,combustion, engines,fuel, enginethunderstorm,grassy,surrounded,walkway, burning</td></tr></table>
+
+(c) Medical Embeddings (Nearest Neighbors): We evaluate nearest neighbors of selected diagnosis, drug and procedure codes (ignoring duplicates and shortening some code names). Metformin, Glimepiride, Pioglitazone and Avandia are diabetic drugs. A contour meter is an instrument to track blood glucose. Advair, Albuterol, Proventil and Spiriva are prescribed to patients with chronic obstructive pulmonary disease (COPD)
+
+<table><tr><td rowspan=1 colspan=1>Code</td><td rowspan=1 colspan=1>Neighboring Codes</td></tr><tr><td rowspan=1 colspan=1>Metformin</td><td rowspan=1 colspan=1>Glimepiride,Avandia,ContourMeter,Pioglitazone</td></tr><tr><td rowspan=1 colspan=1>Spiriva (Bronchodilator)</td><td rowspan=1 colspan=1>Advair,Albuterol,Proventil</td></tr><tr><td rowspan=1 colspan=1>Asbestosis</td><td rowspan=1 colspan=1>Exposure To Asbestos,Coal Workers&#x27;Pneumoconiosis,Ct Scan Chest</td></tr></table>
+
+Table 2: Qualitative evaluation of Jacobian Vectors : In Table 2a and 2b, we evaluate the embeddings of words. In Table 8b and 2c we evaluate embeddings of medical diagnosis codes.
+
+<table><tr><td>Models</td><td>Spearman ρ ×100</td></tr><tr><td>Huang (S)</td><td>58.6</td></tr><tr><td>Huang (M)</td><td>65.7</td></tr><tr><td>C&amp;W</td><td>57</td></tr><tr><td>tf-idf-S</td><td>26.3</td></tr><tr><td>Pruned tf-idf-S</td><td>62.5</td></tr><tr><td>Pruned tf-idf-M</td><td>60.5</td></tr><tr><td>1-M1 7prob Jmean</td><td>61.7</td></tr><tr><td>3-M100 Jmreb 7prob</td><td>59.5</td></tr></table>
+
+(b) SCWS: (S) denotes a single prototype approach versus (M) that denotes a multiprototype approach (that leverages context)
+
+Table 3: Semantic Similarity in Words: The baseline results are taken from (Huang et al., 2012). C&W uses embeddings from the language model of (Collobert & Weston, 2008). Glove corresponds to embeddings by (Pennington et al., 2014). The learning algorithm for our embeddings does not use local context.   
+
+<table><tr><td>Models</td><td>Spearman ρ ×100</td></tr><tr><td>Huang(G)</td><td>22.8</td></tr><tr><td>Huang</td><td>71.3</td></tr><tr><td>Glove</td><td>75.9</td></tr><tr><td>C&amp;W</td><td>55.3</td></tr><tr><td>ESA</td><td>75</td></tr><tr><td>Tiered Pruned tf-idf</td><td>76.9</td></tr><tr><td>1-M1 7prob Jmean</td><td>69.7</td></tr><tr><td>3-M100Jmeab 7prob</td><td>59.6</td></tr></table>
+
+(a) WordSim353: “G” denotes the model in Huang et al. learned only with global context in the document.
+
+context’s Wikipedia page. Then, we use the learned inference network to perform posterior inference to evaluate $\mathcal { T } _ { \mathbf { m e a n } } ^ { \mathrm { l o g } }$ at the corresponding $q ( z | x )$ . This yields a set of contextual Jacobian vectors. We display the nearest neighbors for each word under different contextual Jacobian vectors and find that, while not always perfect, they capture different contextually relevant semantics. The take-away here is that by combining posterior inference in this Bayesian network with our methodology of introspecting the model, one obtains different context-specific representations for the observations despite not having been trained to capture this explicitly.
+
+In Table 8b (appendix), we visualize clusters formed from the embeddings of medical diagnosis codes to find that they exhibit topical coherence. In Table 2c, the nearest neighbors of drugs include other drugs prescribed in conjunction with or as a replacement for the query drug. For diagnosis codes such as “Asbestosis”, the nearest neighbors are symptoms and procedures associated with the disease. Finally, for a qualitative evaluation of Jacobian vectors obtained from a model trained on movie ratings, we refer the reader to the appendix.
+
+Table 4: Medical Relatedness Measure: Evaluating the quality of embedding using medical (NDF-RT and CCS) ontologies. Each column corresponds to a measure of how well the embedding space is amenable to performing analogical reasoning (NDF-RT) or clusters meaningfully (CCS). A higher number is better. SCUIs corresponds to the application of the method developed by (Choi et al., 2016) on data released by (Finlayson et al., 2014). The learning algorithm for our embeddings does not use local context.   
+
+<table><tr><td>Models</td><td>MRMNDF-RT (May Treat)</td><td>MRMNDF-RT (May Prevent)</td><td>MRMccs (Fine Grained)</td><td>MRMccs (Coarse Grained)</td></tr><tr><td>(De Vine et al., 2014)</td><td>53.21</td><td>57.14</td><td>22.63</td><td>24.56</td></tr><tr><td>(Choi et al., 2016)</td><td>59.40</td><td>55.71</td><td>44.80</td><td>47.43</td></tr><tr><td>SCUI</td><td>52.75</td><td>48.57</td><td>34.16</td><td>37.31</td></tr><tr><td></td><td>59.63</td><td>32.86</td><td>31.58</td><td>33.88</td></tr><tr><td>3-M100</td><td>60.32</td><td>38.57</td><td>37.77</td><td>40.87</td></tr></table>
+
+The Semantics of Embeddings: We evaluate the vector space representations that we obtain from $\mathcal { I } _ { \mathbf { m e a n } }$ on benchmarks (such as WordSim353 (Finkelstein et al., 2001) and SCWS (Huang et al., 2012)) that attempt to measure the similarity of words. The algorithmically derived measure of similarity is compared to a human-annotated score (between one and ten) using the Spearman rank correlation. The models that we compare to primarily use local context, which yields a more precise signal about the meanings of particular words. Closest to us in terms of training procedure is (Huang (G)) in Table 3a, whose model we outperform. Finding ways to incorporate local context is fertile ground for future work on models tailor-made for extracting embeddings.
+
+For medical codes, we follow the method in (Choi et al., 2016). The authors build two kinds of evaluations to estimate whether an embedding space of medical diagnosis codes captures medically related concepts well. $\mathbf { M R M } _ { \mathrm { N D F - R T } }$ (Medical Relatedness Measure under NDF-RT) leverages a database (NDF-RT) to evaluate how good an embedding space is at answering analogical queries between drugs and diseases such as uDiabetes ≈ uMetformin − ( $u _ { \mathrm { L u n g C a n c e r } } - u _ { \mathrm { T a r c e v a } } )$ . (Metformin is a diabetic drug and Tarceva is used in the treatment of lung cancer). The evaluation $\mathbf { ( M R M _ { C C S } ) }$ measures if the neighborhood of the diagnosis codes is medically coherent using a predefined medical ontology (CCS) as ground truth. The number computed may be thought of as a measure of precision, where a higher number is better. We refer the reader to the appendix for additional details.
+
+Table 4 details the results on evaluating the medical embeddings. Once again, the baselines we compare (Choi et al., 2016) are variants of Word2Vec that maximize the likelihood of the diagnosis codes conditioned on carefully crafted contexts. Our method performs comparably to the baselines, even though it relies exclusively on global context and was not designed with this task in mind. This setting depicts an instance where Jacobian vectors resulting from a deeper, better-trained model outperform those from a shallow model, highlighting the importance of a method of interpretation agnostic to the structure of the conditional probability functions in the generative model.
+
+Between the three choices of Jacobian vectors, we found that all three perform comparably on the word similarity evaluation with $\mathcal { T } _ { \mathbf { m e a n } } ^ { \mathrm { p r o b } }$ slightly outperforming the others. On the medical data, with we found similar results aside from a few cases where $\mathcal { T } _ { \mathbf { m e a n } } ^ { \mathrm { p r o b } }$ did not perform well. For deeper models, we found that optimizing $\psi ( x )$ improved the quality of the obtained Jacobian vectors on text and medical data. The full versions of Tables 3 and 4 can be found in the appendix.
+
+# 6 DISCUSSION
+
+We explored techniques to improve inference and learning in deep generative models of sparse non-negative data. We also developed and explored a novel, simple, yet effective method to interpret the structure of the non-linear generative model via embeddings obtained from the Jacobian matrix relating latent variables to observations. The embeddings are evaluated qualitatively and quantitatively, and were seen to exhibit interesting semantic structure across a variety of domains. Studying the effects of varying the priors on the latent variables, conditioning on context, and varying the neural architectures that parameterize the conditional distributions suggest avenues for blending ideas from generative modeling and Bayesian inference into building more powerful embeddings for data.
+
+# REFERENCES
+
+Ricardo Baeza-Yates, Berthier Ribeiro-Neto, et al. Modern information retrieval, volume 463. ACM press New York, 1999.
+
+Yoshua Bengio, Rejean Ducharme, Pascal Vincent, and Christian Jauvin. A neural probabilistic ´ language model. JMLR, 2003.
+
+C. Bishop. Pattern Recognition and Machine Learning. Springer New York., 2006.
+
+David M Blei, Andrew Y Ng, and Michael I Jordan. Latent dirichlet allocation. JMLR, 2003.
+
+Leo Breiman. Statistical modeling: The two cultures. Statistical Science, 2001.
+
+Yuri Burda, Roger Grosse, and Ruslan Salakhutdinov. Importance weighted autoencoders. In ICLR, 2015.
+
+Youngduck Choi, Chill Yi-I Chiu, and David Sontag. Learning low-dimensional representations of medical concepts. In AMIA, 2016.
+
+Ronan Collobert and Jason Weston. A unified architecture for natural language processing: Deep neural networks with multitask learning. In ICML, 2008.
+
+Lance De Vine, Guido Zuccon, Bevan Koopman, Laurianne Sitbon, and Peter Bruza. Medical semantic similarity with a neural language model. In CIKM, 2014.
+
+Dumitru Erhan, Yoshua Bengio, Aaron Courville, and Pascal Vincent. Visualizing higher-layer features of a deep network. 2009.
+
+Lev Finkelstein, Evgeniy Gabrilovich, Yossi Matias, Ehud Rivlin, Zach Solan, Gadi Wolfman, and Eytan Ruppin. Placing search in context: The concept revisited. In WWW, 2001.
+
+Samuel G Finlayson, Paea LePendu, and Nigam H Shah. Building the graph of medicine from millions of clinical narratives. Scientific data, 2014.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In AISTATS, 2010.
+
+Geoffrey E Hinton and Ruslan R Salakhutdinov. Replicated softmax: an undirected topic model. In NIPS, 2009.
+
+Geoffrey E Hinton, Peter Dayan, Brendan J Frey, and Radford M Neal. The” wake-sleep” algorithm for unsupervised neural networks. Science, 1995.
+
+R Devon Hjelm, Kyunghyun Cho, Junyoung Chung, Russ Salakhutdinov, Vince Calhoun, and Nebojsa Jojic. Iterative refinement of approximate posterior for training directed belief networks. In NIPS, 2016.
+
+Matthew D Hoffman, David M Blei, Chong Wang, and John William Paisley. Stochastic variational inference. JMLR, 2013.
+
+Eric H. Huang, Richard Socher, Christopher D. Manning, and Andrew Y. Ng. Improving Word Representations via Global Context and Multiple Word Prototypes. In ACL, 2012.
+
+Tommi S Jaakkola and David Haussler. Exploiting generative models in discriminative classifiers. In NIPS, 2007.
+
+Diederik 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.
+
+Diederik P Kingma, Shakir Mohamed, Danilo Jimenez Rezende, and Max Welling. Semi-supervised learning with deep generative models. In NIPS, 2014.
+
+Thomas K Landauer, Peter W Foltz, and Darrell Laham. An introduction to latent semantic analysis. Discourse processes, 1998.
+
+Ken Lang. The 20 newsgroups data set, 2008.
+
+Hugo Larochelle, Yoshua Bengio, Jer´ ome Louradour, and Pascal Lamblin. Exploring strategies for ˆ training deep neural networks. JMLR, 2009.
+
+David D Lewis, Yiming Yang, Tony G Rose, and Fan Li. RCV1: A new benchmark collection for text categorization research. JMLR, 2004.
+
+Shaohua Li, Tat-Seng Chua, Jun Zhu, and Chunyan Miao. Generative topic embedding: a continuous representation of documents. In ACL, 2016.
+
+Yishu Miao, Lei Yu, and Phil Blunsom. Neural variational inference for text processing. In ICML, 2016.
+
+T Mikolov and J Dean. Distributed representations of words and phrases and their compositionality. In NIPS, 2016.
+
+Andriy Mnih and Karol Gregor. Neural variational inference and learning in belief networks. In ICML, 2014.
+
+Netflix. Netflix prize data set. 2009.
+
+Jeffrey Pennington, Richard Socher, and Christopher D Manning. Glove: Global vectors for word representation. In EMNLP, 2014.
+
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In ICML, 2014.
+
+Maja R Rudolph, Francisco JR Ruiz, Stephan Mandt, and David M Blei. Exponential family embeddings. In NIPS, 2016.
+
+Ruslan Salakhutdinov and Hugo Larochelle. Efficient learning of deep boltzmann machines. In AISTATS, 2010.
+
+Charles Spearman. ” general intelligence,” objectively determined and measured. The American Journal of Psychology, 1904.
+
+Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, 2016. URL http://arxiv.org/abs/1605.02688.
+
+Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 2007.
+
+Shengjie Wang, Matthai Plilipose, COM Matthew Richardson, COM Krzysztof Geras, Gregor Urban, and EDU Ozlem Aslan. Analysis of deep neural networks with the extended data jacobian matrix. In ICML, 2016.
+
+# APPENDIX A RCV2: COMPLEXITY OF THE INFERENCE NETWORK
+
+We study the effect of varying the inference network on the two-layer model trained with the RCV2 data. Folding fixed a three-layer DLGM with stochastic dimension 100 (same architecture as $3 - t f i d f$ in Table 1), we learn models on the RCV2 data using $M = 1$ , 100 (we evaluate and report bounds on perplexity using $M = 1 0 0$ ) and display the results in Table 5. When $M = 1$ (the standard procedure for training VAEs and DLGMs), increasing the number of layers in the inference network decreases the quality of the model learned. One possible explanation for this is that the already noisy gradients of the inference network must propagate along a longer path in a deeper inference network, slowing down learning of the parameters $\phi$ which in turn affects the quality of inference. In contrast, increasing hidden dimension of the inference network improves results. Generally, we obtain better results (in both train and validation error) with $M = 1 0 0$ than training with $M = 1$ across the various configurations of the inference network that we tried. Furthermore, we find that when $M = 1 0 0$ , the inference network architecture is less relevant and all models converge to approximately the same result suggesting that the procedure treats the output of the inference network as a crude initialization for the variational parameters and that the subsequent steps of optimization are primarily responsible for gains in learning.
+
+Table 5: Train and Test Perplexity on RCV2: The tables herein show the train and held out bounds on perplexity obtained while varying the structure of the inference network. The top table depicts results for $M = 1$ and the bottom table for $M = 1 0 0$ . The values along the rows and columns depict different parameters for the inference network.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Dimension</td><td colspan="2">1 layer</td><td colspan="2">2 layer</td><td colspan="2">3 layer</td></tr><tr><td>Train</td><td>Validate</td><td>Train</td><td>Validate</td><td>Train</td><td>Validate</td></tr><tr><td rowspan="4">M=1</td><td>100</td><td>337.01</td><td>351.26</td><td>357.28</td><td>371.37</td><td>390.00</td><td>407.15</td></tr><tr><td>400</td><td>322.94</td><td>340.45</td><td>331.36</td><td>347.95</td><td>338.09</td><td>355.64</td></tr><tr><td rowspan="2">Dimension</td><td>1 layer</td><td></td><td></td><td>2 layer</td><td></td><td>3 layer</td></tr><tr><td>Train</td><td>Validate</td><td>Train</td><td>Validate</td><td>Train</td><td>Validate</td></tr><tr><td rowspan="2">M=100</td><td>100</td><td>317.84</td><td>333.64</td><td>318.03</td><td>334.54</td><td>318.04</td><td>335.92</td></tr><tr><td>400</td><td>314.75</td><td>332.35</td><td>314.04</td><td>332.40</td><td>313.95</td><td>332.04</td></tr></table>
+
+APPENDIX B NETFLIX: EMBEDDINGS FOR MOVIES
+
+The Netflix dataset Netflix (2009) comprises movie ratings of 500, 000 users. We treat each user’s ratings as a document and model the numbers ascribed to each movie (from $1 - 5 )$ as counts drawn from the multinomial distribution parameterized as in Eq. 1. We train a three-layer DLGM on the dataset, evaluate $\mathcal { I } _ { \bf m e a n }$ with 100 samples and consider two distinct methods of evaluating the learned embeddings. We cluster the movie embeddings (using spectral clustering with cosine distance to obtain 100 clusters) and depict some of the clusters in Table 6a. We find that clusters exhibit coherent themes such as documentary films, horror and James Bond movies. Other clusters (not displayed) included multiple seasons of the same show such as Friends, WWE wrestling, and Pokemon. In Table 6b, we visualize the neighbors of some popular films. In the examples we visualize, the nearest neighbors include sequels, movies from the same franchise or, as in the case of 12 Angry Men, other dramatic classics.
+
+To compare the effect of using a model to create embeddings versus using the raw data from a large dataset directly, we evaluated nearest neighbors of movies using a simple baseline. For a query movie, we found all users who gave the movie a rating of 3 or above (nominally, they watched and liked the movie). Then, for all those users, we computed the mean ratings they gave to every other movie in the vocabulary and ranked them based on the mean ratings. We display the top five movies obtained using this approach in Table 6. The query words are the same as in Table 6b. For most of the queries, the difference between the two is evident and we simply end up with popular, well-liked movies rather than relevant movies.
+
+Cluster Name   
+Movies   
+
+<table><tr><td rowspan=1 colspan=1>DocumentaryFilms</td><td rowspan=1 colspan=1>Nature:Antarctica,Ken Burns’America:Empire of the Air,Travel the World byTrain:Africa,Deepak Chopra: The Way of the Wizard &amp; Alchemy,The HistoryChannel Presents: Troy: Unearthing the Legend</td></tr><tr><td rowspan=1 colspan=1>Concerts</td><td rowspan=1 colspan=1>Neil Diamond: Greatest Hits Live,Meat Loaf: Bat Out of Hell, Ricky Martin:One Night Only,Beyonce: Live at Wembley,Enigma: MCMXC A.D, SarahBrightman: In Concert</td></tr><tr><td rowspan=1 colspan=1>Horror Movies</td><td rowspan=1 colspan=1>Halloween 5: The Revenge of Michael Myers, Halloween: H2O, Creepshow,Children of the Corn,Poltergeist,Friday the 13th: Part 3,The Omen, Cujo</td></tr><tr><td rowspan=1 colspan=1>JamesBond</td><td rowspan=1 colspan=1>ForYour Eyes Only,Goldfinger, The LivingDaylights,Thunderball, FromRussia With Love,Dr. No</td></tr><tr><td rowspan=1 colspan=2>Hindi Movies    Seeta Aur Geeta,Gupt,Mann, Jeans, Coolie No.1,Mission Kashmir,Rangeela,Baazigar,Daud, Zakhm(a) Clustering Movie Embeddings:We display some of the clusters found from clustering the moviedings.The names were assigned based on salient features of movies in the cluster</td></tr></table>
+
+Cluster Name   
+Movies   
+
+<table><tr><td>Superman II</td><td>Superman: The Movie, Superman II, Superman IV: The Quest for Peace, RoboCop,Batman Returns</td></tr><tr><td>Casablanca</td><td>Citizen Kane,The Treasure of the Sierra Madre,Working with Orson Welles,The Millionairess,Indiscretion of an American Wife,Doctor Zhivago</td></tr><tr><td>Bride of Chucky</td><td>Bride of Chucky,Leprechaun 3,Leprechaun, Wes Craven&#x27;s New Night- mare,Child&#x27;s Play 2: Chucky&#x27;s Back</td></tr><tr><td>The Princess Bride</td><td>The Breakfast Club,Sixteen Candles,Groundhog Day,Beetlejuice, Stand by Me,Pretty in Pink</td></tr><tr><td>12 Angry Men</td><td>To Killä Mockingbird, Rear Window,Mr. Smith Goes to Washington, Inherit the Wind,Vertigo,The Maltese Falcon</td></tr><tr><td colspan="2">(b) Movie Neighbors: We visualize some of the closest neighbors found to movies whose title is displayed the column on the left</td></tr></table>
+
+Movies   
+
+<table><tr><td>Superman II Casablanca</td><td>LOTR:The Two Towers,PotC:The Curse of theBlackPearl,Raiders of the Lost Ark,LOTR: The Fellowship of the Ring To Killa Mockingbird,The Usual Suspects,The Shawshank Redemption,</td></tr><tr><td></td><td>Citizen Kane,The Wizard of Oz</td></tr><tr><td>Bride of Chucky</td><td>The Matrix, Independence Day, The Silence of the Lambs,PotC: The Curse of the Black Pearl, The Sixth Sense</td></tr><tr><td>The Princess Bride</td><td>The Shawshank Redemption,Forrest Gump,LOTR: The Two Towers, LOTR:The Fellowship of the Ring,PotC: The Curse of the Black Pearl</td></tr><tr><td>12 Angry Men</td><td>LOTR: The Fellowship of the Ring,PotC: The Curse of the Black Pear, The Godfather,Forrest Gump,The Shawshank Redemption</td></tr></table>
+
+(c) Movie Neighbors [Baseline]: We visualize some of the closest neighbors to a given query movie. We using a simple baseline that rates every movie based on average scores given by all the users who liked (rating greater than three) the query movie. LOTR (Lord of the Rings), PotC (Pirates of the Caribbean)
+
+Table 6: Qualitative Evaluation of Movie Embeddings: We evaluate $\mathcal { T } _ { \mathrm { m e a n } } ^ { \mathrm { l o g } }$ using 100 Monte-Carlo samples to perform the evaluation in Tables 6a and 6b.
+
+# APPENDIX C WIKIPEDIA: EMBEDDINGS FOR WORD
+
+Table 7 is the full version of Table 3 in the main paper. We find that the three variants of the Jacobian vectors perform comparably across the board. The vectors obtained from shallow log-linear models appear to have the edge. The evaluation on the WordSim and SCWS datasets are done by computing the Spearman rank correlation between human annotated rankings between 1 and 10 and an algorithmically derived measures of word-pair similarity. We first compute the distances between all word pairs. Our measure of similarity is obtained by subtracting the distances from the maximal distance across all word pairs.
+
+<table><tr><td>Models</td><td>Spearman ρ ×100</td></tr><tr><td>Huang(G)</td><td>22.8</td></tr><tr><td>Huang</td><td>71.3</td></tr><tr><td>Glove</td><td>75.9</td></tr><tr><td>C&amp;W</td><td>55.3</td></tr><tr><td>ESA</td><td>75</td></tr><tr><td>Tiered Pruned tf-idf</td><td>76.9</td></tr><tr><td>1-M1 7log Jmean</td><td>65.8</td></tr><tr><td>1-M1 7prob Jmean</td><td>69.7</td></tr><tr><td>1-M1 7.pot</td><td>66.3</td></tr><tr><td>Jmean 1-M100 log</td><td>66.3</td></tr><tr><td>Jmean 1-M100 7prob</td><td></td></tr><tr><td>Jmean 7Pot</td><td>70.9</td></tr><tr><td>1-M100 Jmean</td><td>69.5</td></tr><tr><td>3-M1 log Jmean</td><td>45</td></tr><tr><td>3-M1 7prob Jmean</td><td>46.9</td></tr><tr><td>3-M1 7pot Jmean</td><td>43.9</td></tr><tr><td>3-M100 7log Jmean</td><td>59.6</td></tr><tr><td>3-M100 7prob Jmean</td><td>59.6</td></tr><tr><td>3-M100 7Pot Jmean</td><td>57.8</td></tr></table>
+
+(a) WordSim353: “G” denotes the model in Huang et al. learned only with global context in the document.
+
+<table><tr><td>Models</td><td>Spearman ρ ×100</td></tr><tr><td>Huang (S)</td><td>58.6</td></tr><tr><td>Huang (M)</td><td>65.7</td></tr><tr><td>C&amp;W</td><td>57</td></tr><tr><td>tf-idf-S</td><td>26.3</td></tr><tr><td>Pruned tf-idf-S</td><td>62.5</td></tr><tr><td>Pruned tf-idf-M 7log</td><td>60.5</td></tr><tr><td>1-M1 Jmean 7prob</td><td>59.9</td></tr><tr><td>1-M1 Jmean</td><td>61.7</td></tr><tr><td>1-M1 7pot Jmean</td><td>60.4</td></tr><tr><td>1-M100 7log Jmean</td><td>58.1</td></tr><tr><td>1-M100 .7prob J mean</td><td>61.1</td></tr><tr><td>1-M100 7Pot Jmean</td><td>59.9</td></tr><tr><td>3-M1 7log</td><td>54.9</td></tr><tr><td>mean 3-M1 7prob</td><td>52.2</td></tr><tr><td>L mean 7pot</td><td></td></tr><tr><td>3-M1 mean</td><td>53.2</td></tr><tr><td>3-M100 7log Jmean</td><td>59.6</td></tr><tr><td>3-M100 .7prob Jmean</td><td>59.5</td></tr><tr><td>3-M100 7Pot Jmean</td><td>57.9</td></tr></table>
+
+(b) SCWS: (S) denotes a single prototype approach versus (M) that denotes a multiprototype approach (that leverages context)
+
+Table 7: Semantic Similarity in Words: The baseline results are taken from (Huang et al., 2012). C&W uses embeddings from the language model of (Collobert & Weston, 2008). Glove corresponds to embeddings by (Pennington et al., 2014). Three words: ‘y2k, insufflate, sincere’ from the evaluation datasets were not found in our vocabulary after pre-processing and not used in the evaluation
+
+# APPENDIX D EHR DATA: EMBEDDINGS FOR DIAGNOSIS CODES
+
+For EHR data in particular, the bag-of-diagnosis-codes assumption we make is a crude one since (1) we assume the temporal nature of the patient data is irrelevant, and (2) combining patient statistics over time renders it difficult for the generative model to disambiguate the correlations between codes that correspond to multiple diseases a patient may suffer from. Despite this, it is interesting that the Jacobian vectors still capture much of the meaningful structure among the diagnosis codes (c.f Table 2c, 8b). Here we provide additional details surrounding the evaluating of medical embeddings.
+
+$\mathbf { M R M } _ { \mathbf { C C S } } ( V , G )$ : The Agency for Healthcare Research and Quality’s clinical classification software (CCS) collapses the hierarchical ICD9 diagnosis codes into clinically meaningful categories. The evaluation on CCS checks whether the nearest neighbors of a disease include other diseases related to it (if they are in the same category in the CCS). Using the ICD9 hierarchy, the authors further split the evaluation task into predicting neighbors of fine-grained and coarse grained diagnosis codes.
+
+For a choice of granularity $G \in \{ \mathrm { { f i n e , c o a r s e } } \}$ , $V ( G ) \in V$ denotes the subset of ICD9 codes in the vocabulary. $\mathbb { I } _ { G } ( v ( i ) )$ is one if the $v$ ’s i’th nearest neighbor: $v ( i )$ is in the same group as $v$ according to $G$ .
+
+$$
+\mathbf { M R M } _ { \mathrm { C C S } } ( V , G ) = \frac { 1 } { | V ( G ) | } \sum _ { v \in V ( G ) } \sum _ { k = 1 } ^ { 4 0 } \frac { \mathbb { I } _ { G } ( v ( i ) ) } { \log _ { 2 } ( i + 1 ) }
+$$
+
+$\mathbf { M R M } _ { \mathrm { N D F - R T } } ( V , R )$ : The other evaluation uses the National Drug File Reference Terminology (NDF-RT) to evaluate analogical reasoning. The NDF-RT provides two kinds of relationships $( R )$ between drugs and diseases: May-Treat (if the drug may be used to treat the disease) and May-Prevent. Given $\phi _ { A }$ as the embedding for a code $A$ , this test automates the evaluation of analogies such as $\underbrace { \phi _ { \mathrm { D i a b e t e s } } } _ { r } \approx \underbrace { \phi _ { \mathrm { M e t f o r m i n } } } _ { v } - \underbrace { ( \phi _ { \mathrm { L u n g ~ C a n c e r } } - \phi _ { \mathrm { T a r c e v a } } ) } _ { s }$ . Here $v$ is the query code and $s$ is a representation of the relationship we seek. (Metformin is a diabetic drug and Tarceva is used in the treatment of lung cancer.) The evaluation we perform reports a number proportional to the number of times the neighborhood of $v - s$ contains $r$ for the best value of $s$ (computed from the set of all valid drug-disease relationships in the datasets.)
+
+Given $V ^ { * } \in V$ (concepts for which NDF-RT has at-least one substance with the given relation), $\mathbb { I } _ { R } \left( \cup _ { i = 1 } ^ { 4 0 } ( v - s ) ( i ) \right)$ is one if any of the medical concepts in the top-40 neighborhood of the selected medical concept $v$ satisfies relation $R$ .
+
+$$
+\mathsf { M R M } _ { \mathrm { N D F - R T } } ( V , R ) = \frac { 1 } { | V ^ { * } | } \sum _ { v \in V ^ { * } } \mathbb { I } _ { R } \left( \cup _ { i = 1 } ^ { 4 0 } ( v - s ) ( i ) \right)
+$$
+
+In both cases the choice of 40 was adopted to maintain consistency with (Choi et al., 2016). The evaluation is conducted by taking the average result over all possible seeds $s$ and the best possible seed $s$ for a query.
+
+Table 8a depicts two examples of using the learned embeddings in the Jacobian matrix to answer tasks queries related to drug-disease pairs. Table 8b depicts clusters found in medical diagnosis codes. Table 9 is the extended version of Table 4 in the main paper (where, for the comparison on NDF-RT, we depict the results obtained from the best choice of seed $s$ ).
+
+<table><tr><td rowspan=1 colspan=1>Code 1</td><td rowspan=1 colspan=1>Code 2</td><td rowspan=1 colspan=1>Code 3</td><td rowspan=1 colspan=1>Neighborsof Result</td></tr><tr><td rowspan=1 colspan=1>Evoxac</td><td rowspan=1 colspan=1>Sicca Syndrome</td><td rowspan=1 colspan=1>Methotrexate</td><td rowspan=1 colspan=1>Rheumatoid Arthritis</td></tr><tr><td rowspan=1 colspan=1>Biliary Atresia</td><td rowspan=1 colspan=1>Kidney Transplant</td><td rowspan=1 colspan=1>Leg Varicosity w/ In-flammation</td><td rowspan=1 colspan=1>Ligation of angioaccessarteriovenous fistula</td></tr></table>
+
+(a) Medical Analogies: We can perform analogical reasoning with embeddings of medical codes. If we know a drug used to treat a disease, we can use their relationship in vector space to find unknown drugs associated with a different disease. The queries are of the form Code $1 $ Code $2 \Longrightarrow$ Code $3  ?$ . Sicca syndrome or Sjogren’s disease is an immune disease treated with Evoxac and Methotrexate is commonly used to treat Rheumatoid Arthiritis. “ Leg Varicosity” denotes the presence of swollen veins under the skin. “Ligation of angioaccess arteriovenous fistula” denotes the tying of passage between an artery and a vein.
+
+(b) Medical Embeddings (Clustering): We visualize some topical clusters of diagnosis codes.   
+
+<table><tr><td rowspan=1 colspan=1>Label</td><td rowspan=1 colspan=1>Diagnosis Codes</td></tr><tr><td rowspan=1 colspan=1>Thrombosis</td><td rowspan=1 colspan=1>HxVenous Thrombosis, Compression Of Vein,Renal Vein Thrombosis</td></tr><tr><td rowspan=1 colspan=1>Occular Atrophy</td><td rowspan=1 colspan=1>Optic Atrophy,RetinaLayer Separation, Chronic Endophthalmitis</td></tr><tr><td rowspan=1 colspan=1>Drug Use</td><td rowspan=1 colspan=1>Opioid Dependence,Alcohol Abuse-Continuous,Hallucinogen Dep</td></tr></table>
+
+Table 9: Medical Relatedness Measure: Evaluated using the NDF-RT and CCS ontologies. For the evaluation on NDF-RT, there is a choice of $s$ (Eq 7) to be made. The results are reported both by averaging across all pairs of drugs-diseases used to form $s$ (avg-seed) and the best drug-disease pair (max-seed). SCUIs corresponds to the application of the method developed by (Choi et al., 2016) on data released by (Finlayson et al., 2014)
+
+<table><tr><td>Models</td><td>MRMNDF-RT (May Treat)</td><td>MRMNDF-RT (May Prevent)</td><td>MRMccs (Fine Grained)</td><td>MRMccs (Coarse Grained)</td></tr><tr><td>(De Vine et al.,2014)</td><td>31.34/53.21</td><td>34.47/57.14</td><td>22.63</td><td>24.56</td></tr><tr><td>(Choi et al., 2016)</td><td>36.62/59.40</td><td>28.02/55.71</td><td>44.80</td><td>47.43</td></tr><tr><td>SCUI</td><td>34.89/52.75</td><td>30.95/48.57</td><td>34.16</td><td>37.31</td></tr><tr><td>1-M1 7log Jmean</td><td>4.08/54.36</td><td>19.90/34.29</td><td>30.82</td><td>34.04</td></tr><tr><td>1-M1 .7prob Jmean</td><td>33.87/56.19</td><td>22.41/34.29</td><td>31.76</td><td>35.07</td></tr><tr><td>1-M1 7pot Jmean</td><td>27.45/59.63</td><td>15.02/32.86</td><td>31.58</td><td>33.88</td></tr><tr><td>1-M100 7log Jmean</td><td>33.32/54.82</td><td>19.02/35.71</td><td>33.04</td><td>35.78</td></tr><tr><td>1-M100 .7prob Jmean</td><td>30.70/53.90</td><td>21.66 /34.29</td><td>32.86</td><td>35.66</td></tr><tr><td>1-M100</td><td>28.49/55.73</td><td>15.56/31.43</td><td>32.87</td><td>35.09</td></tr><tr><td>3-M1 7log Jmean</td><td>33.21/52.29</td><td>23.84/42.86</td><td>32.80</td><td>37.58</td></tr><tr><td>3-M1</td><td>12.12/30.28</td><td>7.62 /17.14</td><td>23.42</td><td>26.85</td></tr><tr><td>3-M1 7pot Jmean</td><td>33.30/51.61</td><td>23.47/41.43</td><td>33.02</td><td>37.84</td></tr><tr><td>3-M100 7log Jmean</td><td>37.00/61.47</td><td>23.70/42.86</td><td>37.54</td><td>40.52</td></tr><tr><td>3-M100 7prob Jmean</td><td>9.79/21.33</td><td>4.39 / 11.43</td><td>7.82</td><td>8.60</td></tr><tr><td>3-M100</td><td>36.11/60.32</td><td>22.26/38.57</td><td>37.77</td><td>40.87</td></tr></table>